1
A -Algebraic topology over a field
Fabien Morel
6.11.2006
Contents
0 Introduction 2
1 Unramified sheaves and strongly A1-invariant sheaves 13
1.1 Unramified sheaves of sets . . . . . . . . . . . . . . . . . . . . 13
1.2 Unramified sheaves of groups and strong A1-invariance . . . . 23
1.3 Unramified Z-graded abelian sheaves . . . . . . . . . . . . . . 35
2 Unramified Milnor-Witt K-theories 51
2.1 Milnor-Witt K-theory of fields . . . . . . . . . . . . . . . . . . 51
2.2 Unramified Milnor-Witt K-theories . . . . . . . . . . . . . . . 58
2.3 Milnor-Witt K-theory and strongly A1-invariant sheaves . . . 75
3 A1-homotopy sheaves and A1-homology sheaves 84
1
3.1 Strongly A1-invariance of the sheaves ssAn, n 1 . . . . . . . . 84
3.2 A1-derived category and Eilenberg-MacLane spaces . . . . . . 94
3.3 The Hurewicz theorem and some of its consequences . . . . .101
1 n A1
4 A1-coverings, ssA1(P ) and ss1 (SLn) 114
1
4.1 A1-coverings, universal A1-covering and ssA1 . . . . . . . . . .114
1 n A1
4.2 Basic computation: ssA1(P ) and ss1 (SLn) for n 2 . . . . . .124
1 1
4.3 The computation of ssA1(P ) . . . . . . . . . . . . . . . . . . .128
A Unramified and strongly A1-invariant sheaves over finite fields138
1
0 Introduction
Let k be a commutative field. In this work we prove in the A1-homotopy
theory of smooth k-schemes [50, 39] the analogues of the following facts:
Theorem 1 (Brouwer degree) Let n > 0 be an integer and let Sn denote the
n-sphere. Then for an integer i
ae
0 if i < n
ssi(Sn) =
Z if i = n
Theorem 2 (Hurewicz Theorem) For any pointed connected topological space
X and any integer n 1 the Hurewicz morphism
ssn(X) ! Hn(X)
is the abelianization if n = 1, is an isomorphism if n 2 and X is (n - 1)-
connected, and is an epimorphism if n 3 and X is (n - 2)-connected.
Theorem 3 (Coverings and ss1) Any "reasonable" pointed connected space
X admits a universal pointed covering
X" ! X
It is (up to unique isomorphism) the only pointed simply connected covering
of X. Its automorphism group (as unpointed covering) is ss1(X) and it is a
ss1(X)-Galois covering.
Theorem 4 ss1(P1(R)) = Z and ss1(Pn(R)) = Z=2 for n 2,
ss1(SL2(R)) = Z and ss1(SLn+1(R)) = Z=2 for n 2.
The corresponding complex spaces are simply connected: for n 1 one
has ss1(Pn(C)) = ss1(SLn(C)) = *
Let us denote by Smk the category of smooth separated finite type k-
schemes. Unless otherwise explicitly stated, we will always consider Smk
endowed with the Nisnevich topology (see [39, 40]). By a space over k, or in
short a space, we mean a simplicial sheaf of sets on Smk.
The main achievement of this work is the understanding of the precise
structure of A1-homotopy sheaves of pointed spaces. Once this is done, the
2
analogues of the previous results will easily follow, as well as a bunch of ana-
logues of classical results and computations.
1
Given a space X , we denote by ssA0(X ) the associated sheaf in the Nis-
nevich topology to the presheaf U 7! HomH(k)(U, X ) where H(k) denotes
the A1-homotopy category of smooth k-schemes defined in [50, 39].
1
If X is pointed, given an integer n 1, we denote by ssAn(X ) the asso-
ciated sheaf of groups in the Nisnevich topology to the presheaf of groups
U 7! HomHo(k)( n(U+ ), X ) where Ho(k) denotes the pointed A1-homotopy
category on k (see loc. cit) and the simplicial suspension. This is a sheaf
of abelian groups for n 2.
In classical topology, the underlying structure to the corresponding ho-
motopy "sheaves" is quite simple: ss0 is a "discrete" set, ss1 is a"discrete"
group and the ssn's, n 2, are "discrete" abelian groups.
The following notions are the precise analogues of being "A1-discrete":
Definition 5 1) A sheaf of sets S on Smk (in the Nisnevich topology) is
said to be A1-invariant if for any X 2 Smk, the map
S(X) ! S(A1 x X)
induced by the projection A1 x X ! X, is a bijection.
2) A sheaf of groups G on Smk (in the Nisnevich topology) is said to be
strongly A1-invariant if for any X 2 Smk, the map
HiNis(X; G) ! HiNis(A1 x X; G)
induced by the projection A1 x X ! X, is a bijection for i 2 {0, 1}.
3) A sheaf M of abelian groups on Smk (in the Nisnevich topology) is
said to be strictly A1-invariant if for any X 2 Smk, the map
HiNis(X; M) ! HiNis(A1 x X; M)
induced by the projection A1 x X ! X, is a bijection for any i 2 N.
3
Remark 6 We observe by the very definitions of [39] that a sheaf of set S is
A1-invariant if and only if it is A1-local as a space, that a sheaf of groups G
is strongly A1-invariant if and only if the classifying space B(G) = K(G, 1)
is an A1-local space, and that a sheaf of abelian groups M is strictly A1-
invariant if and only if for any n 2 N the Eilenberg-MacLane space K(M, n)
is A1-local.
The notion of strict A1-invariance is directly taken from [48, 49]; the A1-
invariant sheaves with transfers of Voevodsky are indeed examples of such
sheaves. The Rost's cycle modules [44] give also examples of strictly A1-
invariant sheaves, more precisely of A1-invariant sheaves with transfers by
[11]. Other important examples, which are not of the previous type, are the
sheaf W__ associated to the presheaf of Witt groups X 7! W (X) in charac-
teristic 6= 2 (this is proven in [42]), or the sheaves I_nof unramified powers
of the fundamental ideal used in [33] (still in characteristic 6= 2). In fact
these sheaves can also be defined in characteristic 2 if one considers the Witt
groups of inner product spaces over X studied in [29].
The notion of strong A1-invariance is new and we will meet important ex-
amples of genuine non commutative strongly A1-invariant sheaves of groups.
1 1 A1
For instance ssA1(P ) is non abelian, see 4.3. More generally the ss1 of the
blow-up of Pn, n 2, at several points is highly non-commutative; see [37].
We strongly believe that the notion of A1-fundamental group sheaf should
play a fundamental role in the understanding of A1-connected projective
smooth k-varieties1 in very much the same way as the usual fundamental
group plays a fundamental role in the classification of compact connected dif-
ferentiable manifolds. The rationally connected smooth projective k-varieties
considered in [24] are examples of A1-connected smooth projective k-varieties,
at least with some assumptions on k; see [37].
One of our main global result, which justifies the previous definition, is:
1
Theorem 7 Let X be a pointed space. Then the sheaf of groups ssA1(X ) is
1
strongly A1-invariant and for any n 2 the sheaf of abelian groups ssAn(X )
is strictly A1-invariant.
______________________________
1a space is said to be A1-connected if its ssA10is the point
4
Remark 8 The proof relies very much on the fact that the base is a field,
through Gabber's geometric presentation Lemma [13, 9]. Over a general base
the situation is definitely more complicated, at least when the base scheme
has dimension at least 2 as pointed out by J. Ayoub [1].
Remark 9 Recall from [39] that any space X is the homotopy inverse limit
of its Postnikov tower {P n(X )}n and that if X is pointed, for each n 1
the homotopy fiber of the morphism P n(X ) ! P n-1(X ) is A1-equivalent
1 1
to the Eilenberg-MacLane space K(ssAn(X ), n). The strongly/strictly A -
invariant sheaves and their cohomology play exactly the same role as the
usual homotopy and singular cohomology groups play in classical algebraic
topology.
We are unfortunately unable to prove the analogue structure result for
1
the ssA0 which appears to be the most difficult case:
1 1
Conjecture 10 For any space X the sheaf ssA0(X ) is A -invariant.
Remark 11 This conjecture is easy to check for smooth k-schemes of di-
mension 1. The case of smooth surfaces is already very interesting and
non trivial. To simplify a bit, we assume that k is algebraically closed. Then
if X is a projective smooth k-surface which is birationnally ruled over the
smooth projective curve C of genus g then:
ae
1 * if g = 0
ssA0(X) =
C if g > 0
For surfaces of general type the conjecture is unknown though it is known in
some cases. It is not known for K3 surfaces.
Remark 12 Another interesting general test case is the geometric classi-
fying space Bgm G of a smooth algebraic k-group G [39, 47 ]. From [39],
there exists a natural transformation in X 2 Smk: `G : H1et(X; G) !
HomH(k)(X, Bgm G). We hope (at least if the group G(0)of irreducible com-
ponents of G is of order prime to the characteristic of k) that this should
1
induce an isomorphism on associated sheaves H1et(G) ~= ssA0(Bgm G) in the
Nisnevich topology (in fact most probably also in the Zariski topology). It
doesn't seem to be known in general whether or not this type of sheaves are
A1-invariant.
5
Our results rely on the detailed analysis of unramified sheaves of groups
given in Part 1. Our analysis is done very much in the spirit of a "non-
abelian variant" of Rost's cycle modules [44]. These unramified sheaves can
be described in terms of their sections on function fields of smooth irreducible
k-varieties plus extra structures: "residues" and "specializations".
Our work is entirely elementary except that we use Gabber's geometric
presentation Lemma [13, 9] when k is infinite. It is used in Sections 1.2 and
1.3. To prove Theorem 7 in Section 3.1 we also use Gabber's result to prove
1
that the A1-homotopy sheaves ssAn(X ), n 1, are unramified in the sense
of section 1.1. Using the results of section 1.2 we prove that these sheaves
are strongly A1-invariant. The Appendix provides the necessary properties
which allow us to reduce the case of a finite field to that of infinite fields.
Theorem 14 then implies Theorem 7.
Remark 13 Contrary to Rost's approach [44], the structure involved in our
description of unramified sheaves does not use any notion of "transfers". As
a consequence over a perfect field one may show that the category of Rost
cycle modules (loc. cit.) can be described without using transfers in the
structure; see section 1.3.
One of the achievements in this work is to define the unramified sheaves of
Milnor K-theory on Smk, as well as a bunch of related variants like unramified
Milnor-Witt K-theory, without using any transfers. We will rather deduce the
existence and properties of these transfers in [38]. This yields a completely
new answer to this question which was raised in [5] and settled in [22].
Apart from Gabber's Lemma, one of the main technical tool that we use
is the following result (see Section 3.2). It means that the notions of strong
and strict A1-invariance are in fact the "same" for sheaves of abelian groups.
Theorem 14 Let M be a sheaf of abelian groups on Smk. Then:
M is strongly A1-invariant , M is strictly A1-invariant.
The Hurewicz theorem (see 3.35 and 3.57), and some of its natural con-
sequences, will easily follow from Theorem 7, at least once the notion of A1-
1
chain complex and the corresponding notion of A1-homology sheaves HA*(X )
of a space X are introduced; see Section 3.3. An important consequence of
the Hurewicz theorem is the unstable A1-connectivity theorem (see Theorem
3.38 in Section 3.3):
6
Theorem 15 Let X be a pointed space and n 0 be an integer. If X is
1
simplicially n-connected then it is A1-n-connected, meaning that ssAi(X ) is
trivial for i n.
A stable and much weaker version of this result was obtained in [32]. An
example of simplicially (n-1)-connected pointed space is the n-fold simplicial
suspension n(X ) of a pointed space X . As An - {0} is A1-equivalent to the
simplicial (n - 1)-suspension n-1(Gm^n) (see [39]), it is thus (n - 2)-A1-
connected. In the same way the n-th smash power (P1)^n, which is A1-
equivalent to the simplicial suspension of the previous one [39], is (n - 1)-
A1-connected.
Remark 16 In general, the "correct" A1-connectivity is given by the con-
nectivity of the "corresponding" topological space of real points, through a
real embedding of k, rather than the connectivity of its topological space of
complex points through a complex embedding. This principle2 has been a
fundamental guide to our work. For instance the pointed algebraic "sphere"
(Gm )^n is not A1-connected: it must be considered as a "0-dimensional
twisted sphere". Observe that its space of real points has the homotopy
type of the 0-dimensional sphere S0 = {+1, -1}.
The Hurewicz Theorem implies furthermore that for n 2, the first non-
trivial A1-homotopy sheaf of the (n - 2)-A1-connected sphere An - {0} ~=A1
1
n-1(Gm ^n ) is its ssAn-1and is the free strongly (or strictly by Theorem 14)
A1-invariant sheaf of abelian groups generated by the pointed 0-dimensional
sphere (Gm )^n. This fact fits closely to classical topology as the first non-
trivial homotopy group of the n-dimensional sphere Sn is the free "discrete
abelian group" generated by the pointed 0-dimensional sphere S0. To get
the analogue of Theorem 1, it remains thus to describe the free strongly A1-
invariant sheaf K_MWn on (Gm )^n.
For any irreducible smooth k-scheme X with function field F , the abelian
group of sections K_MWn (X) injects into K_MWn (F ). To define K_MWn we first
define its sections K_MWn(F ) =: KMWn (F ) on a function field F .
Definition 17 Let F be a commutative field. The Milnor-Witt K-theory of
F is the graded associative ring KMW* (F ) generated by the symbols [u], for
______________________________
2which owns much to conversations with V. Voevodsky
7
each unit u 2 F x, of degree +1, and one symbol j of degree -1 subject to
the following relations:
1 (Steinberg relation) For each a 2 F x- {1} : [a].[1 - a] = 0
2 For each pair (a, b) 2 (F x)2 : [ab] = [a] + [b] + j.[a].[b]
3 For each u 2 F x : [u].j = j.[u]
4 Set h := j.[-1] + 2. Then j . h = 0
This object was introduced in a "complicated way" by the author, until
the previous very simple and natural description was found in collaboration
with Mike Hopkins: each relation has a natural A1-homotopic interpretation.
The quotient ring KMW* (F )=(j) is clearly the Milnor K-theory ring KM*(F )
of F introduced by Milnor in [28]. It is not hard to prove (see section 2.1)
that the ring KMW* (F )[j-1] obtained by inverting j is the ring of Laurent
polynomials W (F )[j, j-1] with coefficients in the Witt ring W (F ) of non-
degenerate symmetric bilinear forms on F (see [29], and [45] in characteristic
6= 2). More generally, KMW0 (F ) is the Grothendieck-Witt ring GW (F ) of
non-degenerate symmetric bilinear forms on F . The isomorphism sends the
1-dimensional form (X, Y ) 7! uXY on F to < u >:= j[u]+1 (see section 2.1).
Using residue morphisms in Milnor-Witt K-theory and the results of sec-
tion 1.3, we describe in section 2.2 the unramified sheaf X 7! K_MWn (X),
where, for X irreducible with function field F , K_MWn (X) KMWn (F ) de-
notes the kernel of the residues at points in X of codimension 1. In section
2.3 we prove our main computational result:
Theorem 18 For any field k, for n 1, the morphism of sheaves given by
mapping an n-tuple of units (u1, . .,.un) to its associated symbol [u1] . .[.un]
(Gm )^n ! K_MWn
is the universal one to a strongly A1-invariant sheaf of abelian groups: any
morphism of pointed sheaves (Gm )^n ! M to a strongly A1-invariant sheaf
of abelian groups induces a unique morphism of sheaves of abelian groups
K_MWn ! M
8
In other words, for n 1, the sheaf K_MWn is the free strongly A1-invariant
sheaf of abelian groups generated by (Gm )^n. Using our Hurewicz Theorem
we now obtain the analogue of Theorem 1 we had in mind:
Theorem 19 For n 2 one has a canonical isomorphisms of sheaves
1 n A1 1 ^n MW
ssAn-1(A - {0}) ~=ssn ((P ) ) ~=K_n
It is not hard to compute for (n, m) a pair of integers the abelian group of
morphisms of sheaves of abelian groups from K_MWn to K_MWm: it is K_MWm-n(k),
the isomorphism being induced by the product K_MWn xK_MWm-n! K_MWm. This
implies in particular for n = m the:
Corollary 20 (Brouwer degree) For n 2, the canonical morphism
[An - {0}, An - {0}]Ho(k)~= [(P1)^n, (P1)^n]Ho(k)! KMW0 (k) = GW (k)
is an isomorphism.
We have denoted here by [-, -]Ho(k)the set of morphisms in the pointed
A1-homotopy category Ho(k).
The analogue of the Theory of the Brouwer degree thus assigns to an
A1-homotopy class from an algebraic sphere to itself an element in GW (F );
see [35] for a heuristic discussion in case n = 1. We observe that in case
n = 1 there is only an epimorphism [P1, P1]Ho(k)! GW (k) but the group
[P1, P1]Ho(k)will be entirely understood in Section 4.3.
The class of the Hopf map j 2 [A2- {0}, P1]Ho(k)is indeed closely related
to the element j of the Milnor-Witt K-theory. Our computations thus clearly
stabilize as follows:
Corollary 21 [31, 30] Let SH(k) be the stable A1-homotopy category of P1-
spectra (or T -spectra) over k [50, 31, 30]. Let S0 be the sphere spectrum,
(Gm ) be the suspension spectrum of the pointed Gm , let j : (Gm ) ! S0 be the
(suspension of the) Hopf map and let MGL be the Thom spectrum [50]. For
any integer n 2 Z one has a commutative diagram in which the verticals are
canonical isomorphisms:
[S0, (Gm )^n]SH(k)! [S0, Cone(j) ^ (Gm )^n]SH(k) ~= [S0, MGL ^ (Gm )^n]SH(k)
# o # o # o
KMWn (k) i KMWn (k)=j = KMn(k)
9
Observe that the proof we give here is completely elementary and holds
over any field, as opposed to [31, 30], which use the Milnor conjectures and
were assuming k is perfect of characteristic 6= 2.
Another natural consequence of our work is the theory of A1-coverings
and their relation to A1-fundamental sheaves of groups. This is discussed
in section 4.1. The notion of A1-covering is quite natural: it is a morphism
of spaces having the unique left lifting property with respect to "trivial A1-
cofibrations". The Galois 'etale coverings of order prime to char(k), or the
Gm -torsors are examples of A1-coverings. We will prove the existence of a
universal A1-covering for any pointed A1-connected space X , and more pre-
cisely the exact analogue of Theorem 3.
This theory is in some sense orthogonal, or "complimentary", to the 'etale
theory of the fundamental group: a 0-A1-connected space X has no nontriv-
ial pointed Galois 'etale coverings [37].
In case X is not A1-connected, the 'etale finite coverings are "captured"
1
by the sheaf ssA0(X ) which may have non trivial 'etale covering, like abelian
varieties, which are A1-invariant sheaves.
1
The universal A1-covering and the sheaf ssA1 encode a much more com-
binatorial and geometrical information than the arithmetical information of
the 'etale one. As we already mentioned we hope this combinatorial object
will play a central role in the "A1-surgery classification" approach to projec-
tive smooth A1-connected k-varieties [37].
1 n
We next compute the ssA1 of P , n 2 and of SLn, n 3 in sections 4.2:
Theorem 22 1) For n 2, the canonical Gm -torsor Gm - (An+1 - {0}) !
Pn is the universal A1-covering of Pn, and thus yields an isomorphism
1 n
ssA1(P ) ~=Gm
2) One has a canonical isomorphism
1 MW
ssA1(SL2) ~=K_2
10
and the inclusions SL2 ! SLn, n 3, induce an isomorphism
1
K_MW2=j = K_M2~= ssA1(SLn)
Remark 23 1) In view of [12] it should be interesting to determine the pos-
1
sible ssA1 of linear algebraic groups.
1 A1 A1
2) Of course one has for n 3, ssA1(SLn) = ss1 (SL1 ) = ss1 (GL1 ). We
know from [39, Theorem 3.13 p. 140] that
[ 1(U+ ), GL1 ]Ho(k)= [ 2(U+ ), Z x B(GL1 )]Ho(k)= K2(U)
(where K2 means the Quillen K2). Thus our previous computation recover
the well-known identification between the associated sheaf (in the Zariski or)
Nisnevich topology of K2 and unramified Milnor K-theory in weight 2.
3) Our computations make clear that the Z or Z=2 in the statement
of Theorem 4 have different "motivic" natures, the fundamental groups of
projective spaces being of "weight one" and that of special linear groups of
"weight two".
1 1
We finish our paper with a very explicit description of the sheaf ssA1(P ),
which is the more complicated one! This achieves the proof of the ana-
1 1
logue of Theorem 4. The sheaf of groups ssA1(P ) will be shown to be non-
commutative (!) and is thus not equal to K_MW1 . To describe it, we let
j : A2 - {0} ! P1 be the algebraic Hopf map and we let P1 denote the
space obtained by taking the union of all the Pn's. We have an A1-fibration
sequence (see Section 3.3):
A2 - {0} ! P1 ! P1
which gives a non commutative central extension of sheaves of groups
1 2 A1 1 A1 1
0 ! K_MW2 = ssA1(A - {0}) ! ss1 (P ) ! ss1 (P ) = Gm ! 1
which is explicitly described in Section 4.3.
The results of this paper lead to rather natural applications. To cite
only a few of them, in [38] we address construction of transfers for "finite
A1-covering". There are two main cases: the case of finite 'etale coverings
11
which leads to a new construction of transfers in Milnor K-theory (as well as
Milnor-Witt K-theory) and the case of Gm -torsor Y ! X . The latter yields
a (stable) transfer morphism of the form: (Gm ) ^ (X+ ) ! (Y+ ), whose "real-
points" gives the usual transfer map for the corresponding Z=2-covering of
the real points.
The results of this paper are also used in a fundamental way in our work
on the Euler class for algebraic vector bundles [36]. As we already mentioned,
these results are also the starting point of our study of A1-connected smooth
projective varieties [37].
Part of our results were discussed and announced in [35].
Acknowledgements I want to warmly thank Mike Hopkins and Marc
Levine, for their interest in this work, and for remarks, comments and dis-
cussions which help me to improve it.
Conventions and notations. We fix a base field k. Unless otherwise
explicitly stated, no assumption is made on k, neither concerning the char-
acteristic, nor the perfectness, nor the finiteness.
We denote by Fk the category of finite type separable fields extensions of
k. By a discrete valuation v on F 2 Fk we will always mean a geometric one,
coming from a codimensional 1 point in a smooth model for F . We then let
Ov F denotes its valuation ring, mv Ov its maximal ideal and ~(v) its
residue field.
Recall that Smk denotes the category of smooth separated finite type
k-schemes, also called smooth k-varieties. We will also use the subcategory
S"mk of Smk with the same objects but with only smooth morphisms. It will
always be understood that these categories are endowed with the Nisnevich
topology [40, 39]. Thus "sheaf" always means, unless otherwise explicitly
stated, sheaf in the Nisnevich topology.
We will let Set denote the category of sets, Ab that of abelian groups. A
space is a simplicial object in the category of sheaves of sets on Smk [39].
We denote by Sm0kthe category of essentially smooth k-schemes. Its
objects are k-schemes which are an inverse limit of a left filtering system
(Xff)ffwith transition morphisms Xfi! Xffsmooth affine morphisms be-
tween smooth k-schemes (see [16]). For instance, for each point x 2 X 2 Smk
the local scheme Xx := Spec(OX,x) of X at x, as well as its henselization
12
Xhx:= Spec(OhX,x) are essentially smooth k-schemes. In the same way the
complement of the closed point in Xx or Xhxare essentially smooth over k.
We will very often make the abuse of saying "smooth k-scheme" instead of
essentially smooth k-scheme, if no confusion can arise.
For any scheme X and any integer i we let X(i)denote the set of points
in X of codimension i.
Given a presheaf of sets on Smk, that is to say a functor F : (Smk)op !
Sets, and an essentially smooth k-scheme X = limffXffwe set F (X) :=
colimitffF (Xff). When X = Spec(A) is affine we will also denote this set
simply by F (A).
1 Unramified sheaves and strongly A1-invariant
sheaves
1.1 Unramified sheaves of sets
Definition 1.1 An unramified presheaf of sets S on Smk (resp. on S"m k)
is a presheaf of sets S such that the following holds:
(0) for any X 2 Smk with irreducible components Xff's, ff 2 X(0), the
obvious map S(X) ! ff2X(0)S(Xff) is a bijection.
(1) for any X 2 Smk and any open subscheme U X the restriction
map S(X) ! S(U) is injective if U is everywhere dense in X;
(2) forTany X 2 Smk, irreducible with function field F , the injective map
S(X) ,! x2X(1)S(OX,x) is a bijection (the intersection being computed in
S(F )).
Remark 1.2 An unramified presheaf S (either on Smk or on S"m k) is au-
tomatically a sheaf of sets in the Zariski topology. This follows from (2).
We also observe that with our convention, for S an unramified presheaf, the
formula in (2) also holds for X essentially smooth over k and irreducible
with function field F . We will use these facts freely in the sequel.
Example 1.3 It was observed in [32] that any strictly A1-invariant sheaf on
Smk is unramified in this sense. The A1-invariant sheaves with transfers of
13
[48] as well as the cycle modules3 of Rost [44] give such unramified sheaves.
In characteristic 6= 2 the sheaf associated to the presheaf of Witt groups
X 7! W (X) is unramified by [41] (the sheaf associated in the Zariski topology
is in fact already a sheaf in the Nisnevich topology).
Remark 1.4 Let S be a sheaf of sets in the Zariski topology on Smk (resp.
on S"mk) satisfying properties (0) and (1) of the previous definition. Then
it is unramified if and only if, for any X 2 Smk and any open subscheme
U X the restriction map S(X) ! S(U) is bijective if X - U is everywhere
of codimension 2 in X. We left the details to the reader.
Remark 1.5 Base change. Let S be a sheaf of sets on S"m kor Smk, let
K 2 Fk be fixed and denote by ss : Spec(K) ! Spec(k) the structural
morphism. One may pull-back S to the sheaf S|K := ss*S on S"mK (or SmK
accordingly). One easily checks that the sections on a separable (finite type)
field extension F of K is nothing but S(F ) when F is viewed in Fk. If S is
unramified so is S|K : indeed ss*S is a sheaf and satisfies properties (0) and
(1). We prove (3) using the previous Remark.
Our aim in this subsection is to give an explicit description of unramified
sheaves of sets both on S"mk and on Smk in terms of their sections on fields
F 2 Fk and some extra structure. We start with the simplest case.
Definition 1.6 An unramified F"k-set consists of:
(D1) A functor S : Fk ! Set;
(D2) For any F 2 Fk and any discrete valuation v on F , a subset
S(Ov) S(F )
The previous data are moreover supposed to satisfy the following axioms:
(A1) If i : E F is a separable extension in Fk, and v is a discrete
valuation on F which restricts to a discrete valuation w on E with
ramification index 1 then S(i) maps S(Ow ) into S(Ov) and moreover
______________________________
3these two notions are indeed quite closed by [11]
14
_
if the induced extension i : ~(w) ! ~(v) is an isomorphism, then the
following square of sets is cartesian:
S(OwT) ! S(Ov)T
S(E) ! S(F )
(A2) Let X 2 Smk be irreducible with function field F . If x 2 S(F ), then
x lies in all but a finite number of S(Ox)'s, where x runs over the set
X(1)of points of codimension one.
We first observe that an unramified sheaf of sets S on S"m kdefines an
unramified F"k-set. First, evaluation on the separable field extensions of k
yields a functor:
S : Fk ! Set , F 7! S(F )
For any discrete valuation v on F 2 Fk we observe that S(Ov) is a subset of
S(F ). We now claim that these data satisfy the axioms (A1) and (A2) of
unramified F"k-set.
Axiom (A1) is easily checked by choosing convenient smooth models over
k for the essentially smooth k-schemes Spec(F ), Spec(Ov). To prove axiom
(A2) one observes that any x 2 S(F ) comes, by definition, from an element
x 2 S(U) for U 2 Smk an open subscheme of X. Thus any ff 2 S(F ) lies
in all the S(Ox) for x 2 X(1)lying in U. But clearly there are only finitely
many x 2 X(1)not lying in U.
This construction defines a "restriction" functor from the category of
unramified sheaves of sets on S"mk to that of unramified F"k-sets.
Proposition 1.7 The restriction functor from unramified sheaves on S"m k
to unramified F"k-sets is an equivalence of categories.
Proof. Given an unramified F"k-set S, and X 2 Smk irreducible with
functionTfield F , we define the subset S(X) S(F ) as the intersection
x2X(1)S(Ox) S(F ). We extend it in the obvious way for X not irreducible
so that property (0) is satisfied. Given a smooth morphism f : Y ! X in
Smk we define a map: S(f) : S(X) ! S(Y ) as follows. By property (0)
we may assume X and Y are irreducible with field of fractions E and F
respectfully and f is dominant. The map S(f) is induced by the map S(E) !
15
S(F ) corresponding to the fields extension E F and the observation that
if x 2 X(1)then f-1 (x) is a finite set of points of codimension 1 in Y . We
check that it is a sheaf in the Nisnevich topology using Axiom (A1) and
the characterization of Nisnevich sheaves from [39]. It is clearly unramified.
Finally to show that we have just constructed the inverse to the restriction
functor, we use axiom (A2).
Remark 1.8 From now on in this paper, we will not distinguish between
the notion of unramified F"k-set and that of unramified sheaf of sets on S"mk.
If S is an unramified F"k-set we still denote by S the associated unramified
sheaf of sets on S"mk.
Definition 1.9 An unramified Fk-set S is an unramified F"k-set together
with the following additional data:
(D3) For any F 2 Fk and any discrete valuation v on F such that the
residue field ~(v) is separable over k, a map sv : S(Ov) ! S(~(v)),
called the specialization map associated to v.
These data should satisfy furthermore the following axioms:
(A3) (i) If i : E F is an extension in Fk, and v is a discrete valuation
on F which restricts to a discrete valuation w on E with ramification
index 1, then S(i) maps S(Ow ) to S(Ov) and if the two residue fields
are separable over k the following diagram is commutative:
S(Ow ) ! S(Ov)
# #
S(~(w)) ! S(~(v))
(ii) If i : E F is an extension in Fk, and v a discrete valuation on
F which restricts to 0 on E then the map S(i) : S(E) ! S(F ) has its
image contained in S(Ov).
(iii) if moreover ~(v) is separable over k, then if we let j : E ~(v) de-
notes the induced extension the composition S(E)! S(Ov) sv!S(~(v))
is equal to S(j).
(A4) (i) For any X 2 Smk, any point z 2 X(2)of codimension 2, and for
any point y0 2 X(1) such that z 2 __y0and such that __y02 Smk, the
16
T
map sy0 : S(Oy0) ! S(~(y0)) maps y2(Xz)(1)S(Oy) into S(O_y0,z)
S(~(y0)).
(ii) Moreover if ~(z) is separable over k, the composition
"
S(Oy) ! S(Oz0) ! S(~(z))
y2X(1)
doesn't depend on the choice of y0.
Remark 1.10 The Axiom (A4) has a special"role". When we will con-
struct unramified Milnor-Witt K-theory in Section 2.2 below, this axiom will
appear to be the most difficult to check. In fact the subsection 1.3 is devoted
to develop some technic to check this axiom in special case. In Rost's ap-
proach [44] this axiom follows from the construction of the Rost's complex
for 2-dimensional local smooth k-scheme. However the construction of this
complex (even for dimension 2 schemes) requires transfers.
Now we claim that an unramified sheaf of sets S on Smk defines an
unramified Fk-set. From what we have done before, we already have in
hands an unramified F"k-set S. Now, for any discrete valuation v on F 2 Fk
with residue field ~(v) separable over k, there is an obvious map sv : S(Ov) !
S(~(v)), obtained by choosing smooth models over k for the closed immersion
Spec(~(v)) ! Spec(Ov). These together define the data (D3). We now
claim that these data satisfy the extra-axioms for unramified Fk-sets. Axiom
(A3) is easily checked by choosing convenient smooth models for Spec(F ),
Spec(Ov) or Spec(~(v).
To check the axiom (A4) we use the commutative square:
S(X) S(Oy0)
# #
S(__y0) = S(Oz) S(~(y0))
and property (2).
Theorem 1.11 The restriction functor just constructed from unramified sheaves
of sets on Smk to unramified Fk-sets is an equivalence of categories.
The Theorem follows clearly from the following more precise statement:
17
Lemma 1.12 Given an unramified Fk-set S, there is a unique way to extend
the unramified sheaf of sets S : (S"m k)op ! Set to a sheaf S : (Smk)op ! Set,
such that for any discrete valuation v on F 2 Fk with separable residue field,
the map S(Ov) ! S(~(v)) induced by the sheaf structure is the specialization
map sv : S(Ov) ! S(~(v)). This sheaf is automatically unramified.
Proof. We first define a restriction map s(i) : S(X) ! S(Y ) for a
closed immersion i : Y ,! X in Smk of codimension 1. If Y = qffYffis
the decomposition of Y into irreducible components then S(Y ) = ffS(Yff)
and s(i) has to be the product of the s(iff) : S(X) ! S(Yff). We thus may
assume Y (and X) irreducible. We then claim there exits a (unique) map
s(i) : S(X) ! S(Y ) which makes the following diagram commute
s(i)
S(X) ! S(Y )
\ \
sy
S(OX,y) ! S(~(y))
where y is the generic point of Y . To check this it is sufficient to prove that
for any z 2 Y (1), the image of S(X) through sy is contained in S(OY,z). But
z has codimension 2 in X and this follows from the first part of axiom (A4).
Now we have the following:
Lemma 1.13 Let i : Z ! X be a closed immersion in Smk of codimension
j1 j2 jd
d > 0. Assume there exists a factorization Z ! Y1 ! Y2 ! . .!. Yd = X of
i into a composition of codimension 1 closed immersions, with the Yi closed
subschemes of X each of which is smooth over k. Then the composition
s(jd) s(j2) s(j1)
S(X) ! . .!.S(Y2) ! S(Y1) ! S(Z)
doesn't depend on the choice of the above factorization of i. We denote this
composition by S(i).
Proof. We proceed by induction on d. For d = 1 there is nothing to
prove. Assume d 2. We may easily reduce to the case Z is irreducible with
generic point z. We have to show that the composition
s(jd) s(j2) s(j1)
S(X) ! . .!.S(Y2) ! S(Y1) ! S(Z) S(~(z))
18
doesn't depend on the choice of the flag Z! Y1! . .!.. .!.X. We may
thus replace X by any open neighborhood of z if we want and even by
Spec(OX,z) if necessary.
We first observe that the case d = 2 follows directly from the Axiom
(A4).
In general as OX,z is regular of dimension d there exists such an and a
sequence of elements (x1, . .,.xd) 2 O( ) which generates the maximal ideal
mv of A := OX,z and such that the flag
Spec(A=(x1, . .,.xd)) ! Spec(A=(x2, . .,.xd) ! . .!.Spec(A=(xd)) ! Spec(A)
is the induced flag Z \ ! Y1 \ ! Y2 \ ! . .!. .
We have thus reduced to proving that given z 2 X(d) a point of codi-
mension d, with separable residue field, in a smooth k-scheme X, and with
A = OX,z, and given a sequence (x1, . .,.xd) whose associated flag of closed
subschemes of Spec(A) consists of smooth k-schemes, the composition
S(A) ! S(Spec(A=(xd))) ! . .!.S(Spec(A=(x2, . .,.xd)) ! S(~)
doesn't depend on the choice of (x1, . .,.xd).
As ~(v) is separable over k, by [17, Corollaire (17.12.2)] the conditions on
smoothness on the members of the associated flag to the sequence (x1, . .,.xd)
is equivalent to the fact the family (x1, . .,.xd) reduces to a basis of the ~(*
*v)-
vector space mv=(mv)x2.
As a consequence, if M 2 GLd(A), the sequence M.(xi) also satisfies this
assumption. For instance any permutation on the (x1, . .,.xd) yields an other
such sequence. By the case of codimension 2 which was observed above, we
see that if we permute xi and xi+1 the compositions S(A) ! S(~(v)) are the
same before or after permutation. We get this way that we may permute as
will the xi's.
Now assume that (x01, . .,.x0d) is an other sequence in A satisfying the
same assumption. Write the x0ias linear combination in the xj. We get a
matrix M 2 Md(A) with (x0i) = M.(xj). This matrix reduces in Md(~) to
an invertible matrix by what we just observed above; thus M itself is in-
vertible. Clearly, one may multiply in a sequence (x1, . .,.xd) by a unit of
A an element xi of the sequence without changing the flag (and thus the
19
composition). Thus we may assume det(M) = 1. Now for a local ring A we
know that the group SLd(A) is the group Ed(A) of elementary matrices in A
(see [23, Chapter VI Corollary 1.5.3] for instance). Thus M can be written
as a product of elementary matrices in Md(A).
As we already know that our statement doesn't depend on the ordering
of a sequence, we have reduced to the following claim: given a sequence
(x1, . .,.xd) as above and a 2 A, the (x1 + ax2, x2, . .,.xd) induces the same
composition S(A) ! S(~(v)) as (x1, . .,.xd). But in fact the flags are the
same. This proves our claim.
Now we come back to the proof of the Lemma 1.12. Let i : Z ! X be a
closed immersion in Smk. By what has been recalled in the previous proof, X
can be covered by open subsets U's such that the induced closed immersion
Z \U ! U admits a factorization as in the statement of the previous Lemma
1.13. Thus for each such U we get a canonical map sU : S(U) ! S(Z \ U).
But applying the same Lemma to the intersections U \ U0, with U0 an other
such open subset, we see that the sU are compatible and define a canonical
map: s(i) : S(X) ! S(Z).
Let f : Y ! X be any morphism between smooth (quasi-projective)
k-schemes. Then f is the composition Y ,! Y xk X ! X of the closed
immersion (given by the graph of f) f : Y ,! Y xk X and the smooth
projection pX : Y xk X ! X. We set
s(pX ) s( f)
s(f) := S(X) ! S(Y xk X) ! S(Y )
To check that this defines a functor on (Smk)op is not hard. First given
a smooth morphism ss : X0 ! X and a closed immersion i : Z ! X in
Smk, denote by i" : Z0 ! X0 the inverse image of i through ss and by
ss0 : Z0 ! Z the obvious smooth morphism. Then the following diagram is
clearly commutative
s(ss) 0
S(X) ! S(X )
# s(i) # s(i0)
s(ss0) 0
S(Z) ! S(Z )
20
g
Then, to prove the functoriality, one takes two composable morphism Z !
f
Y ! X and contemplates the diagram
Z ,! Z xk Y ,! Z xk Y xk X
|| # #
Z ! Y ,! Y xk X
|| || #
Z ! Y ! X
Then one realizes that applying S and s yields a commutative diagram,
proving the claim. Now the presheaf S on Smk is obviously an unramified
sheaf on Smk as these properties only depend on its restriction to S"mk.
Remark 1.14 Again, from now on in this paper, we will not distinguish
between the notion of unramified Fk-set and that of unramified sheaf of sets
on Smk. If S is an unramified Fk-set we still denote by S the associated
unramified sheaf of sets on Smk.
Remark 1.15 The proof of Lemma 1.12 also shows the following. Let S and
E be sheaves of sets on Smk, with S unramified and E satisfying conditions
(0) and (1) of unramified presheaves. Then to give a morphism of sheaves
: E ! S is equivalent to give a natural transformation OE : E|Fk ! S|Fk
such that:
1) for any discrete valuation v on F 2 Fk, the image of E(Ov) E(F )
through OE is contained in S(Ov) S(F );
2) if moreover the residue field of v is separable over k, then the induced
square commutes:
E(Ov) sv! E(~(v))
# OE # OE
S(Ov) ! S(~(v))
We left the details to the reader.
A1-invariant unramified sheaves.
Lemma 1.16 1) Let S be an unramified sheaf of sets on S"m k. Then S is
A1-invariant if and only if it satisfies the following:
For any k-smooth local ring A of dimension 1 the canonical map
S(A) ! S(A1A) is bijective.
21
2) Let S be an unramified sheaf of sets on Smk. Then S is A1-invariant
if and only if it satisfies the following:
For any F 2 Fk the canonical map S(F ) ! S(A1F) is bijective.
Proof. 1) One implication is clear. Let's prove the other one. Let
X 2 Smk be irreducible with function field F . In the following commutative
square
S(X) ! S(A1X)
# #
S(F ) ! S(F (T ))
each map is injective. We observe that S(A1X) ! S(F (T ) factors as S(A1X) !
S(A1F) ! S(F (T ). By our assumption S(F ) = S(A1F); this proves that
S(A1X) is contained inside S(F ). Now it is sufficient to prove that for any
x 2 X(1)one has the inclusion S(A1X) S(OX,x) S(F ). But S(A1X)
S(A1OX,x) S(F (T ), and our assumption gives S(OX,x) = S(A1OX,x). This
proves the claim.
2) One implication is clear. Let's prove the other one. Let X 2 Smk be
irreducible with function field F . In the following commutative square
S(A1X) S(A1F)
# ||
S(X) S(F )
each map is injective but maybe the left vertical one. The latter is thus also
injective which clearly implies the statement.
Remark 1.17 Given an unramified sheaf S of sets on S"mk with Data (D3),
and satisfying the property that for any F 2 Fk, the map S(F ) ! S(F (T ))
is injective, then S is an unramified Fk-group if and only if its extension to
k(T ) is an unramified Fk(T)-set.
Indeed, given a smooth irreducible k-scheme X, a point x 2 X of codi-
mension d, then X|k(T) is still irreducible k(T )-smooth and __x|k(T) is irre-
ducible and has codimension d in X|k(T). Moreover the maps M(X) !
M(X|k(T)), M(Xx) ! M((X|k(T))_x|k(T)), etc.. are injective. So to check
the Axioms involving equality between morphisms, etc..., it suffices to check
them over k(T ) for M|k(T). This allows us to reduce the checking of several
Axioms like (A4) to the case k is infinite.
22
1.2 Unramified sheaves of groups and strong A1-invariance
Our aim in this section is to study unramified sheaves of groups G on Smk
(or on S"mk), as well as their potential strong A1-invariance property, as well
as the comparison between H1 in Zariski and Nisnevich topology. By an
"unramified sheaf of groups" we mean a sheaf of groups on Smk (or on S"mk)
whose underlying sheaf of sets is unramified in the sense of the previous sec-
tion.
Let G be such an unramified sheaf of groups on Sm k (or S"mk). For any
discrete valuation v on F 2 Fk we introduce the pointed set
H1v(Ov; G) := G(F )=G(Ov)
and we observe this is a left G(F )-set. More generally for y a point of codi-
mension 1 in X 2 Smk, we set H1y(X; G) = H1y(OX,y; G). By axiom (A2),
is X is irreducible with function field F the induced left action of G(F ) on
y2X(1)H1y(X; G) preserves the weak-product
0y2X(1)H1y(X; G) y2X(1)H1y(X; G)
where the weak-product 0y2X(1)H1y(X; G) means the set of families for which
all but a finite number of terms are the base point of H1y(X; G). By def-
inition, the isotropy subgroup of this action of G(F ) on the base point of
0y2X(1)H1y(X; G) is exactly G(X) = \y2X(1)G(OX,y). We will summarize this
property by saying that the diagram (of groups, action and pointed set)
1 ! G(X) ! G(F ) ) 0y2X(1)H1y(X; G)
is "exact" (the double arrow refereing to a left action).
Definition 1.18 For any point z of codimension 2 in a smooth k-scheme
X, we denote by H2z(X; G) the orbit set of 0y2X(1)H1y(X; G) under the left
z
action of G(F ), where F 2 Fk denotes the field of functions of Xz.
Now for an irreducible smooth k-scheme X with function field F we may
define an obvious "boundary" G(F )-equivariant map
0y2X(1)H1y(X; G) ! z2X(2)H2z(X; G) (1.1)
23
by collecting together the compositions, for each z 2 X(2):
0y2X(1)H1y(X; G) ! 0y2X(1)H1y(X; G) ! H2z(X; G)
z
It is not clear in general whether or not the image of the boundary map
is always contained in the weak product 0z2X(2)H2z(X; G). We will use the
following Axiom depending on G and an integer d which completes (A2):
(A2') For any smooth k-scheme X of dimension d, the image of the bound-
ary map (1.1) is contained in the weak product 0z2X(2)H2z(X; G).
Remark 1.19 Given an unramified sheaf G of groups on S"m kwith Data
(D3), and satisfying the property that for any F 2 Fk, the map G(F ) !
G(F (T )) is injective, then G satisfies (A2') if and only if its extension to
k(T ) does. This is done along the same lines as in Remark 1.17.
We assume from now on that G satisfies (A2'). Altogether we get for X
smooth over k, irreducible with function field F , a "complex" C*(X; G) of
groups, action, and pointed sets:
1 ! G(X) G(F ) ) 0y2X(1)H1y(X; G) ! z2X(2)H2z(X; G)
We set for X 2 Smk: G(0)(X) := 0x2X(0)G(~(x)), G(1)(X) := 0y2X(1)H1y(X; G)
and G(2)(X) := 0z2X(2)H2z(X; G). The correspondence X 7! G(i)(X), i 2,
can be obviously extended to an unramified presheaf of groups on S"mk, which
we still denote by G(i). Note that G(0)is a sheaf in the Nisnevich topology.
However for G(i), i 2 {1, 2} it is not the case in general, these are only shea*
*ves
in the Zariski topology, as any unramified presheaf.
The complex C*(X; G) : 1 ! G(X) ! G(0)(X) ) G(1)(X) ! G(2)(X) will
play in the sequel the role of the (truncated) analogue for G of the Cousin
complex of [9] or of the complex of Rost considered in [44].
Definition 1.20 Let 1 ! H G ) E ! F be a sequence with G a group
acting on the set E which is pointed (as a set not as a G-set), with H G a
subgroup and E ! F a G-equivariant map of sets, with F endowed with the
trivial action. We shall say this sequence is exact if the isotropy subgroup of
the base point of E is H and if the "kernel" of the pointed map E ! F is
equal to the orbit under G of the base point of E.
We shall say that it is exact in the strong sense if moreover the map
E ! F induces an injection into F of the (left) quotient set G\E F .
24
By construction C*(X; G) is exact in the strong sense, for X smooth local
of dimension 2.
Let us denote by Z1(-; G) G(1)the sheaf theoretic orbit of the base
point under the action of G(0)in the Zariski topology on S"mk. We thus have
an exact sequence of sheaves on S"mk in the Zariski topology
1 ! G G(0)) Z1(-; G) ! *
As it is clear that H1Zar(X; G(0)) is trivial (the sheaf G(0)being flasque), th*
*is
yields for any X 2 Smk an exact sequence (of groups and pointed sets)
1 ! G(X) G(0)(X) ) Z1(X; G) ! H1Zar(X; G) ! *
in the strong sense.
Remark 1.21 If X is an (essentially) smooth k-scheme of dimension 1,
we thus get a bijection H1Zar(X; G) = G(0)(X)\G(1)(X). For instance, when X
is a smooth local k-scheme of dimension 2, and if V X is the complement
of the closed point, a smooth k-scheme of dimension 1, we thus get a bijection
H2z(X; G) = H1Zar(V ; G)
Beware that here the Zariski topology is used. This gives a "concrete" inter-
pretation of the "strange" extra cohomology set H2z(X; G).
For X 2 Smk let us denote by K1(X; G) 0y2X(1)H1y(X; G) the kernel
of the boundary map 0y2X(1)H1y(X; G) ! z2X(2)H2z(X; G). The correspon-
dence X 7! K1(X; G) is a sheaf in the Zariski topology on S"m k. There is
an obvious injective morphism of sheaves in the Zariski topology on S"m k:
Z1(-; G) ! K1(-; G). As C*(X; G) is exact for any k-smooth local X of di-
mension 2, Z1(-; G) ! K1(-; G) induces a bijection for any (essentially)
smooth k-scheme of dimension 2.
Remark 1.22 If X is an (essentially) smooth k-scheme of dimension 2,
we thus get that the H1 of the complex C*(X; G) is H1Zar(X; G).
Now we introduce the following axiom on G:
25
(A5) (i) For any separable finite extension E F in Fk, any discrete valu-
ation v on F which restricts to a discrete valuation w_on E with rami-
fication index 1, and such that the induced extension i: ~(w) ! ~(v)
is an isomorphism, the commutative square of groups
G(Ow ) G(E)
# #
G(Ov) G(F )
induces a bijection H1v(Ov; G) ! H1w(Ow ; G).
(ii) For any 'etale morphism X0 ! X between smooth local k-schemes
of dimension 2, with closed point respectfully z0 and z, inducing an
isomorphism on the residue fields ~(z) ~=~(z0), the pointed map
H2z(X; G) ! H2z0(X0; G)
has trivial kernel.
Remark 1.23 In case G is an abelian sheaf of groups, the point (ii) of this
axioms implies more: using the Mayer-Vietoris exact sequence, we easily see
that in fact the map (a group homomorphism indeed) H2z(X; G) ! H2z0(X0; G)
involved in the previous Lemma is also surjective, thus an isomorphism.
Lemma 1.24 Let G be as above. Then the following conditions are equiva-
lent:
(i) the Zariski sheaf X 7! K1(X; G) is a sheaf in the Nisnevich topology
on S"mk;
(ii) for any smooth k-scheme X of dimension 2 the comparison map
H1Zar(X; G) ! H1Nis(X; G) is a bijection;
(iii) G satisfies Axiom (A5)
Proof. (i) ) (ii). Under (i) we know that X 7! Z1(X; G) is a sheaf in the
Nisnevich topology on smooth k-schemes of dimension 2 (as Z1(X; G) !
K1(X; G) is an isomorphism on smooth k-schemes of dimension 2). The
exact sequence in the Zariski topology 1 ! G G(0) ) Z1(-; G) ! *
considered above is then also an exact sequence of sheaves in the Nisnevich
topology. The same reasoning as above easily implies (ii), taking into account
that H1Nis(X; G(0)) is also trivial (easy and left to the reader).
26
(ii) ) (iii). Assume (ii). Let's prove (A5) (i). With the assumptions
given the square
Spec(F ) ! Spec(Ov)
# #
Spec(E) ! Spec(Ow )
is a distinguished square in the sense of [39]. Using the corresponding Mayer-
Vietoris type exact sequence and the fact by (ii) that H1(X; G) = * for any
smooth local scheme X yields immediately that G(E)=G(Ow ) ! G(F )=G(Ov)
is a bijection.
Now let's prove (A5) (ii). Set V = X - {z} and V 0= X0- {z0}. The
square
V 0 X0
# #
V X
is distinguished. From the discussion preceding the Lemma and the inter-
pretation of H2z(X; G) as H1Zar(V ; G), the kernel in question is thus the set *
*of
(isomorphism classes) of G-torsors over V (indifferently in Zariski and Nis-
nevich topology as H1Zar(V ; G) ~= H1Nis(V ; G) by (ii) ) which become trivial
over V 0; but such a torsor can thus be extended to X0 and by a descent
argument in the Nisnevich topology, we may extend the torsor on V to X.
Thus it is trivial because X is local.
(iii) ) (i). Now assume Axiom (A5). We claim that Axiom (A5) (i)
gives exactly that X 7! G(1)(X) is a sheaf in the Nisnevich topology. (A5)
(ii) is easily seen to be what exactly what is needed to imply that K1(-; G)
is a sheaf in the Nisnevich topology.
The monomorphism of Zariski sheaves Z1(-; G) ! K1(-; G) is G(0)-
equivariant.
Lemma 1.25 Assume G satisfies (A5). Fix an integer d 0. The following
conditions are equivalent:
(i) For any smooth k-scheme X of dimension d the map Z1(X; G) !
K1(X; G) is bijective;
(ii) For any local smooth k-scheme U of dimension d the map Z1(U; G) !
K1(U; G) is bijective;
(iii) For any local smooth k-scheme U of dimension d with function
field F , the complex C*(U; G) : 1 ! G(U) ! G(F ) ) G(1)(U) ! G(2)(U) is
exact.
27
When this conditions are satisfied, for any smooth k-scheme X of dimen-
sion d the comparison map H1Zar(X; G) ! H1Nis(X; G) is a bijection.
Proof. (i) , (ii) is clear as both are Zariski sheaves. (ii) ) (iii) is
proven exactly as in the proof of of (ii) in Lemma 1.24. (iii) ) (i) is also
clear using the given expressions of the two sides.
If we assume these conditions are satisfied, then
1 1 1 1
G(0)(X)\Z (X; G) = HZar(X; G) ! HNis(X; G) = G(0)(X)\K (X; G)
is a bijection. The last equality follows from the fact that K1(; G) is a Nis-
nevich sheaf and the (easy) fact that H1Nis(X; G(0)) is also trivial.
Now we will use one more extra Axiom concerning G and related to A1-
invariance properties:
(A6) For any localization U of a smooth k-scheme at some point u of
codimension 1, the "complex":
1 ! G(A1U) G(0)(A1U) ) G(1)(A1U) ! G(2)(A1U)
is exact. Moreover the morphism G(U) ! G(A1U) is an isomorphism.
Observe that if G satisfies (A6) it is A1-invariant by Lemma 1.16 (as G
is assumed to be unramified). Observe also that if G satisfies Axioms (A2')
and (A5), then we know by Lemma 1.24 that H1Nis(A1X; G) = H1Zar(A1X; G) =
H1(A1X; G) for X smooth of dimension 1.
Our main result in this subsection is the following. Observe that in this
statement we need to assume that G is an unramified sheaf of groups on Smk
(and not only on S"m k). The reason comes from the proof of Lemma 1.30
which uses at some point a restriction to a smooth divisor.
Theorem 1.26 Assume k is infinite. Let G be an unramified sheaf of groups
on Smk that satisfies Axioms (A2'), (A5) and (A6). Then it is strongly
A1-invariant. Moreover, for any smooth k-scheme X, the comparison map
H1Zar(X; G) ! H1Nis(X; G)
is a bijection.
28
Remark 1.27 1) When k is a finite field one can show that an unramified
sheaf of groups on Smk which satisfies Axioms (A2'), (A5) and (A6) is
also strongly A1-invariant. However we can't prove that the comparison map
is a bijection, we only know that its kernel is trivial. We won't use these fac*
*ts.
2) We will show conversely that a strongly A1-invariant sheaf of groups G
on Smk, k any field, is always unramified. This is proven when k is perfect in
the Appendix, Theorem A.1. The case when k is infinite is done in Corollary
3.8. We prove moreover in Theorem 3.9 that if k is infinite G satisfies axioms
(A2'), (A5) and (A6).
We thus obtain in the case k is infinite, an equivalence between the cate-
gory of strongly A1-invariant sheaves of groups on Smk and that of unramified
sheaves of groups on Smk satisfying axioms (A2'), (A5) and (A6).
We believe that the result should still hold over a finite field.
To prove theorem 1.26 we fix an unramified sheaf of groups G on Smk
which satisfies the Axioms (A2'), (A5) and (A6).
We now introduce two properties depending on G and an integer d 0:
(H1) (d) For any local smooth k-scheme of dimension d the complex
1 ! G(U) G(0)(U) ) G(1)(U) ! G(2)(U) is exact.
(H2) (d) For any localization U of a smooth k-scheme at some point u
of codimension d, the "complex":
1 ! G(A1U) G(0)(A1U) ) G(1)(A1U) ! G(2)(A1U)
is exact.
(H1) (d) is a reformulation of (ii) of Lemma 1.25. It is a tautology in
case d 2. (H2) (d1) holds by Axiom (A6).
Lemma 1.28 Let d 0 be an integer.
1) (H1) (d) ) (H2) (d)
2) If k is infinite: (H2) (d) ) (H1) (d+1)
29
Proof of Theorem 1.26 Assume that k is infinite. Lemma 1.28 implies
by an easy induction that properties (H1)(d) and (H2)(d) hold, for any
any d. Lemmas 1.25 and 1.29 below easily imply, form those, the statement
of the Theorem.
Lemma 1.29 Assume G is A1-invariant. Fix an integer d 0. The follow-
ing conditions are equivalent:
(i) For any smooth k-scheme X of dimension d the map
1 1 1 1 1 1
G(0)(X)\Z (X; G) = HZar(X; G) ! HZar(AX ; G) = G(0)(A1X)\Z (AX ; G)
is bijective;
(ii) For any local smooth k-scheme U of dimension d
1 1
G(0)(A1U)\Z (AU ; G) = *
Proof. The implication (i) ) (ii) is clear as for U a smooth local k-
scheme H1Zar(U; G) = G(0)(U)\Z1(U; G) is trivial. The implication (ii) ) (i)
is proven as follows. (ii) means that H1Zar(A1U; G) = * for any local smooth
k-scheme U. Fix X 2 Smk and denote by ss : A1X ! X the projection. To
prove (i) for X it is sufficient to prove that the pointed simplicial sheaf of
sets Rss*(B(G|A1X)) has trivial ss0. Indeed, its ss1 sheaf is ss*(G|A1X) = G|X
because G is A1-invariant. If the ss0 is trivial, B(G|X ) ! Rss*(B(G|A1X)) is
a simplicial weak equivalence which implies the result. Now to prove the
ss0Rss*((B(G|A1X))) is trivial, we just observe that its stalk a point x 2 X is
H1Zar(A1Xx; G) which is trivial by assumption.
Proof of Lemma 1.28 Let d 2 be an integer (else there is nothing to
prove).
Let us prove 1). Assume that (H1) (d) holds. Let U be an irreducible
smooth k-scheme with function field F . Let us study the following diagram
whose middle row is C*(A1U; G), whose bottom row is C*(U; G) and whose
30
top row is C*(A1F; G):
G(F ) G(F (T )) i 0y2(A1 (1)H1y(A1F; G)
F)
[ || "
G(A1U) G(F (T )) ) 0y2(A1 (1)H1y(A1U; G)! 0 1 (2)H2z(A1U; G)
U) z2(AU)
|| [ " "
G(U) G(F ) ) 0y2U(1)H1y(U; G) ! 0z2U(2)H2z(U; G)
(1.2)
The top horizontal row is exact by Axiom (A6). Assume U is local of di-
mension d. The bottom horizontal row is exact by (H1) (d). The middle
vertical column can be explicited as follows. The points y of codimension 1
in A1Uare of two types: either the image of y is the generic point of U or it is
a point of codimension 1 in U; the first set is clearly in bijection with (A1F)*
*(1)
and the second one with U(1)through the map y 2 U(1)7! y[T ] := A1_y A1U.
For y of the first type, it is clear that the set H1y(A1U; G) is the same as
H1y(A1F; G). As a consequence, 0y2(A1 (1)H1y(A1U; G) is exactly the product of
U)
0y2(A1 (1)H1y(A1F; G) and of 0 (1)H1y[T](A1U; G).
F) y2U
To prove (H2)(d) we have exactly to prove the exactness of the middle
horizontal row in (1.2) and more precisely that the action of G(F (T )) on
K1(A1U; G) is transitive.
Take ff 2 K1(A1U; G). As the top horizontal row is exact, there is a
g 2 G(F (T )) such that g.ff lies in 0y2U(1)H1v[T](A1U; G) 0y2(A1 (1)H1y(A1*
*U; G),
U)
which is the kernel of the vertical G(F (T ))-equivariant map 0y2(A1 (1)H1y(A1*
*U; G) !
U)
0y2(A1 (1)H1y(A1F; G)
F)
Thus g.ff lies in K1(A1U; G) \ 0y2U(1)H1y[T](A1U; G) 0y2(A1 (1)H1y(A1U; *
*G).
U)
Now the obvious inclusion K1(U; G) K1(A1U; G) \ 0y2U(1)H1y[T](A1U; G) is a
bijection. Indeed, from part 1) of Lemma 1.30 below, 0y2U(1)H1y(U; G)
0y2U(1)H1y[T](A1U; G) is injective and is exactly the kernel of the composition
of the boundary map 0y2U(1)H1y[T](A1U; G) ! z2(A1U)(2)H2z(A1U; G) and the
projection
z2(A1U)(2)H2z(A1U; G) ! y2U(1),z2(A1_y)(1)H2z(A1U; G)
This shows that K1(A1U; G)\ 0y2U(1)H1y[T](A1U; G) is contained in 0y2U(1)H1y(U*
*; G).
31
But now, the right vertical map in (1.2), z2U(2)H2z(U; G) ! z2(A1U)(2)H2z(*
*A1U; G),
is induced by the correspondence z 2 U(2)7! A1_z A1Uand the corresponding
maps on H2z(-; G). By part 2) of Lemma 1.30 below, this map has trivial
kernel. This easily implies that K1(A1U; G) \ 0y2U(1)H1y[T](A1U; G) is contain*
*ed
in K1(U; G), proving our claim.
Thus g.ff lies in K1(U; G). Now by (H1) (d) we know there is an h 2 G(F )
with hg.ff = * as required.
Let us now prove 2). Assume (H2) (d) holds. Let's prove (H1) (d+1).
Let X be an irreducible smooth k-scheme (of finite type) of dimension d+1
with function field F , let u 2 X 2 Smk be a point of codimension d + 1 and
denote by U its associated local scheme, F its function field. We have to check
the exactness at the middle of G(F ) ) 0y2U(1)H1y(U; G) ! 0z2U(2)H2z(U; G).
Let ff 2 K1(U; G) 0y2U(1)H1y(U; G). We want to show that there ex-
ists g 2 G(F ) such that ff = g.*. Let us denote by yi 2 U the points of
codimension one in U where ff is non trivial. Recall that for each y 2 U(1),
H1y(U; G) = H1y(X; G) where we still denote by y 2 X(1)the image of y in
X. Denote by ffX 2 0y2X(1)H1y(X; G) the canonical element with same sup-
port yi's and same components as ff. ffX may not be in K1(X; G), but, by
Axiom (A2'), its boundary its trivial except on finitely many points zj of
codimension 2 in X. Clearly these points are not in U(2), thus we may, up
to removing the closure of these zj's, find an open subscheme 0in X which
contains u and the yi's and such that the corresponding element induced by
ff ff 0 2 0y2 0(1)H1y(X; G) is in K1( 0; G).
As k is infinite, by Gabber's presentation Lemma [13, 9] there exists an
open subscheme in 0, containing u and the yi's and an 'etale morphism
! A1V, with V k-smooth of dimension d, such that if Y denotes the
reduced closed subscheme whose generic points are the yi, the composition
Y ! ! A1V is still a closed immersion (and such that the composition
Y ! ! A1V! V is a finite morphism).
As U is the localization of at u, the 'etale morphism U ! A1Vinduces
32
a morphism of complexes of the form:
G(F ) - 0y2U(1)H1y(U; G) ! 0z2U(2)H2z(U; G)
" " "
G(E(T )) - 0y2(A1 (1)H1y(A1V; G)! 0 1 (2)H2z(A1V; G)
V) z2(AV)
where E is the function field of V . Let y0ibe the images of the yi in A1V;
these are points of codimension 1 and have the same residue field (because
Y ! A1V is a closed immersion). By the axiom (A5)(i), we see that for
each i, the map H1y0i(A1V; G) ! H1yi(U; G) is a bijection so that there exists
in the bottom complex an element ff0 2 0y2(A1 (1)H1y(G) whose image is ff.
V)
The boundary of this ff0 is trivial. To show this, observe that if z 2 (A1V)(2)
is not contained in Y , then the boundary of ff0 has a trivial component in
H2z(A1V; G). Moreover, if z 2 (A1V)(2)lies in the image of Y in A1V, there is,
by construction, a unique point z0 of codimension 2 in , lying in Y and
mapping to z. It has moreover the same residue field as z. The claim now
follows from (A5)(ii).
By the inductive assumption (H2) (d) we see that ff0 is of the form h.*
in 0y2(A1 (1)H1y(A1V; G) with h 2 G(E(T )). But if g denotes the image of h
V)
in G(F ) we have ff = g.*, proving our claim.
Lemma 1.30 Let G be an unramified sheaf of groups on Smk satisfying
(A2'), (A5) and (A6).
1) Let v be a discrete valuation on F 2 Fk. Denote by v[T ] the discrete
valuation in F (T ) corresponding to the kernel of Ov[T ] ! ~(v)(T ). Then the
map
H1v(Ov; G) ! H1v[T](A1Ov; G)
is injective and its image is exactly the kernel of
H1v[T](A1Ov; G) ! 0z2(A1~(v))(1)H2z(A1Ov; G)
where we see z 2 (A1~(v))(1)as a point of codimension 2 in A1Ov.
2) For any k-smooth local scheme U of dimension 2 with closed point u,
the "kernel" of the map
H2u(G) ! H2u[T](G)
is trivial.
33
Proof. Part 1) follows immediately from the fact that we know from our
axioms the exactness of each row of the Diagram (1.2) is exact for U smooth
local of dimension 1.
To prove 2) we shall use the interpretation of H2z(U; G), for U smooth local
of dimension 2 with closed point z, as H1Zar(V ; G), with V the complement of
the closed point u. By Lemma 1.24, we know that H1Zar(V ; G) ~=H1Nis(V ; G).
Pick up an element ff of H2u(U; G) = H1Nis(V ; G) which becomes trivial in
H2u[T](A1U; G) = H1Nis(VT; G), where VT = (A1U)u[T]-u0, u0denoting the generic
point of A1u A1U. This means that the G-torsor over V become trivial_over
VT. As VT is the inverse limit of the schemes of the form - \ u0, where
runs over the open subschemes of A1Uwhich contains u0, we_see that there
exists such an for which_the pull-back of ff to - \ u0is already trivial.
As contains u0, \ u0 A1~(u)is a non empty dense subset; in case ~(u)
__
is infinite, we thus know that there exists a ~(u)-rational point z in \ u0
lying over u. As ! U is smooth, it follows from [17, Corollaire 17.16.3 p.
106] that there exists an immersion U0 ! such that U0 ! U is 'etale and
whose image contains z. This immersion is then a_closed immersion, and
up to shrinking a bit U0 we may assume that \ u0\ U0 = {z}. Thus the
cartesian square
U0 - z ! U0
# #
V ! U
is a distinguished square [39]. And the pull-back of ff to U0 - z is trivial.
Extending it to U0 defines a descent data which defines an extension of ff to
U; thus as any element of H1Zar(U; G) = H1Nis(U; G) ff is trivial we get our
claim.
Gm -loop sheaves. Recall the following construction, used for instance
by Voevodsky in [48]. Given a presheaf of groups G on Smk, we let G-1
denote the presheaf of groups given by
X 7! Ker(G(Gm x X) ev1!G(X))
Observe that if G is a sheaf of groups, so is G-1, and that if G is unramified,
so is G-1.
Lemma 1.31 If G is a strongly A1-invariant sheaf of groups, so is G-1.
34
Proof. One might prove this using our description of those strongly
A1-invariant sheaf of groups given in our first section. We propose another
argument. Let B(G) be the simplicial classifying space of G (see [39] for
instance). Choose a fibrant resolution B(G) of B(G). We study the pointed
function space
RHom o(Gm , B(G)) := Hom o(Gm , B(G))
We observe it is fibrant and automatically A1-local, as B(G) is. Moreover
its ss1 sheaf is clearly G-1 and its higher homotopy sheaves vanish. Thus
the connected component of RHom o(Gm , B(G)) is B(G-1). We know claim
that this space is in fact 0-connected. For this we observe that the simplicial
set of sections Hom o(Gm , B(G))(F ) over a field F 2 Fk is 0-connected. To
prove this it is sufficient to prove that H1((Gm )F ; G) = * is the point. This
follows from the computation of the set H1((Gm )F ; G) using the complex
described in the first section, and the fact that H1(A1F; G) = *. Now by [32,
Lemma 6.1.3] this fact implies that the space itself is 0-connected.
1.3 Unramified Z-graded abelian sheaves
In this section we want to give some criteria which imply the Axioms (A4)
for some type of unramified abelian sheaves. Our method is inspired by Rost
[44] but avoids the use of transfers. This section (and part of this paper)
grew up in fact from our willingness to construct unramified Milnor-Witt K-
theory (as well as Milnor K-theory) without any transfers: using the result
of this section, this is achieved in the next section.
Let M* be a functor Fk ! Ab* to the category of Z-graded abelian groups.
Important convention for this section We will make everywhere in
this subsection the following additional assumption: M* is extended from
a perfect subfield k0 k over which k is of finite type. This means that
there is a Z-graded functor M0*on Fk0 such that M is isomorphic to the
extension M0|k; see Remark 1.5. This means that the value M*(F ) over F is
M0*(F |k0), where F is considered as a separable finite type extension of k0.
Unless there is a possibility of confusion, we will not mention however the
data M0*, k0, etc... which will be always understood. One of the advantages
35
of this assumption is that if F is any finite type extension of k, maybe not
separable, we may evaluate anyway M* on F and simply denote by M*(F )
the group M0*(F |k0). When we will say that M satisfies Axiom "Lambda",
we will really mean otherwise explicitly expressed that M0*satisfies Axiom
"Lambda", etc... In the same way, when we assume that M* is endowed with
some Datum, we really mean that M0 is endowed with this structure (over
k0).
We will assume throughout this section that M* is endowed with the fol-
lowing extra structures.
(D4) (i) For any F 2 Fk a structure of Z[F x=(F x2)]-module on M*(F ),
which we denote by (u, ff) 7!< u > ff 2 Mn(F ) for u 2 F x and for
ff 2 Mn(F ). This structure should be functorial in the obvious sense in
Fk.
(D4) (ii) For any F 2 Fk and any n 2 Z, a map F x x Mn-1(F ) !
Mn(F ), (u, ff) 7! [u].ff, functorial (in the obvious sense) in Fk.
(D4) (iii) For any discrete valuation v on F 2 Fk and uniformizing
element ss a graded epimorphism of degree -1
@ssv: M*(F ) ! M*-1(~(v))
which is functorial, in the obvious sense, with respect to extensions E ! F
such that v restricts to a discrete valuation on E, with ramification index 1,
if we choose as uniformizing element an element ss in E.
We assume furthermore that the following axioms hold:
(B0) For (u, v) 2 (F x)2 and ff 2 Mn(F ), one has
[uv]ff = [u]ff+ < u > [v]ff
and moreover [u][v]ff = - < -1 > [v][u]ff.
(B1) For a k-smooth integral domain A with field of fractions F , for any
ff 2 Mn(F ), then for all but only finitely many point x 2 Spec(A)(1), one has
36
that for any uniformizing element ss for x, @ssx(ff) 6= 0.
(B2) For any discrete valuation v on F 2 Fk with uniformizing element ss
one has @ssv([u]ff) = [__u]@ssv(ff) 2 Mn(~(v)) and @ssv(< u > ff) =< __u> @ssv(*
*ff) 2
M(n-1)(~(v)), for any unit u in (Ov)x and any ff 2 Mn(F ).
(B3) For any field extension E F 2 Fk and for any discrete valuation
v on F 2 Fk which restricts to a discrete valuation w on E, with ramifica-
tion index e, let ss 2 Ov be a uniformizing element for v and ae 2 Ow be a
uniformizing element for w. Write ae = usse, with u a unit in Ov. Then one
has for ff 2 M*(E), @ssv(ff|F ) = effl< __u> (@aew(ff))|~(v)2 M*(~(v)).
Here we set for any integer n,
Xn
nffl= < (-1)(i-1)>
i=1
We observe that as a particular case of (B3) we may choose E = F so
that e = 1 and we get that for any any discrete valuation v on F 2 Fk, any
uniformizing element ss, and any unit u 2 Oxv, then one has @ussv(ff) =< __u>
@ssv(ff) 2 M(n-1)(~(v)) for any ff 2 Mn(F ).
Thus in case Axiom (B3) holds, the kernel of the surjective homomor-
phism @ssvonly depends on the valuation v, not on any choice of ss. In that
case we then simply denote by
M*(Ov) M*(F )
this kernel. Axiom (B1) is then exactly equivalent to Axiom (A2) for un-
ramified F"k-sets. The following is easy:
Lemma 1.32 Assume M* satisfies Axioms (B1), (B2) and (B3). Then
it satisfies (in each degree) the axioms for a unramified F"k-abelian group.
Moreover, it satisfies Axiom (A5) (i).
We assume from now on (in this section) that M* satisfies Axioms (B0),
(B1), (B2) and (B3). Thus we may (and will) consider each Mn as a sheaf
of abelian groups on S"mk.
37
We recall that we denote, for any discrete valuation v on F 2 Fk, by
H1v(Ov, Mn) the quotient group Mn(F )=Mn(Ov) and by @v : Mn(F ) !
H1v(Ov, Mn) the projection. Of course, if one chooses a uniformizing ele-
ment ss, one gets an isomorphism `ss: M(n-1)(~(v)) ~= H1v(Ov, Mn) with
@v = `ssO @ssv.
For each discrete valuation v on F 2 Fk, and any uniformizing element
ss set
sssv: M*(F ) ! M*(~(v)) , ff 7! @ssv([ss]ff)
Lemma 1.33 Assume M* satisfies Axioms (B0), (B1), (B2) and (B3).
Then for each discrete valuation v the homomorphism sssv: M*(Ov) M*(F )
doesn't depend on the choice of a uniformizing element ss.
Proof. From Axiom (B0) we get for any unit u 2 Ox , any uniformizing
element ss and any ff 2 Mn(F ): [uss]ff = [u]ff+ < u > [ss]ff. Thus if moreover
ff 2 M(Ov), one has sussv(ff) = @ussv([uss]ff) = @ussv([u]ff) + @ussv(< u > [ss*
*]ff) =
@ussv(< u > [ss]ff), as by Axiom (B2) @ussv([u]ff) = [__u]@ussv(ff) = [__u]0 = *
*0. But
by the same Axiom (B2), @ussv(< u > [ss]ff) =< __u> @ussv([ss]ff), which by
Axiom (B3) is equal to < __u>< __u> @ssv([ss]ff) = @ssv([ss]ff). This proves the
claim.
We will denote by
sv : M*(Ov) ! Mn(~(v))
the common value of all the sssv's. In this way M* is endowed with a datum
(D3).
We introduce the following Axiom:
(HA) (i) For any F 2 Fk, the following diagram
P @P
(P)
0 ! M*(F ) ! M*(F (T )) -! P2A1FM*-1(F [T ]=P ) ! 0
is a short exact sequence. Here P runs over the set of irreducible unitary
polynomials, and (P ) means the associated discrete valuation.
38
(HA) (ii) For any ff 2 M(F ), one has @T(T)([T ]ff|F(T)) = ff.
This axiom is obviously related to the Axiom (A6), as it immediately
implies that for any F 2 Fk, M(F ) ! M(A1F) is an isomorphism and
H1Zar(A1F; M) = 0.
We next claim:
Lemma 1.34 Let M* satisfies all the previous Axioms, including Axioms
(HA) (i) and (HA) (ii) then Axioms (A1) (ii), (A3) (i) and (A3) (ii)
hold.
Proof. The first part of Axiom (A1) (ii) follows from Axiom (B4). For
the second part we choose a uniformizing element ss in Ow , which is still a
uniformizing element for Ov and the square
@ssv
M*(F ) - ! M(*-1)(~(v)
" "
@ssw
M*(E) - ! M(*-1)(~(w)
is commutative by our definition (D4) (iii). Moreover the morphism M*(E) !
M*(F ) preserve the product by ss by (D4) (i).
To prove Axiom (A3) we proceed as follows. By assumption we have
E Ov F . Choose a uniformizing element ss of v. We consider the ex-
tension E(T ) F induced by T 7! ss. The restriction of v is clearly the
valuation defined by T on E[T ]. The ramification index is 1. Using the
previous point, we see that we can reduce to the case E F is E E(T )
and v = (T ). In that case, the claim follows from our Axioms (HA) (i) and
(HA) (ii).
From now on, we assume that M* satisfies all the Axioms previously met
in this subsection. We observe that by construction the Axiom (A5) (i) is
clear.
Fix a discrete valuation v on F 2 Fk. We denote by v[T ] the discrete
valuation on F (T ) defined by the divisor Gm |~(v) Gm |Ov whose open com-
plement is Gm |F . Choose a uniformizing element ss for v. Observe that
39
ss 2 F (T ) is still a uniformizing element for v[T ].
We want to analyze the following commutative diagram in which the
horizontal rows are short exact sequences (given by Axiom (HA)):
P P
P@(P)
0 ! M*(F ) ! M*(F (T )) -! P2(A1F)(1)M*-1(F [T ]=P ) !*
* 0
# @ssv # @ssv[T] # P,Q@ss,PQ
P Q
Q@(Q)
0 ! M*-1(~(v)) ! M*-1(~(v)(T )) -! Q2(A1~(v))(1)M*-2(~(v)[T ]=Q)!*
* 0
(1.3)
and where the morphisms @ss,PQ: M*(F [T ]=P ) ! M*-1(~(v)[T ]=Q) are de-
fined by the diagram.
For this we need the following Axiom:
(B4) Let v be discrete valuation on F 2 Fk and let ss be a uniformizing
element. Let P 2 (A1F)(1)and Q 2 (A1~(v))(1)be fixed.
(i) If the closed point Q 2 A1~(v) A1Ovis not in the divisor DP A1Ov
with generic point P 2 A1F A1Ovthen the morphism @ss,PQis zero.
(ii) If Q is in DP A1Ovand if the local ring ODP,Q is a discrete valuation
ring with ss as uniformizing element then
___
P 0 Q
@ss,PQ= - < -___ > @Q : M*(F [T ]=P ) ! M*-1(~(v)[T ]=Q)
Q0
We will set U = Spec(Ov) in the sequel. We first observe that (A1U)(1)=
(A1F)(1)q {v[T ]}, where as usual v[T ] means the generic point of A1~(v) A1U.
For each P 2 (A1F)(1), there is a canonical isomorphism M*-1(F [T ]=P ) ~=
H1P(A1U; M*), as P itself is a uniformizing element for the discrete valuation
(P ) on F (T ). For v[T ], there is also a canonical isomorphism M*-1(~(v)[T ])*
* ~=
H1v[T](A1U; M*) as ss is also a uniformizing element for the discrete valuation
v[T ] on F (T ).
Using the previous isomorphisms, we see that the beginning of the com-
40
plex C*(A1U; M*) (see Section 1.2) is isomorphic to
@ssv[T]+P P@P(P) i j
0 ! M*(A1U) ! M*(F (T )) - ! M*-1(~(v)(T )) P2(A1F)(1)M*-1(F [T ]=P )
The diagram (1.3) can be used to compute theicokernel of the previous j
morphism @ : M*(F (T )) ! M*-1(~(v)(T )) P2(A1F)(1)M*-1(F [T ]=P ).
Indeed the epimorphism @0
P Q P ss,P
Q @(Q)- P,Q@Q
M*-1(~(v)(T )) ( P M*-1(F [T ]=P )) -! Q2(A1~(v))(1)M*-2(~(v)[T ]=*
*Q)
composed with @ is trivial, and the diagram
0
M*(F (T )) !@ M*-1(~(v)(T )) ( P M*-1(F [T ]=P ))@! Q M*-2(~(v)[T ]=Q) ! 0
(1.4)
is an exact sequence: this is just an obvious reformulation of the properties
of (1.3).
Now fix Q0 2 (A1~(v))(1). Let (A1F)(1)0be the set of P 's such that Q0 lies *
*in
the divisor DP of A1Udefined by P .
Lemma 1.35 Assume M* satisfies all the previous Axioms. The obvious
quotient
i j@0Q
M*(F (T )) !@ M*-1(~(v)(T )) P2(A1 (1)M*-1(F [T ]=P )! M*-2(~(v)[T ]=Q0 ) !*
* 0
F)0
of the previous diagram is also an exact sequence.
Proof. Using the snake Lemma, it is sufficient to prove that the image
of the composition P62(A1 (1)M*-1(F [T ]=P ) P2(A1)(1)M*-1(F [T ]=P ) !
U)0 U
Q2(A1~(v))(1)M*-2(~(v)[T ]=Q is exactly Q2(A1~(v))(1)-{Q0}M*-2(~(v)[T ]=Q. Ax-
iom (B4)(i) readily implies that this image is contained in
Q2(A1~(v))(1)-{Q0}M*-2(~(v)[T ]=Q).
Now we want to show that the image entirely reaches each M*-2(~(v)[T_]=Q,_
Q 6= Q0. For any such Q, there is a P , irreducible, such that Q is ffP , for
some unit ff 2 ~(v)x . Thus Q lies over DP , but not Q0. Moreover, (ss, P ) is
a system of generators of the maximal ideal of the local dimension 2 regular
41
ring (Ov[T ])(Q), thus (Ov[T ]=P )(Q)is a discrete valuation ring with uniformi*
*z-
ing element the image of ss. By Axiom (B4)(ii) now, we conclude that @ss,PQ
is onto, proving the claim.
Now let X be a local smooth k-scheme of dimension 2 with closed point
z and function field E. Recall from the beginning of section 1.2 that we
y2X(1)@y
denote by H2z(X; M) the cokernel of the sum of the residues M*(E) -!
y2X(1)H1y(X; M*). We thus have a canonical exact sequence of the form:
y2X(1)@y y2X(1)@yz
0 ! M*(X) ! M*(E) -! y2X(1)H1y(X; M*) -! H2z(X; M*) ! 0
(1.5)
where the homomorphisms denoted @yzare defined by the diagram. This di-
agram is the complex C*((A1U)0; M*).
For X the localization (A1U)0 of A1Uat some closed point Q0 2 A1~(v), with
U = Spec(Ov) where v is a discrete valuation on some F 2 Fk, we thus get
immediately:
Corollary 1.36 Assume M* satisfies all the previous Axioms. The complex
C*((A1U)0; M*) is canonically isomorphic to exact sequence:
i j
0 ! M*((A1U)Q ) ! M*(F (T )) ! M*-1(~(v)(T )) P2(A1 (1)M*-1(F [T ]=P )
F)0
! M*-2(~(v)[T ]=Q) ! 0
This isomorphism provides in particular a canonical isomorphism
M*-2(~(v)[T ]=Q0) ~=H2Q0(A1U; M*)
Corollary 1.37 Assume M* satisfies all the previous Axioms. For each n,
the unramified sheaves of abelian groups (on S"mk) Mn satisfies Axiom (A2').
Proof. From Remark 1.19, it suffices to check this when k is infinite.
Now assume X is a smooth k-scheme. Let y 2 X(1)be a point of codi-
mension 1. We wish to prove that given ff 2 H1y(X; M*), there are only
finitely many z 2 X(2)such that @yz(ff) is non trivial. But as k is infinite, by
Gabber's Lemma, there is an open neighborhood X of y and an 'etale
42
morphism ! A1V, for V some open subset of an affine space over k, such
that the morphism __y\ ! A1Vis a closed immersion.
The complement __y- __y\ is a closed subset everywhere of > 0-dimension
and thus contains only finitely many points of codimension 1 in __y.
For any z 2 (__y\ )(1), the 'etale morphism ! A1Vobviously induces a
commutative square
@yz 2
H1y(X; M*) ! Hz(X; M*)
" o " o
@yz 2 1
H1y(A1V; M*) ! Hz(AV ; M*)
(because __y\ ! A1Vis a closed immersion), we reduce to proving the claim
for the image of y in A1V, which clearly follows from our previous results.
Now that we know that M* satisfies Axiom (A2'), for X a smooth k-
scheme with function field E we may define as in section 1.2 a (whole) com-
plex C*(X; M*) of the form
y2X(1)@y y,z@yz
0 ! M*(X) ! M*(E) -! y2X(1)H1y(X; M*) -! z2X(2)H2z(X; M*)
(1.6)
We thus get as an immediate consequence:
Corollary 1.38 Assume M* satisfies all the previous Axioms. For any dis-
crete valuation v on F 2 Fk, setting U = Spec(Ov), the complex C*(A1U; M*)
is canonically isomorphic to the exact sequence (1.4):
i j
0 ! M*(A1U) ! M*(F (T )) ! M*-1(~(v)(T )) P2(A1F)(1)M*-1(F [T ]=P )
! Q2(A1U)(1)M*-2(~(v)[T ]=Q) ! 0
Consequently, the complex C*(A1U; M*) is an exact complex, and in particular,
for each n, the unramified sheaves of abelian groups (on S"m k) Mn satisfies
Axiom (A6).
Proof. Only the statement concerning Axiom (A6) is not completely
clear: we need to prove that Mn(U) ! Mn(A1U) is an isomorphism for U a
smooth local k-scheme of dimension 1. The rest of the Axiom is clear.
This claim is clear by Axiom (HA) for U of dimension 0. We need to prove
43
it for U of the form Spec(Ov) for some discrete valuation v on some F 2 Fk
(observe that for the moment M* only defines an unramified sheaf on S"mk,
and we can only apply point 1) of Lemma 1.16. But this statement follows
rather easily by contemplating the diagram (1.3).
We next prepare the statement of our last Axiom. Let X be a local
smooth k-scheme of dimension 2, with field of functions F and closed point
z. Consider the complex C*(X; M*) associated to X in (1.5). By definition
we have a short exact sequence:
0 ! M*(F )=M*(X) ! y2X(1)H1y(X; M*) ! H2z(X; M*) ! 0
Let y0 2 X(1)be such that __y0is smooth over k.
The properties of the induced morphism
M*(F )=M*(X) ! y2X(1)-{y0}H1y(X; M*) (1.7)
will play a very important role. We first observe:
Lemma 1.39 Assume M* satisfies all the previous Axioms (including (B4)).
Suppose furthermore that k is infinite. Let X be a local smooth k-scheme of
dimension 2, with field of functions F and closed point z, let y0 2 X(1)be
such that __y0is smooth over k. Then the homomorphism (1.7) is onto.
Proof. We first observe (without using that k is infinite) that this prop-
erty is true for any localization of a scheme of the form A1Uat a point z of
codimension 2, with U = Spec(Ov), for some discrete valuation v on F . If
__y 1 __ 1
i0s A~(v)this is just Axiom (HA). If y0 is not A~(v)we observe that the
complex C*((A1U)z; M*):
y2((A1)z)1@y
M(F (T )) - !U y2((A1U)z)(1)H1y(X; M) ! H2z(A1~(v); M*) ! 0
is isomorphic to the one of Corollary 1.36. By Axiom (B4)(ii) we deduce
that the map @yz: H1y0(X; M) ! H2z(A1~(v); M*) is surjective. This clearly
implies the statement.
To prove the general case we use Gabber's Lemma (and that k is infinite).
Let ff be an element in y2X(1)-{y0}H1y(X; M). Let y1, .., yr be the points
in the support of ff. There exists an 'etale morphism X ! A1U, for some
local smooth scheme U of dimension 1, and with function field K, such that
44
__y 1
!iAU is a closed immersion for each i. But then use the commutative
square
y2X1-{y0}@y
M*(F ) -! y2X(1)-{y0}H1y(X; M*)
" "
y2((A1)z)1-{y0}@y
M*(K(T )) U-! y2((A1U)z)(1)-{y0}H1y(A1U; M*)
We now conclude that ff = iffi, with ffi 2 H1yi(X; M*) ~= H1yi(A1U; M*),
i 2 {1, . .,.r} comes from an element from the bottom right corner. The
isomorphism H1yi(X; M*) ~=H1yi(A1U; M*) is a consequence of our definition of
H1y(-; M*) and (D4)(iii). The bottom horizontal morphism is onto by the
first case we treated. Thus ff lies in the image of our morphism.
Now for our X local smooth k-scheme of dimension 2, with field of func-
tions F and closed point z, with y0 2 X(1)such that __y0is smooth over k,
choose a uniformizing element ss of y0 (in OX,y0). This produces by defi-
nition an isomorphism M*-1(~(y0)) ~= H1y0(X; M*). Now the kernel of the
morphism (1.7) is clearly contained in M*-1(~(y0)) ~=H1y0(X; M*). We may
now state our last Axiom:
(B5) Let X be a local smooth k-scheme of dimension 2, with field of
functions F and closed point z, let y0 2 X(1) be such that __y0is smooth
over k. Choose a uniformizing element ss of y0 (in OX,y0). Then the kernel
of the morphism (1.7) is (identified to a subgroup of M*-1(~(y0))) equal to
M*-1(Oy0,z) M*-1(~(y0)).
Remark 1.40 Thus if M* satisfies Axiom (B5) one gets an exact sequence
0 ! M*-1(Oy0,z) ! M*(F )=M*(X) ! y2X(1)-{y0}H1y(X; M*)
If k is infinite, Lemma 1.39 shows that it is in fact a short exact sequence.
We don't know whether this is still true over a finite field.
Lemma 1.41 Assume that M* satisfies all the previous Axioms of this sec-
tion, including (B4), (B5). Assume the field k is infinite.
45
1) Let X be a local smooth k-scheme of dimension 2, with field of functions
F and closed point z, let y0 2 X(1)be such that __y0is smooth over k. Choose
a uniformizing element ss of OX,y0. Then the homomorphism M*-1(~(y0)) ~=
@y0z2
H1y0(X; M) ! Hz(X; M) induces an isomorphism
y0,ss: M*-1(~(y0))=M*-1(Oy0,z) = H1z(__y0; M*-1) ~=H2z(X; M)
2) Assume f : X0 ! X is an 'etale morphisms between smooth local k-
schemes of dimension 2, with closed points respectfully z0 and z and with the
same residue field ~(z) = ~(z0). Then the induced morphism H2z(X; M*) !
H2z0(X0; M*) is an isomorphism. In particular, M* satisfies Axiom (A5) (ii).
Proof. 1) We know from the previous Remark that the sequence 0 !
M*-1(Oy0) ! M*(F )=M*(X) ! y2X(1)-{y0}H1y(X; M*) ! 0 is a short exact
sequence. By the definition of H2z(X; M) given by the short exact sequence
(1.5), this provides a short exact sequence of the form
0 ! M*-1(Oy0,z) ! M*-1(~(y0)) ! H2z(X; M) ! 0
and produces the required isomorphism y0,ss.
2) Choose y0 2 X(1)such that __y0is smooth over k and a uniformizing
element ss 2 OX,y0. Clearly the pull back of y0 to X0 is still a smooth divisor
denoted by y00, and the image of ss is a uniformizing element for Oy00. Then
the following diagram clearly commutes
__ y00,ss0
H1z0(y00; M*-1) ! H2z0(X0; M)
" "
y,ss 2
H1z(__y0; M*-1) ! Hz(X; M*)
Thus all the morphisms in this diagram are isomorphisms.
Definition 1.42 Let M* be a functor Fk ! Ab* endowed with data (D4)
(i), (D4) (ii) and (D4) (iii); we will be saying that M* is a Z-graded A1k-
module if it satisfies moreover the Axioms (B0), (B1), (B2), (B3), (HA),
(B4) and if M*|k(T)satisfies (B5).
Theorem 1.43 Let M* be a Z-graded A1k-module. Then endowed with the
sv's constructed in Lemma 1.33, for each n, Mn is an unramified Fk-set in the
46
sense of Definition 1.9. By Lemma 1.12 it thus defines an unramified sheaf
of abelian groups on Smk. This unramified sheaf of abelian groups satisfies
Axioms (A2'), (A6) and its base change to any infinite field F 2 Fk satisfies
(A5).
Corollary 1.44 Let M* be a Z-graded A1k-module. Then for each n, Mn is
a strongly A1-invariant sheaf.
Proof. If k is infinite this follows from the previous Theorem and The-
orem 1.26. If k is finite this is proven in Theorem A.8.
Proof of Theorem 1.43. The previous results (Lemmas 1.32 and 1.34)
have already established that Mn is an unramified sheaf of abelian groups on
S"mk, satisfying all the Axioms for unramified sheaves on Smk except Axiom
(A4). Axiom (A2') is proven in Corollary 1.37. Axiom (A5)(i) is clear and
Axiom (A5)(ii) holds if k is infinite by Lemma 1.41. Axiom (A6) holds by
Corollary 1.38. The only remaining point is Axiom (A4). But by Remark
1.17 to prove (A4) in general it is sufficient to treat the case k is infinite.
We assume from now on in this proof that k is infinite.
We start by checking the first part of Axiom (A4). Let X = Spec(A) be
a local smooth k-scheme of dimension 2 with closed point z and function field
F . Let y0 2 X(1)be such that __y0is smooth over k. Choose a pair (ss0, ss1)
of generators for the maximal ideal of A, such that ss0 defines y0. Clearly
___ss __ __
2 O1(y0) is a uniformizing element for z 2 O(y0).
We consider the complex (1.5) of X with coefficients in M* and the in-
duced commutative square:
y2X(1)-{y0}@y
M*(F ) -! y2X(1)-{y0}H1y(X; M*)
# @y0 # - y2X(1)-{y0}@yz
@y0z 2
H1y0(X; M*) -! Hz(X; M*)
We put this square at the top of the commutative square
@y0z 2
H1y0(X; M*) -! Hz(X; M*)
# o __ # o
@ss1z
M*-1(~(y0)) -! M*-2(~(z))
where H1y0(X; M*) !~ M*-1(~(y0)) is the inverse to the canonical isomor-
phism `ss0induced by ss0, and where H2z(X; M*) !~ M*-2(~(z)) is obtained
47
by composing the inverse to the isomorphism y0,ss0obtained by the previous
lemma and `__ss1.
Now we add on the left top corner the morphism M*-1(OX,y0) ! M*(F ),
ff 7! [ss0] ff. We thus get a commutative square of the form:
[ss0].- y2X(1)-{y0}@y 1
M*-1(Oss0) ! M*(F ) -! y2X(1)-{y0}Hy(X; M*)
# @ss0y0 __ #
@ss1z
M*-1(~(y0)) -! M*-2(~(z))
(1.8)
As for y 6= y0, ss0 is unit in OX,y we see that if ff 2 \y2X(1)M*(Oy) the
[ss0]- y2X(1)-{y0}@y
image of ff through the composition M*-1(Oy0) ! M*(F ) - !
y2X(1)-{y0}H1y(X; M*) is zero. By the commutativity of the above diagram
this shows that_the image of such an ff through sy0 = @ss0y0([y0].-) lies in the
kernel of @ss1z. But this kernel is M*-1(O__y0,z) and this proves the first par*
*t of
Axiom (A4) (for M*-1 thus) for M*.
Now we prove the second part of Axiom (A4). Assume that ~(z) is
separable over k. Let y1 2 X(1)be such that __y1is smooth over k and different
from __y0. Clearly the intersection __y0\ __y1is the point z as a closed subset*
*. If
__y __ (1)
a0nd y1 do not intersect transversally,_we_may choose a y2 2 X which
will intersects transversally both y0 and y1. Thus we may this way reduce to
the case, that __y0and __y1do intersect transversally.
Choose ss1 2 A which defines __y1. Clearly (ss0, ss1) generate the maximal
ideal of A. Now we want to prove that the two morphisms \y2X(1)M*(Oy) !
M*-2(~(z)) obtained by using y0 is the same as the one obtained by using
y1.
We contemplate the complex (1.5) for X and expand the equation @O@ = 0
for the elements of the form [ss0][ss1]ff with ff 2 \y2X(1)M*(Oy). From our
axioms it follows that if y 6= y0 and y 6= y1 then @y([ss0][ss1]ff) = 0. Now
`y1
@ss1y1([ss0][ss1]ff) is [___ss0]sy1(ff) 2 M*-1(~(y1)) ~= H1y1(X; M*) and @ss0y0*
*([ss0][ss1]ff)
`y0
is (using Axiom (B0)) - < -1 > [___ss1]sy0(ff) 2 M*-1(~(y0)) ~= H1y0(X; M*).
Now we compute the last boundary morphism and find that the sum
__ss __ss
y1,ss1O `__ss0(sz O0sy1(ff)) + y0,ss0O `__ss1(- < -1 > sz O1sy0(ff)) = 0
48
vanishes in H2z(X; M) (as @ O @ = 0). Lemma 1.45 below exactly yields, from
this, the required equality sz O sy1(ff) = sz O sy0(ff).
Lemma 1.45 Assume that M* is as above. Assume the field k is infinite.
Let X = Spec(A) be a local smooth k-scheme of dimension 2, with field of
functions F and closed point z. Let (ss0, ss1) be elements of A generating the
maximal ideal of A and let y0 2 X(1)the divisor of X corresponding to ss0
and y1 2 X(1) that corresponding to ss0. Assume both are smooth over k.
Then the composed isomorphism
`__ss1 __ y0,ss0
M*-2(~(v)) ~= H1z(y0; M*-1) ~= H2z(X; M)
is equal to < -1 > times the isomorphism
`__ss0 __ y1,ss1
M*-2(~(v)) ~= H1z(y1; M*-1) ~= H2z(X; M)
Proof. We first observe that if f : X0 ! X is an 'etale morphism, with
X0 smooth local of dimension two, with closed point z0 having the same
residue field as z, and if y00and y01denote respectfully the pull-back of y0
and y1, then the elements (ss0, ss1) of A0= O(X0) satisfy the same conditions.
Clearly, by the previous Lemma, the assertion is true for X if and only if it
is true for X0, because the `ss's and y,ss's are compatible. Now there is a
Nisnevich neighborhood of z: ! X and an 'etale morphism ! (A2~(z))(0,0)
which is also an 'etale neighborhood and such that (ss0, ss1) corresponds to the
coordinates (T0, T1). In this way we reduce to the case X = (A2~(z))(0,0)and
(ss0, ss1) = (T0, T1).
Now one reapplies exactly the same computation as in the proof of the
Theorem to elements of the form [T0][T1](ff|F(T0,T1)) 2_M*(F_(T0, T1)) with
ff 2 M*-2(F ). Now the point is that using our axioms sT0(0,0)OsY1(ff|F(T0,T1))*
* =
__
sT0(0,0)(ff|F(T0)) = ff and the same holds for the other term. We thus get from
the proof the equality, for each ff 2 M*-2(F )
Y1,T1O `__T0(ff) = Y0,T0O `__T1(< -1 > ff)
which proves our claim.
49
Let M* be a Z-graded A1k-module. Observe that for any discrete valuation
v on F 2 Fk the image of (Ov)x x M(*-1)(Ov) ! M*(F ), (u, ff) 7! [u]ff lies
in M*(Ov). This produces for each n 2 Z a morphism of sheaves on Smk:
Gm x M(*-1)! M*.
Lemma 1.46 The previous morphism of sheaves induces for any n, an iso-
morphism (Mn)-1 ~=M(n-1).
Proof. This easily follows from the short exact sequence
@TD0
0 ! Mn(F ) = Mn(A1F) ! Mn(Gm |F ) -! Mn-1(F ) ! 0
given by Axiom (HA) (i).
Remark 1.47 Conversely assume k is perfect. Given a Z-graded abelian
sheaf M* on Smk, consisting of strongly A1-invariant sheaves, together with
isomorphisms (Mn)-1 ~=M(n-1), then using our result in Section 3, one may
show that evaluation on fields yields a functor Fk ! Ab* to Z-graded abelian
groups together with Data (D4) (i), (D4) (ii) and (D4) (iii) satisfying
Axioms (B0), (B1), (B2), (B3), (HA), (B4) and (B5). In this way we
get an equivalence of categories.
Remark 1.48 By the results of Section 3) any strongly A1-invariant sheaf
is strictly A1-invariant. The category HMk of homotopy modules over k
(see also [11]) consisting of Z-graded strictly A1-invariant abelian sheaves
M* on Smk, together with isomorphisms (Mn)-1 ~= M(n-1), is the heart of
the homotopy t-structure on the stable A1-homotopy category of P1-spectra
over k. this is proven in [31, 30] over a perfect field k.
Remark 1.49 Our approach can be used also to analyze Rost cycle modules
[44], at least over a perfect field k. Let Mk be the full subcategory of Z-grad*
*ed
A1k-modules (or equivalently of the category HMk introduced in the previous
remark) be the full subcategory consisting of those M* satisfying < u >= 1
for each u 2 F x. Those M* have a trivial Z[F x]-module structure. Observe
that in that case the residue morphisms @ssvbecome canonical (independent of
ss). Then Rost's Axioms implies the existence of an obvious forgetful functor
from his category of cycle modules over k to Mk. This can be shown to be an
equivalence of category (using for instance [11] or by direct inspection using
our construction of transfers in [38]. This means that in the concept of cycle
50
module, one may forget the transfers (but should keep trike of consequences
like Axioms (B4) and (B5)).
One gets back Gersten complexes from [32, 9] and canonical transfers by
[38]. It is clear that Rost's complexes are isomorphic (canonically) to the
associated Gersten complexes.
In general (relaxing the assumption that the Z[F x]-module structure is
trivial, one needs some work to prove that the Gersten (or Cousin) complex
from [9] for M* is indeed explicitly constructed like in Rost using residues,
normalization process and transfers. This is of course conjectured to be true
(and known in some case like [46]).
2 Unramified Milnor-Witt K-theories
Our aim in this section is to compute (or describe), for any integer n > 0,
the free strongly A1-invariant sheaf, which we denote by Zst-A1(n) on the
n-th smash power of Gm . As we will prove in Section 3 that any strongly
A1-invariant sheaf of abelian groups is also strictly A1-invariant, this is also
the free strictly A1-invariant sheaf on (Gm )^n. We will make a free use of the
previous section.
2.1 Milnor-Witt K-theory of fields
The following definition was found in collaboration with Mike Hopkins:
Definition 2.1 Let F be a commutative field. The Milnor-Witt K-theory of
F is the graded associative ring KMW* (F ) generated by the symbols [u], for
each unit u 2 F x, of degree +1, and one symbol j of degree -1 subject to
the following relations:
1 (Steinberg relation) For each a 2 F x- {1} : [a].[1 - a] = 0
2 For each pair (a, b) 2 (F x)2 : [ab] = [a] + [b] + j.[a].[b]
3 For each u 2 F x : [u].j = j.[u]
4 Set h := j.[-1] + 2. Then j . h = 0
51
These Milnor-Witt K-theory groups were introduced by the author in
a different (and more complicated) way, until the previous presentation was
found with Mike Hopkins. The advantage of this presentation was made clear
1
in our computations of the stable ssA0 in [31, 30] as the relations all have ve*
*ry
natural explanations in the stable A1-homotopical world. To perform these
computations in the unstable world and also to produce unramified Milnor-
Witt K-theory sheaves in a completely elementary way, over any field (any
characteristic) we will need to use an "unstable" variant of that presentation
in Lemma 2.4.
Remark 2.2 The quotient ring KMW* (F )=j is the Milnor K-theory KM*(F )
of F defined in [28]: indeed if j is killed, the symbol [u] becomes additive.
Observe precisely that j controls the failure of u 7! [u] to be additive in
Milnor-Witt K-theory.
With all this in mind, it is natural to introduce the Witt K-theory of F
as the quotient KW*(F ) := KMW* (F )=h. It was studied in [34] and will also
be used in our computations below. In loc. cit. it was proven that the non-
negative part is the quotient of the ring T ensW(F) (I(F )) by the Steinberg
relation << u >> . << 1 - u >>. This can be shown to still hold in
characteristic 2.
Proceeding along the same line, it is easy to prove that the non-negative
part KMW 0(F ) is isomorphic to the quotient of the ring T ensKMW0(F)(KMW1 (F ))
by the Steinberg relation [u].[1 - u]. This is related to our old definition of
KMW* (F ).
We will need at some point a presentation of the group of weight n Milnor-
Witt K-theory. The following one will suffice for our purpose. One may give
some simpler presentation but we won't use it:
Definition 2.3 Let F be a commutative field. Let n be an integer. We
let K"MWn (F ) denote the abelian group generated by the symbols of the form
[jm , u1, . .,.ur] with m 2 N, r 2 N, and n = r - m, and with the ui's unit in
F , and subject to the following relations:
1n (Steinberg relation) [jm , u1, . .,.ur] = 0 if ui+ ui+1 = 1, for some i.
2n For each pair (a, b) 2 (F x)2 and each i: [jm , . .,.ui-1, ab, ui+1, . .]*
*.=
[jm , . .,.ui-1, a, ui+1, . .].+ [jm , . .,.ui-1, b, ui+1, . .].
+[jm+1 , . .,.ui-1, a, b, ui+1, . .]..
52
4n For each i, [jm+2 , . .,.ui-1, -1, ui+1, . .].+2[jm+1 , . .,.ui-1, ui+1, .*
* .].=
0
The following lemma is straightforward:
Lemma 2.4 For any field F , any integer n, the correspondence [jm , u1, . .,*
*.un] 7!
jm [u1] . .[.un] induces an isomorphism
K"MWn (F ) ~=KMWn (F )
Proof. The proof consists in expressing the possible relations between
elements of degree n. That is to say the element of degree n in the two-
sided ideal generated by the relations of Milnor-Witt K-theory, except the
number 3, which is encoded in our choices. We left the details to the reader.
Now we establish some elementary but useful facts. For any unit a 2 F x,
we set < a >= 1 + j[a] 2 KMW0 (F ). Observe then that h = 1+ < -1 >.
Lemma 2.5 Let (a, b) 2 (F x)2 be units in F . We have the followings for-
mulas:
1) [ab] = [a]+ < a > .[b] = [a]. < b > +[b];
2) < ab > = < a > . < b >; KMW0 (F ) is central in KMW* (F );
3) < 1 >= 1 in KMW0 (F ) and [1] = 0 in KMW1 (F );
4) < a > is a unit in KMW0 (F ) whose inverse is < a-1 >;
5) [a_b] = [a]- < a_b> .[b]. In particular one has: [a-1] = - < a-1 > .[a].
Proof. 1) is obvious. One obtains the first relation of 2) by applying j
to relation 2 and using relation 3. By 1) we have for any a and b: < a >
.[b] = [b]. < a > thus the elements < a > are central.
Multiplying relation 4 by [1] (on the left) implies that (< 1 > -1).(<
-1 > +1) = 0 (observe that h = 1+ < -1 >). Using 2 this implies that
< 1 >= 1. By 1) we have now [1] = [1]+ < 1 > .[1] = [1] + 1.[1] = [1] + [1];
thus [1] = 0. 4) follows clearly from 2) and 3). 5) is an easy consequence of
1) 2) 3) and 4).
Lemma 2.6 1) For each n 1, the group KMWn (F ) is generated by the
products of the form [u1]. . ...[un], with the ui 2 F x.
2) For each n 0, the group KMWn (F ) is generated by the products
of the form jn. < u >, with u 2 F x. In particular the product with j:
KMWn (F ) ! KMWn-1(F ) is always surjective if n 0.
53
Proof. An obvious observation is that the group KMWn (F ) is generated
by the products of the form jm .[u1]. . ...[u`] with m 0, ` 0, `-m = n and
with the ui's units. The relation 2 can be rewritten j.[a].[b] = [ab] - [a] - [*
*b].
This easily implies the result using the fact that < 1 >= 1.
Remember that h = 1+ < -1 >. Set ffl := - < -1 >2 KMW0 (F ).
Observe then that relation 4 in Milnor-Witt K-theory can also be rewritten
ffl.j = j.
Lemma 2.7 1) For a 2 F x one has: [a].[-a] = 0 and < a > + < -a >= h;
2) For a 2 F x one has: [a].[a] = [a].[-1] = ffl[a][-1] = [-1].[a] = ffl[-1]*
*[a];
3) For a 2 F x and b 2 F x one has [a].[b] = ffl.[b].[a];
4) For a 2 F x one has < a2 >= 1.
Corollary 2.8 The graded KMW0 (F )-algebra KMW* (F ) is ffl-graded commu-
tative: for any element ff 2 KMWn (F ) and any element fi 2 KMWm (F ) one
has
ff.fi = (ffl)n.mfi.ff
Proof. It suffices to check this formula on the set of multiplicative gen-
erators F xq {j}: for products of the form [a].[b] this is 3) of the previous
Lemma. For products of the form [a].j or j.j, this follows from the relation
3 and relation 4 (reading ffl.j = j) in Milnor-Witt K-theory.
Proof of Lemma 2.7. We adapt [28]. Start from the equality (for
a 6= 1) -a = _1-a_1-a-1. Then [-a] = [1 - a]- < -a > .[1 - a-1]. Thus
[a].[-a] = [a][1 - a]- < -a > .[a].[1 - a-1] = 0- < -a > .[a].[1 - a-1] =
< -a >< a > [a-1][1 - a-1] = 0
by 1 and 1) of lemma 2.5. The second relation follows from this by applying
j2 and expanding.
As [-a] = [-1]+ < -1 > [a] we get
0 = [a].[-1]+ < -1 > [a][a]
so that [a].[a] = - < -1 > [a].[-1] = [a].[-1] because 0 = [1] = [-1]+ <
-1 > [-1]. Using [-a][a] = 0 we find [a][a] = - < -1 > [-1][a] = [-1][a].
54
Finally expanding
0 = [ab].[-ab] = ([a]+ < a > .[b])([-a]+ < -a > [b])
gives
0 =< a > ([b][-a]+ < -1 > [a][b])+ < -1 > [-1][b]
as [-a] = [a]+ < a > [-1] we get
0 =< a > ([b][a]+ < -1 > [a][b]) + [b][-1]+ < -1 > [-1][b]
the last term is 0 by 3) so that we get the third claim.
The fourth one is obtained by expanding [a2] = 2[a] + j[a][a]; now due to
point 2) we have [a2] = 2[a] + j[-1][a] = (2 + j[-1])[a] = h[a]. Applying j
we thus get 0.
Let us denote (in any characteristic) by GW (F ) the Grothendieck-Witt
ring of isomorphism classes of non-degenerate symmetric bilinear forms [29]:
this is the group completion of the commutative monoid of isomorphism
classes of non-degenerate symmetric bilinear forms for the direct sum.
For u 2 F x, we denote by < u >2 GW (F ) the form on the vector space of
rank one F given by F 2! F , (x, y) 7! uxy. By the results of loc. cit., these
< u > generate GW (F ) as a group. The following Lemma is (essentially)
[29, Lemma (1.1) Chap. IV]:
Lemma 2.9 [29] The group GW (F ) is generated by the elements < u >,
u 2 F x, and the following relations give a presentation of GW (F ):
(i) < u(v2) >=< u >;
(ii) < u > + < -u >= 1+ < -1 >;
(iii) < u > + < v >=< u + v > + < (u + v)uv > if (u + v) 6= 0.
When char(F ) 6= 2 the first two relations imply the third one and one
obtains the standard presentation of the Grothendieck-Witt ring GW (F ),
see [45, ]. If char(F ) = 2 the third relation becomes 2(< u > -1) = 0.
We observe that the subgroup (h) of GW (F ) generated by the hyperbolic
plan h = 1+ < -1 > is actually an ideal (use the relation (ii)). We let
W (F ) be the quotient (both as a group or as a ring) GW (F )=(h) and let
W (F ) ! Z=2 be the corresponding mod 2 rank homomorphism; W (F ) is
55
the Witt ring of F [29], and [45] in characteristic 6= 2. Observe that the
following commutative square of commutative rings
GW (F ) ! Z
# # (2.1)
W (F ) ! Z=2
is cartesian. The kernel of the mod 2 rank homomorphism W (F ) ! Z=2 is
denoted by I(F ) and is called the fundamental ideal of W (F ).
It follows from our previous results that u 7!< u >2 KMW0 (F ) satisfies all
the relations defining the Grothendieck-Witt ring. Only the last one requires
a comment. As the symbol < u > is multiplicative in u, we may reduce to
the case u + v = 1 by dividing by < u + v > if necessary. In that case, this
follows from the Steinberg relation to which one applies j2. We thus get a
ring epimorphism (surjectivity follows from Lemma 2.6)
OE0 : GW (F ) i KMW0 (F )
For n > 0 the multiplication by jn : KMW0 (F ) ! KMW-n(F ) kills h (because
h.j = 0 and thus we get an epimorphism:
OE-n : W (F ) i KMW-n(F )
Lemma 2.10 For each field F , each n 0 the homomorphism OE-n is an
isomorphism.
Proof. Following [3], let us define by Jn (F ) the fiber product In(F )xin(F)
KMn(F ), where we use the Milnor epimorphism sn : KMn(F )=2 i in(F ),
with in(F ) := In(F )=I(n+1)(F ). For n 0, In(F ) is understood to be
W (F ). Now altogether the J*(F ) form a graded ring and we denote by
j 2 J-1 (F ) = W (F ) the element 1 2 W (F ). For any u 2 F x, denote by
[u] 2 J1(F ) I(F ) x F x the pair (< u > -1, u). Then the four relations
hold in J*(F ) which produces an epimorphism KMW* (F ) i J*(F ). For n > 0
the composition of epimorphisms W (F ) ! KMW-n(F ) ! J-n (F ) = W (F ) is
the identity. For n = 0 the composition GW (F ) ! KMW0 (F ) ! J0(F ) =
GW (F ) is also the identity. The Lemma is proven.
Corollary 2.11 The canonical morphism of graded rings KMW* (F ) ! W (F )[j, j-*
*1]
induced by [u] 7! j-1(< u > -1) induces an isomorphism KMW* (F )[j-1] =
W (F )[j, j-1].
56
Remark 2.12 For any F let I*(F ) denote the graded ring consisting of the
powers of the fundamental ideal I(F ) W (F ). We let j 2 I-1 (F ) = W (F )
be the generator. Then the product with j acts as the inclusions In(F )
In-1 (F ). We let [u] =< u > -1 2 I(F ) be the opposite to the Pfister
form << u >>= 1- < u >. Then these symbol satisfy the relations of
Milnor-Witt K-theory [34] and the image of h is zero. We obtain in this way
an epimorphism KW*(F ) i I*(F ), [u] 7!< u > -1 = - << u >>. This
ring I*(F ) is exactly the image of the morphism KMW* (F ) ! W (F )[j, j-1]
considered in the Corollary above.
We have proven that this is always an isomorphism in degree 0. In
fact this remains true in degree 1, see Corollary 2.47 for a stronger version.
In fact it was proven in [34] (using [?] and Voevodsky's proof of the Milnor
conjectures) that
KW*(F ) i I*(F ) (2.2)
is an isomorphism in characteristic 6= 2. Using Kato's proof of the analogues
of those conjectures in characteristic 2 [21] we may extend this result for any
field F .
From that we may also deduce (as in [34]) that the obvious epimorphism
KW*(F ) i J*(F ) (2.3)
is always an isomorphism.
Here is a very particular case of the last statement, but completely ele-
mentary:
Proposition 2.13 Let F be a field for which any unit is a square. Then the
epimorphism
KMW* (F ) ! KM*(F )
is an isomorphism in degrees 0, and the epimorphism
KMW* (F ) ! KW*(F )
is an isomorphism in degrees < 0. In fact In(F ) = 0 for n > 0 and
In(F ) = W (F ) = Z=2 for n 0. In particular the epimorphisms (eq:kwi)
and (eq:kmwj) are isomorphisms.
57
Proof. The first observation is that < -1 >= 1 and thus 2j = 0 (fourth
relation in Milnor-Witt K-theory). Now using Lemma 2.14 below we see that
for any unit a 2 F x, j[a2] = 2j[a] = 0, thus as any unit b is a square, we get
that for any b 2 F x, j[b] = 0. This proves that the second relation of Milnor-
Witt K-theory gives for units (a, b) in F : [ab] = [a] + [b] + j[a][b] = [a] + *
*[b].
The proposition now follows easily from these observations.
Lemma 2.14 Let a 2 F x and let n 2 Z be an integer. Then the following
formula holds in KMW1 (F ):
[an] = nffl[a]
where for n 0, we nffl2 KMW0 (F ) is defined as follows
Xn
nffl= < (-1)(i-1)>
i=1
(and satisfies for n > 0 the relation nffl=< -1 > (n - 1)ffl+ 1) and where for
n 0, nffl:= - < -1 > (-n)ffl.
Proof. The proof is quite straightforward by induction: one expands
[an] = [an-1] + [a] + j[an-1][a] as well as [a-1] = - < a > [a] = -([a] +
j[a][a]).
2.2 Unramified Milnor-Witt K-theories
In this section we will define for each n 2 Z an explicit sheaf K_MWn on Smk
called unramified Milnor-Witt K-theory in weight n, whose sections on any
field F 2 Fk is the group KMWn (F ). In the next section we will prove that
for n > 0 this sheaf K_MWn is the free strongly A1-invariant sheaf generated
by (Gm )^n.
Residue homomorphisms. Recall from [28], that for any discrete val-
uation v on a field F , with valuation ring Ov F , and residue field ~(v),
one can define a unique homomorphism (of graded groups)
@v : KM*(F ) ! KM*-1(~(v))
called "residue" homomorphism, such that
@v({ss}{u2} . .{.un}) = {___u2} . .{.__un}
58
for any uniformizing element ss and units ui 2 Oxv, and where __udenotes the
image of u 2 Ov \ F x in ~(v).
In the same way, given a uniformizing element ss, one has:
Theorem 2.15 There exists one and only one morphism of graded groups
@ssv: KMW* (F ) ! KMW*-1(~(v))
which commutes to product by j and satisfying the formulas:
@ssv([ss][u2] . .[.un]) = [___u2] . .[.__un]
and
@ssv([u1][u2] . .[.un]) = 0
for any units u1, ..., un of Ov.
Proof. Uniqueness follows from the following Lemma as well as the for-
mulas [a][a] = [a][-1], [ab] = [a] + [b] + j[a][b] and [a-1] = - < a > [a] =
-([a] + j[a][a]). The existence follows from Lemma 2.16 below.
To define the residue morphism @ssvwe use the method of Serre [28]. Let
, be a variable of degree 1 which we adjoin to KMW* (~(v)) with the relation
,2 = ,[-1]; we denote by KMW* (~(v))[,] the graded ring so obtained.
Lemma 2.16 Let v be a discrete valuation on a field F , with valuation ring
OvF and let ss be a uniformizing element of v. The map
Z x Oxv= F x ! KMW* (~(v))[,]
(ssn.u) 7! ss(ssn.u) := [__u] + (nffl< __u>).,
and j 7! j satisfies the relation of Milnor-Witt K-theory and induce a mor-
phism of graded rings:
ss: KMW* (F ) ! KMW* (~(v))[,]
Proof. We first prove the first relation of Milnor-Witt K-theory. Let
ssn.u 2 F x with u in Oxv. We want to prove ss(ssn.u) ss(1 - ssn.u) = 0 in
KMW* (~(v))[,]. If n > 0, then 1 - ssn.u is in Oxv and clearly by definition
59
ss(1 - ssn.u) = 0. If n = 0, then write 1 - u = ssm .v with v a unit in
Ov. If m > 0 the symmetric reasoning allows to conclude. If m = 0, then
ss(u) = [__u] and ss(1 - u) = [1 - __u] in which case the result is also clea*
*r.
It remains to consider the case n < 0. Then ss(ssn.u) = [__u]+(nffl< __u>),.
Moreover we write (1 - ssn.u) as ssn(-u)(1 - ss-n u-1) and we observe that
(-u)(1-ss-n u-1) is a unit on Ov so that ss(1-ssn.u) = [-__u]+nffl< -__u> ,.
Expanding ss(ssn.u) ss(1 - ssn.u) we find [__u][-__u] + nffl< __u> ,[-__u] + n*
*ffl<
-__u> [__u][,] + (nffl)2 < -1 > ,2. We observe that [__u][-__u] = 0 and that
(nffl)2 < -1 > ,2 = (nffl)2[-1] < -1 > , = nffl< -1 > ,[-1] because
(nffl)2[-1] = nffl[-1] (this follows from Lemma 2.14 : (nffl)2[-1] = nffl[(-1)n*
*] =
2 n 2 n n
[(-1)n ] = [(-1) ] as n - n is even). Thus ss(ss .u) ss(1 - ss .u) = nffl{??},
where the expression {??} is
< -__u> ([__u] - [-__u])+ < -1 > [-1]
But [__u] - [-__u] = [__u] - [__u] - [-1] - j[__u][-1] = - < __u> [-1] thus < -*
*__u>
([__u] - [-__u]) = - < -1 > [-1], proving the result.
We now check relation 2 of Milnor-Witt K-theory. Expanding we find
that the coefficient which doesn't involve , is 0 and the coefficient of , is
nffl< __u> +mffl< __v> -nffl< -__u> (< __v> -1) + mffl< __v> (< u > -1)
+nfflmffl< __u_v> (< -1 > -1)
A careful computation (using < __u> + < -__u>=< 1 > + < -1 >=< ___uv>
+ < -___uv> yields that this term is
nffl+ mffl- nfflmffl+ < -1 > nfflmffl
which is shown to be (n + m)ffl. The last two relations of the Milnor-Witt
K-theory are very easy to check.
We now proceed as in [28], we set for any ff 2 KMWn (F ):
ss(ff) := sssv(ff) + @ssv(ff).,
The homomorphism @ssvso defined is easily checked to have the required
properties. Moreover sssv: KMW* (F ) ! KMW* (~(v)) is clearly a morphism of
rings, and as such is the unique one mapping j to j and ssnu to [__u].
60
Proposition 2.17 We keep the previous notations and assumptions. For
any ff 2 KMW* (F ):
1) @ssv([-ss].ff) =< -1 > sssv(ff);
2) @ssv([u].ff) = [__u]@ssv(ff) for any u 2 Oxv.
3) @ssv(< u > .ff) =< __u> @ssv(ff) for any u 2 Oxv.
Proof. We observe that, for n 1, KMWn (F ) is generated as group by
elements of the form jm [ss][u2] . .[.un+m ] or of the form jm [u1][u2] . .[.un*
*+m ],
with the ui's units of Ov and with n + m 1. Thus it suffices to check the
formula on these elements. This is quite straightforward.
Remark 2.18 A heuristic but useful explanation of this "trick" of Serre is
the following. Spec(F ) is the open complement in Spec(Ov) of the closed
point Spec(~(v)). If one had a tubular neighborhood for that close immer-
sion, there should be a morphism E( v) - {0} ! Spec(F ) of the complement
of the zero section of the normal bundle to Spec(F ) ; the map `ssis the
map induced in cohomology by this "hypothetical" morphism. Observe that
choosing ss corresponds to trivializing v, in which case E( v) - {0} becomes
(Gm )Spec(~(v)). Then the ring KMW* (~(v))[,] is just the ring of sections of
KMW* on (Gm )Spec(~(v)). The "funny" relation ,2 = ,[-1] which is true for
any element in KMW* (F ), can also be explained by the fact that the re-
duced diagonal (Gm )Spec(~(v))! (Gm )^2Spec(~(v))is equal to the multiplication
by [-1].
Lemma 2.19 For any fields extension E F and for any discrete valuation
on F which restricts to a discrete valuation w on E with ramification index
e. Let ss be a uniformizing element of v and ae a uniformizing element of w.
Write it ae = usse with u 2 Oxv. Then for each ff 2 KMW* (E) one has
@ssv(ff|F ) = effl< __u> (@aew(ff))|~(v)
Proof. We just observe that the square (of rings)
KMW* (F ) !ss KMW* (~(v))[,]
" "
ae MW
KMW* (E) ! K* (~(w))[,]
where is the ring homomorphism defined by [a] 7! [a|F ] for a 2 ~(v) and
, 7! [__u] + effl< __u> , is commutative. It is sufficient to check the commuta-
tivity in degree 1. This is not hard.
61
Using the residue homomorphism and the previous Lemma one may define
for any discrete valuation v on F the subgroup K_MWn(Ov) KMWn (F ) as the
kernel of @ssv. From our previous Lemma (applied to E = F , e = 1), it is clear
that the kernel doesn't depend on ss, only on v. We define H1v(Ov; K_MWn)
as the quotient group KMWn (F )=KRn(Ov). Once we choose a uniformiz-
ing element ss we get of course a canonical isomorphism KMWn (~(v)) =
H1v(Ov; K_MWn).
Remark 2.20 One very important feature of residue homomorphisms is
that in the case of Milnor K-theory, these residues homomorphisms don't
depend on the choice of ss, only on the valuation, but in the case of Milnor-
Witt K-theory, they do depend on the choice of ss: for u 2 Ox , as one has
@ssv([u.ss]) = @ssv([ss]) + j.[__u] = 1 + j.[__u].
This property of independence of the residue morphisms on the choice of
ss is a general fact (in fact equivalent) for the Z-graded unramified sheaves
M* considered above for which the Z[F x=F x2]-structure is trivial, like Milnor
K-theory. These are called "oriented": in the spirit of Remark 2.18.
Remark 2.21 To make the residue homomorphisms "canonical" (see [3, 4,
46] for instance), one defines for a field ~ and a one dimensional ~-vector spa*
*ce
L, twisted Milnor-Witt K-theory groups: KMW* (~; L) = KMW* (~) Z[~x]Z[L-
{0}], where the group ring Z[~x ] acts through u 7!< u > on KMW* (~) and
through multiplication on Z[L - {0}]. The canonical residue homomorphism
is of the following form
@v : KMW* (F ) ! KMW*-1(~(v); mv=(mv)2)
with @v([ss].[u2] . .[.un]) = [___u2] . .[.__un] __ss, where mv=(mv)2 is the *
*cotangent
space at v (a one dimensional ~(v)-vector space).
The following result and its proof follow closely Bass-Tate [5]:
Theorem 2.22 Let v be a discrete valuation ring on a field F . Then the
subring
K_MW*(Ov) KMW* (F )
is as a ring generated by the elements j and [u] 2 KMW1 (F ), with u 2 Oxva
unit of Ov.
Consequently, the group K_MWn (Ov) is generated by symbols [u1] . .[.un]
with the ui's in Oxvfor n 1 and by the symbols j-n < u > with the u's in
Oxvfor n 0
62
Proof. The last statement follows from the first one as in Lemma 2.6.
We consider the quotient graded abelian group Q* of KMW* (F ) by the sub-
ring A* generated by the elements and j 2 KMW-1(F ) and [u] 2 KMW1 (F ),
with u 2 Oxv a unit of Ov. We choose a uniformizing element ss. The
valuation morphism induces an epimorphism Q* ! KMW*-1(~(v)). It clearly
suffices to check that this is an isomorphism. We will produce an epimor-
phism KMW*-1(~(v)) ! Q* and show that the composition KMW*-1(~(v)) !
Q*! KMW*-1(~(v)) is the identity.
We construct a KMW* (~(v))-module structure on Q*(F ). Denote by E*
the graded ring of endomorphisms of the graded abelian group Q*(F ). First
the element j still acts on Q* and yields an element j 2 E-1. Let a 2 ~(v)x
be a unit in ~(v). Choose a lifting "ff2 Oxv. Then multiplication by "ffclearly
induces a morphism of degree +1, Q* ! Q*+1. We first claim that it doesn't
depend on the choice of "ff. Let "ff0= fif"fbe another lifting so that u 2 Oxv
is congruent to 1 mod ss. Expending ["ff0] = ["ff] + [fi] + j["ff][fi] we see *
*that
it is sufficient to check that for any a 2 F x, the product [fi][a] lies in the
subring A*. Write a = ssn.u with u 2 Oxv. Then expending [ssn.u] we end up
to checking the property for the product [fi][ssn], and using Lemma 2.14 we
may even assume n = 1. Write beta = 1 - ssn.v, with n > 0 and v 2 Oxv.
Thus we have to prove that the products of the above form [1 - ssn.v][ss]
are in A*. For n = 1, the Steinberg relation yields [1 - ss.v][ss.v] = 0.
Expending [ss.v] = [ss](1 + j[v]) + [v], implies [1 - ss.v][ss](1 + j[v]) is in*
* A*.
But by Lemma 2.7, 1 + j[v] =< v > is a unit of A*, with inverse itself.
Thus [1 - ss.v][ss] 2 A*. Now if n 2, 1 - ssn.v = (1 - ss) + ss(1 - ssn-1v) =
n-1 x
(1 - ss)(1 + ss(1-ss__1-ss)) = (1 - ss)(1 - ssw), with w 2 Ov . Expending, we g*
*et
[1 - ssn.v][ss] = [1 - ss][ss] + [1 - ssw][ss] + j[1 - ss][1 - ssw][ss] = [1 - *
*ssw][ss].
Thus the result holds in general.
We thus define this way elements [u] 2 E1. We now claim these ele-
ments (together with j) satisfy the four relations in Milnor-Witt K-theory:
this is very easy to check, by the very definitions. Thus we get this way
a KMW* (~(v))-module structure on Q*. Pick up the element [ss] 2 Q1 =
KMW1 (F )=A1. Its image through @ssvis the generator of KMW* (~(v)) and
the homomorphism KMW*-1(~(v)) ! Q*, ff 7! ff.[ss] provides a section of
@ssv: Q* ! KMW*-1(~(v)). This is clear from our definitions.
It suffices now to check that KMW*-1(~(v)) ! Q* is onto. Using the fact
that any element of F can be written ssnu for some unit u 2 Oxv, we see that
KMW* (F ) is generated as a group by elements of the form jm [ss][u2] . .[.un] *
*or
63
jm [u1] . .[.un], with the ui's in Oxv. But the latter are in A* and the former
are clearly, modulo A*, in the image of KMW*-1(~(v)) ! Q*.
Remark 2.23 In fact one may also prove as in loc. cit. the fact that the
morphism ssdefined in the Lemma 2.16 is onto and its kernel is the ideal
generated by j and the elements [u] 2 KMW1 (F ) with u 2 Oxva unit of Ov
congruent to 1 modulo ss. We will not give the details here, we do not use
these results.
Theorem 2.24 For any field F the following diagram is a (split) short exact
sequence of KMW* (F )-modules:
@P(P)
0 ! KMWn (F ) ! KMWn (F (T )) - ! P KMWn-1(F [T ]=P ) ! 0
(where P runs over the set of unitary irreducible polynomials of F [T ]).
Proof. It it is again very much inspired from [28]. We first observe that
the morphism KMW* (F ) ! KMW* (F (T )) is a split monomorphism; from our
@T(T)([T][-)
previous computations we see that KMW* (F (T )) - ! KMW* (F ) provides
a retraction.
Now we define a filtration on KMW* (F (T )) by sub-rings Ld's
L0 = KMW* (F ) L1 . . .Ld . . .KMW* (F (T ))
such that Ld is exactly the sub-ring generated by j 2 KMW-1(F (T )) and
all the elements [P ] 2 KMW1 (F (T )) with P 2 F [T ] - {0} of degree less
orSequal to d. Thus L0 is indeed KMW* (F ) KMW* (F (T )). Observe that
MW MW
d Ld = K* (F (T )). Observe that each Ld is actually a sub K* (F )-
algebra.
Also observe that using the relation [a.b] = [a]+[b]+j[a][b] that if [a] 2 Ld
and [b] 2 Ld then so are [ab] and [a_b]. As a consequence, we see that for
n 1, Ld(KMWn (F (T ))) is the sub-group generated by symbols [a1] . .[.an]
such that each aiitself is a fraction which involves only polynomials of degree
d. In degree 0, we see in the same way that Ld(KMWn (F (T ))) is the
sub-group generated by symbols < a > jn with a a fraction which involves
only polynomials of degree d.
It is also clear that for n 1, Ld(KMWn (F (T ))) is generated as a group
by elements of the form jm [a1] . .[.an+m ] with the ai of degree d.
64
Lemma 2.25 1) For n 1, Ld(KMWn (F (T ))) is generated by the elements
of L(d-1)(KMWn (F (T ))) and elements of the form jm [a1] . .[.an+m ] with a1 of
degree d and the ai's, i 2 of degree (d - 1).
2) Let P 2 F [T ] be a unitary polynomial of degree d > 0. Let G1, ...,
Gi be be polynomials of degrees (d - 1). Finally let G be the rest of the
Euclidean division of j2{1,...,i}Gj by P , so that G has degree (d-1). Then
one has in the quotient group KMW2 (F (T ))=Ld-1 the equality
[P ][G1 . .G.i] = [P ][G]
Proof. 1) We proceed as in Milnor's paper. Let f1 and f2 be polynomials
of degree d. We may write f2 = -af1+g, with a 2 F xa unit and g of degree
(d - 1). If g = 0, the we have [f1][f2] = [f1][a(-f1)] = [f1][a] (using the
relation [f1, -f1] = 0). If g 6= 0 then as in loc. cit. we get 1 = af1_g+ f2*
*_g
and the Steinberg relation yields [af1_g][f2_g] = 0. Expanding with j we get:
([f1] - [g_a] - j[g_a][af1_g])[f2_g] = 0, which readily implies (still in KMW2 *
*(F (T ))):
g f2
([f1] - [__])[__ ] = 0
a g
But expanding the right factor now yields
g f2
([f1] - [__])([f2] - [g] - j[g][__ ]) = 0
a g
which implies (using again the previous vanishing):
g
([f1] - [__])([f2] - [g]) = 0
a
We see that [f1][f2] can be expressed as a sum of symbols in which at most
one of the factor as degree d, the other being of smaller degree. An easy
induction proves 1).
2) We first establish the case i = 2. We start with the Euclidean division
G1G2 = P Q + G. We get from this the equality 1 = __G__G1.G2+ _PQ__G1.G2which
gives [_PQ__G1.G2][__G__G1.G2] = 0. We expand the left term as [_PQ__G1.G2] =< *
*__Q__G1.G2> [P ]+
[__Q__G1.G2]. We thus obtain [P ][__G__G1.G2] = - < __Q__G1.G2> [__Q__G1.G2][__*
*G__G1.G2] but the right
65
hand side is clearly in L(d-1)(observe Q has degree (d-1)) thus [P ][__G__G1.*
*G2] 2
L(d-1) KMW2 (F (T )). Now [__G__G1G2] = [G] - [G1G2] - j[G1G2][__G__G1G2]. T*
*hus
[P ][__G__G1.G2] = [P ][G]-[P ][G1G2]+ < -1 > j[G1G2][P ][__G__G1G2]. This show*
*s that
modulo L(d-1), [P ][G] - [P ][G1G2] is zero, as required.
For the case i 3 we proceed by induction. Let j2{2,...,i}Gj = P.Q + G0
be the Euclidean division of j2{2,...,i}Gj by P with G0 of degree (d - 1).
Then the rest G of the Euclidean division by P of G1 . .G.iis the same
as the rest of the Euclidean division of G1G0 by P . Now [P ][G1 . .G.i] =
[P ][G1]+[P ][G2 . .G.i]+j[P ][G2 . .G.i][G1]. By the inductive assumption this
is equal, in KMW2 (F (T ))=Ld-1, to [P ][G1]+[P ][G0]+j[P ][G0][G1] = [P ][G0G1*
*].
By the case 2 previously proven we thus get in KMW2 (F (T ))=Ld-1,
[P ][G1 . .G.i] = [P ][G1G0] = [P ][G]
which proves our claim.
Now we continue the proof of Theorem 2.24 following Milnor's proof of
[28, Theorem 2.3]. Let d 1 be an integer and let P 2 F [T ] be a uni-
tary irreducible polynomial of degree d. We denote by KP Ld=L(d-1)the
sub-graded group generated by elements of the form jm [P ][G1] . .[.Gn] with
the Gi of degree (d - 1). For any polynomial G of degree (d - 1), the
multiplication by ffl[G] induces a morphism:
ffl[G]. : KP ! KP
jm [P ][G1] . .[.Gn] 7! ffl[G]jm [P ][G1] . .[.Gn] = jm [P ][G][G1] . .[.Gn]
of degree +1. Let EP be the graded associative ring of_graded endomorphisms
of KP . We claim that the map (F [T ]=P )x ! (EP )1, (G) 7! ffl[G]. (where G
has degree (d - 1)) and the element j 2 (EP )-1 (corresponding to the
multiplication by j) satisfy the four relations of the Milnor-Witt K-theory.
Let us check the Steinberg relation. Let G 2 F [T ] be of degree (d - 1).
Then so is 1-G and the relation (ffl[G].)O(ffl[1-G].) = 0 2 EP is clear. Let us
check relation 2. We let H1 and H2 be polynomials of degree_ (d_- 1). Let _
G be the rest of division of H1H2 by P . By definition ffl[(H1)(H2)]. is ffl[(G*
*)]..
But by the part 2) of the Lemma we have (in KP KMWm (F (T ))=L(d-1)):
_
ffl[(G)].(jm [P ][G1] . .[.Gn]) = jm [P ][G][G1] . .[.Gn] = jm [P ][H1H2][G1] .*
* .[.Gn]
which easily implies the claim. The last two relations are easy to check.
66
We thus obtain a morphism of graded ring KMW* (F [T ]=P ) ! EP . By
letting KMW* (F [T ]=P ) act on [P ] 2 Ld=L(d-1) KMW1 (F (T ))=L(d-1)we
obtain a graded homomorphism
KMW* (F [T ]=P ) ! KP Ld=L(d-1)
which is clearly an epimorphism. By the first part of the Lemma, we see that
the induced homomorphism
P KMW* (F [T ]=P ) ! Ld=L(d-1) (2.4)
is an epimorphism. Now using our definitions, one checks as in [28] that for
P of degree d, the residue morphism @P vanishes on L(d-1)and that moreover
the composition
P @P
P KMW* (F [T ]=P ) i Ld(KMWn (F (T )))=L(d-1)(KMWn (F (T ))) -P! P KMW* (F *
*[T ]=P )
is the identity. As in loc. cit. this implies the Theorem, with the observation
that the quotients Ld=Ld-1 are KMW* (F )-modules and the residues maps are
morphisms of KMW* (F )-modules.
Remark 2.26 We observe that the previous Theorem in negative degrees is
exactly [28, Theorem 5.3].
Now we fixe a base field k. We will make constant use of the results of
Section 1.3. We endow the functor F 7! KMW* (F ), Fk ! Ab* with Data
(D4) (i), (D4) (ii) and (D4) (iii). The datum (D4) (i) comes from
the KMW0 (F ) = GW (F )-module structure on each KMWn (F ) and the datum
(D4) (ii) comes from the product F xxKMWn (F ) ! KMW(n+1)(F ). The residue
homomorphisms @ssvgives the Data (D4) (iii). We observe of course that
these Data are extended from the prime field of k.
Axioms (B0), (B1) and (B2) are clear from our previous results. The
Axiom (B3) follows at once from Lemma 2.19.
Axiom (HA) (ii) is clear. The Theorem 2.24 establishes Axiom (HA)
(i).
For any discrete valuation v on F 2 Fk, and any uniformizing element
ss, define morphisms of the form @yz: KRn(~(y)) ! KRn-1(~(z)) for any y 2
67
(A1F)(1)and z 2 (A1~(v))(1)fitting in the following diagram:
0 ! KMW* (F ) ! KMW* (F (T )) ! y2(A1F)(1)KMW*-1(~(y))! 0
# @ssv # @ssv[T] # y,z@ss,yz
0 ! KMW*-1(~(v)) ! KMWn-1(~(v)(T ))! z2A1~(v)KR*-2(~(v)) ! 0
(2.5)
The following Theorem establishes Axiom (B4).
Theorem 2.27 Let v be a discrete valuation on F 2 Fk, let ss be a uni-
formizing element. Let P 2 Ov[T ] be an irreducible primitive polynomial,
and Q 2 ~(v)[T ] be an irreducible unitary polynomial.
(i) If the closed point Q 2 A1~(v) A1Ovis not in the divisor DP then the
morphism @ss,PQis zero.
(ii) If Q is in DP A1Ovand if the local ring ODP,Q is a discrete valuation
ring with ss as uniformizing element then
___
P 0 Q
@ss,PQ= - < -___ > @Q
Q0
Proof. Let d 2 N be an integer. We will say that Axiom (B4) holds
in degree d if for any field F Fk, any irreducible primitive polynomial
P 2 Ov[T ] of degree d, any unitary irreducible Q 2 ~(v)[T ] then: if Q
doesn't lie in the divisor DP , the homomorphism @PQis 0 on KMW* (F [T ]=P )
and if Q lies in DP and that the local ring O_y,zis a discrete valuation ring
with ss as uniformizing element , then the homomorphism @PQis equal to -@ssQ.
We use Remark 1.17 to reduce to the case the base field k is infinite.
We now proceed by induction on d to prove that Axiom (B4) holds in
degree d for any d. For d = 0 this is trivial. For d = 1 this is very easy.
We will use:
Lemma 2.28 Assume ~(v) is infinite. Let P be a primitive irreducible poly-
nomial of degree d in F [T ]. Let Q be a unitary irreducible polynomial in
~(v)[T ].
68
__ __
Assume either that P is prime to Q, or that Q divides P and that the
local ring ODP,Q is a discrete valuation_ring with_uniformizing element ss.
Then the elements of the form jm [G1 ] . .[.Gn], where all the Gi's are
irreducible_elements in Ov[T ] of degree < d, such_that, either G1 is equal
to ss or G1 is prime to Q, and for any i 2, Gi is prime to Q, generate
KMW* (F [T ]=P ) as a group.
___ ___
Proof. First the symbols of the form jm [G1 ] . .[.Gn] with the Gi irre-
ducible elements of degree < d of Ov[T ] clearly generate the Milnor-Witt
K-theory of f[T ]=P as a group.
__
1) We first assume that P is prime to Q. It suffices to check that those
element above are expressible_in_terms of symbols of the form of the Lemma._
Pick up one such jm [G1 ] . .[.Gn]. Assume that there exists i such that Gi is
divisible by Q (otherwise there is nothing to prove), for instance G1.
As the field ~(v) is infinite, there is an ff 2 Ov such that G1(ff) is a unit
in Oxv. Then there exists a unit u in Oxv and an integer v (actually the
valuation of P (ff) at ss) such that P + ussvG is divisible by_T_- ff in Ov[T ].
Write P +_ussvG1_= (T - ff)H1. Observe that Q_which_divides G1 and is
prime to P must be prime to both T - __ffand H1 .
Observe that (T-ff)_ussvH1 = _P_ussv+ G1 is the Euclidean division of (T-ff)*
*_ussvH1 by
P . By Lemma 2.25 one has in KMW* (F (T )), modulo Ld-1
(T - ff)
jm [P ][G1][G2] . .[.Gn] = jm [P ][________H1][G2] . .[.Gn]
ussv
Because @PDP vanishes on Ld-1, applying @PDP to the previous congruence
yields the equality in KMW* (F [T ]=P )
___ ___ m (T - ff)___ ___
jm [G1 ] . .[.Gn] = j [________H1 ][G2] . .[.Gn]
ussv
___ (T-ff) ___ (T-ff) ___
Expanding [(T-ff)_ussvH1] as [_____ussv] + [H1 ] + j[_____ussv][H1 ] shows that*
* we may
strictly reduce the number of Gi's whose mod ss reduction is divisible by Q.
This proves our first claim (using the relation [ss][ss] = [ss][-1] we may inde*
*ed
assume that only G1 is maybe equal to ss).
__
2) Now assume that Q divides P and that the local ring ODP,Q is a
discrete valuation ring with uniformizing element ss. By our assumption, any
69
non-zero element in the discrete valuation ring ODP,Q = (Ov[T ]=P )Q can be
written as __
R
ssv___
S
with R and S polynomials in Ov[T ] of degree < d whose mod ss reduction
in ~(v)[T ] is_prime_to_Q._ From this, it follows easily that the symbols of
the form jm [G1 ] . .[.Gn], with the Gi's being either a polynomial in Ov[T ] of
degree < d whose mod ss reduction in ~(v)[T ] is prime to Q, either equal to
ss.
The Lemma is proven.
Now let d > 0 and assume the claim is proven in degrees < d, for all
fields. Let P be a primitive irreducible polynomial of degree d in Ov[T ]. Let
Q be a unitary irreducible polynomial in ~(v)[T ].
___
Under our inductive assumption, we may compute @ss,PQ(jm [G1] . .[.Gn])
for any sequence G1, .., Gn as in the Lemma._
Indeed,_the symbol_jm [P ][G1] . .[.Gn] 2 KMWn-mhas residue at P the sym-
bol jm [G1 ] . .[.Gn]. All its other potentially non trivial residues concern
irreducible polynomials of degree < d. By the (proof of) Theorem 2.24, we
know that there exists an ff 2 Ld-1(KMWn-m(F (T )) such that
___
jm [P ][G1] . .[.Gn] + ff
___ *
* ___
has only one non vanishing residue, which is at P , and which equals jm [G1 ] .*
* .[.Gn].
Then clearly the support of ff (which means the set of points of codimen-
sion one in A1F where ff has a non trivial residue) consists of the divisors
defined by the Gi's (P doesn't appear). But those doesn't contain Q.
Using the commutative diagram which defines the @PQ's, we may compute
___ ___
@ss,PQ(jm [G1 ] . .[.Gn]) as
X
@QQ(@ssv(jm [P ][G1] . .[.Gn]+ff)) = @QQ(@ssv(jm [P ][G1] . .[.Gn])+ @ss,GiQ*
*(@GiDGi(ff))
i
P ss,G G
By our inductive assumption, i@Q i(@DiGi(ff)) = 0 because the supports
Gi do not contain Q.
70
We then have two cases:
1) G1 is not ss. Then
@ssv(jm [P ][G1] . .[.Gn]) = 0
___ ___
as every element lies in Oxv[T]. Thus in that case, @ss,PQ(jm [G1 ] . .[.Gn]) *
*= 0
which is compatible with our claim.
2) G1 = ss. Then
@ssv(jm [P ][ss][G2] . .[.Gn]) = - < -1 > @ssv(jm [ss][P ][G2] . .[.Gn])
__ ___ ___
= - < -1 > jm [P ][G2 ] . .[.Gn]
__
Applying_@QQ yields 0 if P is prime to Q, as all the terms are units. If
P = QR, then R is a unit in (A1~v)Q by our assumptions. Expending [QR] =
[Q] + [R] + j[Q][R], we get
___ ___ m ___ ___ __ ___ ___
@ss,PQ(jm [G1 ] . .[.Gn]) = - < -1 > j ([G2 ] . .[.Gn] + j[R ][G2 ] . .[.Gn*
*])
__ m ___ ___
= - < -R > j [G2 ] . .[.Gn]
__ __0
It remains to observe that R = P_Q0.
By the previous Lemma the symbols we used generate KMW* (F [T ]=P ).
Thus the previous computations prove the Theorem.
Now we want to prove Axiom (B5). Let X be a local smooth k-scheme
of dimension 2, with field of functions F and closed point z, let y0 2 X(1)
be such that __y0is smooth over k. Choose a uniformizing element ss of OX,y0.
Denote by Kn(X; y0) the kernel of the map
y2X(1)-{y0}@y
KMWn (F ) -! y2X(1)-{y0}H1y(X; K_MWn) (2.6)
By definition K_MWn (X) Kn(X; y0). The morphism @ssy0: KMWn (F ) !
KMWn-1(~(y0)) induces an injective homomorphism Kn(X; y0)=K_MWn (X)
KMWn-1(~(y0)).
We first observe:
Lemma 2.29 Assume k is infinite. Keep the previous notations and as-
sumptions. Then K_MWn-1(Oy0) Kn(X; y0)=K_MWn (X) KMWn-1(~(y0)).
71
Proof. As k is infinite, we may apply Gabber's lemma to y0, and in
this way, we see (by an easy diagram chase) that we can reduce to the case
X = (A1U)z where U is a smooth local k-scheme of dimension 1. As Theorem
2.27 implies Axiom (B4), we know by ?? that the following complex
y2X(1)@y
0 ! K_MWn(X) ! KMWn (F ) -! y2X(1)H1y(X; K_MWn) ! H2z(X; K_MWn) ! 0
Moreover, we know from there that for __y0smooth, the morphism H1y(X; K_MWn) !
H2z(X; K_MWn) can be "interpreted" as the residue map. Its kernel is thus
K_MWn-1(Oy0) KMWn-1(~(y0)) ~=H1y(X; K_MWn). The exactness of the previous
complex implies then at once that in that case
Kn(X; y0)=K_MWn (X) = K_MWn-1(Oy0)
proving the statement.
Our last objective is now to show that in fact K_MWn-1(Oy0) = Kn(X; y0)=K_MW*
*n (X)
KMWn-1(~(y0)). To do this we observe that by Lemma 1.39, for k infinite, the
morphism (2.6) above is an epimorphism. Thus the previous statement is
equivalent to the fact that the diagram
y2X(1)-{y0}@y
0 ! K_MWn-1(Oy0) ! KMWn (F )=K_MWn (X) -! y2X(1)-{y0}H1y(X; K_MWn) ! 0
is a short exact sequence or in other words that the epimorphism
y2X(1)-{y0}@y
n(X; y0) : KMWn (F )=K_MWn (X)+K_MWn-1(Oy0) -! y2X(1)-{y0}H1y(X; K_M*
*Wn)
(2.7)
is an isomorphism. We also observe that the group KMWn (F )=K_MWn (X) +
K_MWn-1(Oy0) doesn't depend actually on the choice of a local parametrization
of __y0.
Theorem 2.30 Assume k is infinite. Let X be a local smooth k-scheme of
dimension 2, with field of functions F and closed point z, let y0 2 X(1)be
such that __y0is smooth over k. Then the epimorphism n(X; y0)(2.7) is an
isomorphism.
Proof. We know from Axiom (B1) (that is to say Theorem 2.27) and
Lemma ?? that the assertion is true for X a localization of A1U at some
codimension 2 point, where U is a smooth local k-scheme of dimension 1.
72
P
Lemma 2.31 Given any element ff 2 KMWn (F ), write it as ff = iffi,
where the ffi's are pure symbols. Let Y X be the union of the hypersurfaces
defined by each factor of each pure symbol ffi. Let X ! A1U be an 'etale
morphism with U smooth local of dimension 1, with field of functions E,
such that Y ! A1Uis a closed immersion. Then for each i there exists a pure
symbol fii 2 KMWn (E(T )) which maps to ffi modulo K_MWn(X) KMWn (F ).
As a consequence, if @y(ff) 6= 0 in H1y(X; K_MWn) for some y 2 X(1)then
y 2 Y and @y(ff) = @y(fi) =2 H1y(X; K_MWn) = H1y(A1U; K_MWn).
Proof. Let us denote by ssj the irreducible elements in the factorial ring
O(U)[T ] corresponding to the irreducible components of Y A1U. Each
ffi = [ff1i] . .[.ffni] is a pure symbol in which each term ffsidecomposes as
a product ffsi= usiff0siof a unit usiin O(X)x and a product ff0siof ssj's
(this follows from our choices and the factoriality property of A := O(X).
Thus ff0iis in the image of KMWn (E(T )) ! KMWn (F ). Now by construction,
A=( ssj) = B=( ssj), where B = O(U)[T ]. Thus one may choose unit vsiin
s
Bx with wsi:= ui_vsi 1[ ssj].
Now set fisi= vsiff0si, fii := [fi1i] . .[.fini]. Then we claim that fii map*
*s to ffi
modulo K_MWn(X) KMWn (F ). In other words, we claim that [ff1i] . .[.ffni] -
[fi1i] . .[.fini] lies in K_MWn(X) which means that each of its residue at any *
*point
of codimension one in X vanishes. Clearly, by construction the only non-zero
residues can only occur at each ssj.
We end up in showing the following: given elements fis 2 A - {0},
s 2 {1, . .,.n} and ws 2 Ax which is congruent to 1 modulo each irreducible
element ss which divides one of the fis, then for each such ss, @ss([fi1] . .[.*
*fin]) =
@ss([w1fi1] . .[.wnfin]). We expand [w1fi1] . .[.wnfin] as [w1][w2fi2] . .[.wnf*
*in] +
[fi1][w2fi2] . .[.wnfin]+j[w1][fi1][w2fi2]_.s.[.wnfin].sNow using Proposition 2*
*.17
and the fact that wi = 1, we immediately get @ss([w1fi1] . .[.wnfin]) =
@ss([fi1][w2fi2] . .[.wnfin]) which gives the result. An easy induction gives t*
*he
result. This proof can obviously be adapted for pure symbols of the form
jn[ff].
Now the theorem follows easily from the Lemma. Let __ff2 KMWn (F )=K_MWn (X)+
K_MWn-1(Oy0) be in the kernel of n(X; y0). Assume ff 2 KMWn (F ) represents
__ff. By Gabber's Lemma there exists an 'etale morphism X ! A1
U with U_
smooth local of dimension 1, with field of functions E, such that Y [y0 ! A1U
is a closed immersion, where Y is obtained by writing ff as a sum of pure
73
symbols ffi's. By the previous Lemma, we may find fii in KMWn (E(T )) map-
ping_to ff modulo K_MWn (X) yo ffi. Let fi be the sum of the fii's. Then
fi2 KMWn (E(T ))=K_MWn ((A1U)z)_+ K_MWn-1(Oy0) is also in the kernel of our
morphism n((A1U)z; y0). Thus fi= 0 and so __ff= 0.
Unramified KR -theories. We will now slightly generalize our con-
struction by allowing some "admissible" relations in KMW* (F ). An admis-
sible set of relations R is the datum for each F 2 Fk of a graded ideal
R*(F ) KMW* (F ) with the following properties:
(1) For any extension E F in Fk, R*(E) is mapped into R*(F );
(2) For any discrete valuation v on F 2 Fk, any uniformizing element ss,
@ssv(R*(F )) R*(~(v));
(3) For any F 2 Fk the following sequence is a short exact sequence:
P P
P@DP
0 ! R*(F ) ! R*(F (T )) - ! P R*-1(F [t]=P ) ! 0
The third one is clearly usually more difficult to check.
Given an admissible relation R, for each F 2 Fk we simply denote by
KR*(F ) the quotient graded ring KMW* (F )=R*(F ). The property (1) above
means that we get this way a functor
Fk ! Ab*
This functor is moreover clearly endowed with data (D4) (i) and (D4) (ii)
coming from the KMW* -algebra structure. The property (2) defines the data
(D4) (iii). The axioms (B0), (B1), (B2), (B3) are immediate conse-
quences from those for KMW* . Property (3) implies axiom (HA) (i). Axiom
(HA) (ii) is clear. Axioms (B4) and (B5) are also consequences from the
corresponding axioms just established for KMW* . We thus get as in Theorem
1.43 a Z-graded strongly A1-invariant sheaf, denoted by K_R* with isomor-
phisms (K_Rn)-1 ~=K_Rn-1. There is obviously a structure of Z-graded sheaf of
algebras over K_MW*.
Lemma 2.32 Let R* KMW* (k) be a graded ideal. For any F 2 Fk, denote
by R*(F ) := R*.KMW* (F ) the ideal generated by R*. Then R*(F ) is an ad-
missible relation on KMW* . We denote the quotient simply by KMW* (F )=R*.
74
Proof. Properties (1) and (2) are easy to check. We claim that the
property (3) also hold: this follows from Theorem 2.24 which states that the
morphisms and maps are KMW* (F )-module morphisms.
Example 2.33 For instance we may take an integer n and R* = (n)
KMW* (k); we obtain mod n Milnor-Witt unramified sheaves. For R* = (j)
the ideal generated by j, this yields Milnor K-theory. For R* = (n, j) this
yields mod n Milnor K-theory. For R = (h), this yields Witt K-theory, for
R = (j, `) this yields mod ` Milnor K-theory.
Example 2.34 Let RI*(F ) be the kernel of the epimorphism KMW* (F ) i
I*(F ), [u] 7!< u > -1 = - << u >> described in [34], see also Remark
2.12. Recall from the Remark 2.12 that KMW* (F )[j-1] = W (F )[j, j-1] and
that I*(F ) is the image of KMW* (F ) ! W (F )[j, j-1]. Now by our previous
results the morphism KMW* (F ) ! W (F )[j, j-1] induces a morphism of Z-
graded A1k-modules. We now conclude using the Lemma 2.35 below.
Let OE : M* ! N* be a morphism (in the obvious sense) of Z-graded
A1k-modules. Denote for each F 2 Fk by Im(OE)*(F ) (resp. Ker(OE)*(F )) the
image (resp. the kernel) of OE(F ) : M*(F ) ! N*(F ). This extends easily to
a functor Fk ! Ab*. The data (D4) (i), (D4) (ii) and (D4) (iii) on both
M* and N* clearly induce data of the same nature on Im(OE)* and Ker(OE)*.
Lemma 2.35 Let OE : M* ! N* be a morphism of Z-graded A1k-modules.
Then Im(OE)* and Ker(OE)* are Z-graded A1k-modules.
Proof. The only difficulty is to check axiom (HA) (i). It is in fact very
easy to check using the axioms (HA) (i) and (HA) (ii) for M* and N*.
Indeed (HA) (ii) provides a splitting of the short exact sequences of (HA)
(i) for M* and N* which are compatible. One get the axiom (HA) (i) for
Im(OE)* and Ker(OE)* using the snake lemma.
2.3 Milnor-Witt K-theory and strongly A1-invariant
sheaves
In this section, k is again any commutative field. Fix a natural number n 1.
Recall from [39] that (Gm )^n denotes the n-th smash power of the pointed
space Gm . We first construct a canonical morphism of pointed spaces
oen : (Gm )^n ! K_MWn
75
(Gm )^n is a priori the associated sheaf to the naive presheaf n : X 7!
(Ox (X))^n but in fact:
Lemma 2.36 The presheaf n : X 7! (O(X)x )^n is an unramified sheaf.
Proof. It is as a presheaf clearly unramified in the sense of our definition
1.1 thus automatically a sheaf in the Zariski topology. One way further check
it is a sheaf in the Nisnevich topology as well by checking Axiom (A1). Each
time we use the following easy observation. Let Effbe a family of pointed
subsets in a pointed set E. Then \ff(Eff)^n = (\ffEff)^n inside E^n.
Fix an irreducible X 2 Smk with function field F . There is a tauto-
logical symbol map (O(X)x )^n (F x)^n ! KMWn (F ) that takes a sym-
bol (u1, . .,.un) 2 (O(X)x )^n to the corresponding symbol in [u1] . .[.un] 2
KMWn (F ). But clearly this symbol [u1] . .[.un] 2 KMWn (F ) lies in K_MWn(X),
that is to say each of its residues at points of codimension 1 in X is 0. This
follows at once from the definitions and elementary formulas for the residues.
This defines a morphism of sheaves on S"m k. Now to show that this
extends to a morphism of sheaves on Smk, using the equivalence of categories
of Theorem 1.11 (and its proof) we end up to show that our symbol maps
commutes to restriction maps sv, which is also clear from the elementary
formulas we proved in Milnor-Witt K-theory. In this way we have obtained
our canonical symbol map
oen : (Gm )^n ! K_MWn
From the previous section we know that K_MWn is a strongly A1-invariant
sheaf.
Our aim in this section is to prove:
Theorem 2.37 Let n 1. The morphism oen is the universal morphism
from (Gm )^n to a strongly A1-invariant sheaf of abelian groups. In other
words, given a morphism of pointed sheaves OE : (Gm )^n ! M, with M
a strongly A1-invariant sheaf of abelian groups, then there exists a unique
morphism of sheaves of abelian groups : K_MWn ! M such that O oen = OE.
Remark 2.38 The statement is wrong if we release the assumption that M
is a sheaf of abelian groups. The free strongly A1-invariant sheaf of groups
76
generated by Gm will be seen in ?? to be non commutative. For n = 2, it is
a sheaf of abelian groups. For n > 2 it is not known to us.
The statement is also clearly false for n = 0: (Gm )^0 is just Spec(k)+ , th*
*at
is to say Spec(k) with a base point added, and the free strongly A1-invariant
generated by Spec(k)+ is Z, not K_MW0 . To see a analogous presentation of
K_MW0 see Theorem 2.46 below.
Roughly, the idea of the proof is to first use Lemma 2.4 to show that
OE : (Gm )^n ! M induces on fields F 2 Fk a morphism KMWn (F ) ! M(F )
and then to use our work on unramified sheaves in section 1.1 to observe this
induces a morphism of sheaves.
Theorem 2.39 Let M be a strongly A1-invariant sheaf, let n 1 be an
integer, and let OE : (Gm )^n ! M be a morphism of pointed sheaves. For any
field F 2 Fk, there is unique morphism
(F ) : KMWn (F ) ! M(F )
such that for any (u1, . .,.un) 2 (F x)n, n(F )([u1, . .,.un]) = OE(u1, . .,.u*
*n).
Preliminaries. We will freely use some notions and some elementary
results from [39].
Let M be a sheaf of groups on Smk. Recall that we denote by M-1 the
sheaf M(Gm ), and for n 0, by M-n the n-th iteration of this construc-
tion. To say that M is strongly A1-invariant is equivalent to the fact that
K(M, 1) is A1-local [39]. Indeed from loc. cit., for any pointed space X , we
have HomHo(k)(X ; K(M, 1)) ~=H1(X ; M) and HomHo(k)( (X ); K(M, 1)) ~=
"M(X)). Here we denote for M is (a strongly A1-invariant) sheaf of abelian
groups and X a pointed space by M"(X ) the kernel of the evaluation at the
base point of M(X ) ! M(k), so that M(X ) splits as M(k) M"(X ).
We also observe that because M is assumed abelian, the map (from
"pointed to base point free classes")
HomHo(k)( (X ); K(M, 1)) ! HomH(k)( (X ); K(M, 1))
is a bijection.
77
From Lemma 1.31 and its proof we know that in that case, RHom o(Gm ; K(M, *
*1))
is canonically isomorphic to K(M-1, 1) and that M-1 is also strongly A1-
invariant. We also know that R s(K(M, 1) ~=M.
As a consequence, for a strongly A1-invariant sheaf of abelian group M,
the evaluation map
HomHo(k)( ((Gm )^n), K(M, 1)) ! M-n (k)
is an isomorphism of abelian groups.
Now for X and Y pointed spaces, the cofibration sequence X _ Y !
X x Y ! X ^ Y splits after applying the suspension functor . Indeed,
as (X x Y) is a co-group object in Ho(k) the (ordered) sum of the two
morphism (X x Y) ! (X ) _ (Y) = (X _ Y) gives a left inverse
to (X ) _ (Y) ! (X x Y). This left inverse determines an Ho(k)-
isomorphism (X ) _ (Y) _ (X ^ Y) ~= (X x Y).
We thus get canonical isomorphisms:
"M(X x Y) = M"(X ) M"(Y) M"(X ^ Y)
and analogously
H1(X x Y; M) = H1(X ; M) H1(Y; M) H1(X ^ Y; M)
As a consequence, the product ~ : Gm x Gm ! Gm on Gm induces in
Ho(k) a morphism (Gm x Gm ) ! Z(Gm ) which using the above splitting
decomposes as
(~) = : (Gm ) _ (Gm ) _ ((Gm )^2) ! (Gm )
The morphism ((Gm )^2) ! (Gm ) so defined is denoted j. It can be shown
to be isomorphic in Ho(k) to the Hopf map A2 - {0} ! P1.
Let M be a strongly A1-invariant sheaf of abelian groups. We will denote
by
j : M-2 ! M-1
the morphism of strongly A1-invariant sheaves of abelian groups induced by
j.
78
In the same way let : (Gm ^Gm ) ~= (Gm ^Gm ) be the twist morphism
and for M a strongly A1-invariant sheaf of abelian groups, we still denote by
: M-2 ! M-2
the morphism of strongly A1-invariant sheaves of abelian groups induced by
.
Lemma 2.40 Let M be a strongly A1-invariant sheaf of abelian groups.
Then the morphisms j O and j
M-2 ! M-1
are equal.
Proof. This is a direct consequence of the fact that ~ is commutative.
As a consequence, for any m 1, the morphisms of the form
M-m-1 ! M-1
obtained by composing m times morphisms induced by j doesn't depend on
the chosen ordering. We thus simply denote by jm : M-m-1 ! M-1 this
canonical morphism.
Proof of Theorem 2.39 By Lemma 2.6 1), the uniqueness is clear. By
a base change argument analogous to [32, Corollary 5.2.7], we may reduce to
the case F = k.
>From now on we fix a morphism of pointed sheaves OE : (Gm )^n ! M,
with M a strongly A1-invariant sheaf of abelian groups. We first observe that
OE determines and is determined by the Ho(k)-morphism OE : ((Gm )^n) !
K(M, 1), or equivalently by the associated element OE 2 M-n (k).
For any symbol (u1, . .,.ur) 2 (kx )r, r 2 N, we let S0 ! (Gm )^r be the
(ordered) smash-product of the morphisms [ui] : S0 ! Gm determined by ui.
For any integer m 0 such that r = n + m, we denote by [jm , u1, . .,.ur] 2
M(k) ~=HomHo(k)( (S0), K(M, 1)) the composition
jm ^n OE
jm O ([u1, . .,.un]) : (S0) ! ((Gm )^r) ! ((Gm ) ) ! K(M, 1)
The theorem now follows from the following:
79
Lemma 2.41 The previous assignment (m, u1, . .,.ur) 7! [jm , u1, . .,.ur] 2
M(k) satisfies the relations of Definition 2.3 and as a consequence induce a
morphism
(k) : KMWn (k) ! M(k)
Proof. The proof of the Steinberg relation 1n will use the following
stronger result by P. Hu and I. Kriz:
Lemma 2.42 (Hu-Kriz [19]) The canonical morphism of pointed sheaves
(A1 - {0, 1})+ ! Gm ^ Gm , x 7! (x, 1 - x) induces a trivial morphism
"(A1 - {0, 1}) ! (Gm ^ Gm ) (where " means unreduced suspension4) in
Ho(k).
For any a 2 kx -{1} the suspension of the morphism of the form [a, 1-a] :
S0 ! (Gm )^2 factors in Ho(k)) through " (A1 - {0, 1}) ! (Gm ^ Gm ) as
the morphism Spec(k) ! Gm ^ Gm factors itself through A1 - {0, 1}. This
clearly implies the Steinberg relation in our context as the morphism of the
form ([ui, 1-ui]) : (S0) ! ((Gm )^2) appears as a factor in the morphism
which defines the symbol [jm , u1, . .,.ur], with ui+ ui+1 = 1, in M(k).
Now, to check the relation 2n, we observe that the pointed morphism
[a][b] ~
[ab] : S0 ! Gm factors as S0 ! Gm x Gm ! Gm . Taking the suspension
and using the above splitting which defines j, yields that
([ab]) = ([a]) _ ([b]) _ j([a][b]) : (S0) ! (Gm )
in the group HomHo(k)( (S0), (Gm )) whose law is denoted by _. This
clearly implies relation 2n.
Now we come to check the relation 4n. For any a 2 kx , the morphism
a : Gm ! Gm given by multiplication by a is not pointed (unless a = 1).
However the pointed morphism a+ : (Gm )+ ! Gm induces after suspension
(a+ ) : S1 _ (Gm ) ~= ((Gm )+ ) ! (Gm ). We denote by < a >: (Gm ) !
(Gm ) the morphism in Ho(k) induced on the factor (Gm ). We need:
Lemma 2.43 1) For any a 2 kx , the morphism M-1 ! M-1 induced by
< a >: (Gm ) ! (Gm ) is equal to Id + j O [a].
______________________________
4observe that if k = F2, A1 - {0, 1} has no rational point
80
2) The twist morphism 2 HomHo(k)( (Gm ^ Gm ), (Gm ^ Gm )) and
the inverse, for the group structure, of IdGm ^ < -1 >~=< -1 > ^IdGm have
the same image in the set HomH(k)( (Gm ^ Gm ), (Gm ^ Gm )).
Remark 2.44 In fact the map
HomHo(k)( (Gm ^Gm ), (Gm ^Gm )) ! HomH(k)( (Gm ^Gm ), (Gm ^Gm ))
is a bijection. Indeed we know that (Gm ^Gm )) is A1-equivalent to A2-{0}
and also to SL2 because the morphism SL2 ! A2-{0} (forgetting the second
column) is an A1-weak equivalence. As SL2 is a group scheme, the classical
argument shows that this space is A1-simple. Thus for any pointed space
1
X , the action of ssA1(SL2)(k) on HomHo(k)(X , SL2) is trivial. We conclude
because as usual, for any pointed spaces X and Y, with Y A1-connected, the
map HomHo(k)(X , Y) ! HomH(k)(X , Y) is the quotient by the action of the
1
group ssA1(Y)(k).
Proof. 1) The morphism a : Gm ! Gm is equal to the composition
[a]xId ~
Gm ! Gm x Gm ! Gm . Taking the suspension, the previous splittings
give easily the result.
2) Through the Ho(k)-isomorphism (Gm ^ Gm ) ~= A2 - {0}, the twist
morphism becomes the opposite of the permutation isomorphism (x, y) 7!
(y, x). This follows easily from the definition of this isomorphism using the
Mayer-Vietoris square
Gm x Gm A1 x Gm
\ \
Gm x A1 A2 - {0}
and the fact that our automorphism on A2- {0} permutes the top right and
bottom left corner.
Consider the action of GL2(k) on A2 - {0}. As any matrix in SL2(k) is
a product of elementary matrices, the associated automorphism`A2 - {0}'~=
0 1
A2 - {0} is the identity in H(k). As the permutation matrix is
` ' ` ' 1 0
-1 0 1 0
congruent to or modulo SL2(k), we get the result.
0 1 0 -1
81
Proof of Theorem 2.37 By Lemma 2.45 below, we know that for any
smooth irreducible X with function field F , the restriction map M(X)
M(F ) is injective.
As K_MWn is unramified, the Remark 1.15 of section 1.1 shows that to
produce a morphism of sheaves : K_MWn ! M it is sufficient to prove that
for any discrete valuation v on F 2 Fk the morphism (F ) : KMWn (F ) !
M(F ) maps K_MWn (Ov) into M(Ov) and in case the residue field ~(v) is
separable, that some square is commutative (see Remark 1.15).
But by Theorem 2.22, we know that the subgroup K_MWn(Ov) of KMWn (F )
is the one generated by symbols of the form [u1, . .,.un], with the ui 2 Oxv.
The claim is now trivial: for any such symbol there is a smooth model X of
Ov and a morphism X ! (Gm )^n which induces [u1, . .,.un] when composed
with (Gm )^n ! K_MWn. But know composition with OE : (Gm )^n ! M gives
an element of M(X) which lies in M(Ov) M(F ) which is by definition the
image of [u1, . .,.un] through (F ). A similar argument applies to check the
commutativity of the square of the Remark 1.15: one may choose then X
so that there is a closed irreducible Y X of codimension 1, with OX,jY =
Ov F . Then the restriction of ([u1, . .,.un]) M(Ov) is just induced by
the composition Y ! X ! (Gm )^n ! M, and this is also compatible with
the sv in Milnor-Witt K-theory.
Lemma 2.45 Let M be an A1-invariant sheaf of pointed sets on Smk. Then
for any smooth irreducible X with function field F , the kernel of the restric-
tion map M(X) M(F ) is trivial.
In case M is a sheaf of groups, we see that the restriction map M(X) !
M(F ) is injective.
Proof. This follows from [32, Lemma 6.1.4] which states that LA1(X=U)
is always 0-connected for U non-empty dense in X. Now the kernel of
M(X) ! M(U) is covered by HomHo(k)(X=U, M), which is trivial as M
is his own ss0 and LA1(X=U) is 0-connected.
We know deal with K_MW0. We observe that there is a canonical morphism
of sheaves of sets Gm =2 ! K_MW0 , U 7!< U >, where Gm =2 means the
cokernel in the category of sheaves of abelian groups of Gm !2 Gm .
Theorem 2.46 The canonical morphism of sheaves Gm =2 ! K_MW0 is the
universal morphism of sheaves of sets to a strongly A1-invariant sheaf of
82
abelian groups. In other words K_MW0 is the free strongly A1-invariant sheaf
on the space Gm =2.
Proof. Let M be a strongly A1-invariant sheaf of abelian groups. Denote
by Z[S] the free sheaf of abelian groups on a sheaf of sets S. When S is
pointed, then the latter sheaf splits canonically as Z[S] = Z Z(S) where
Z(S) is the free sheaf of abelian groups on the pointed sheaf of sets S, meaning
the quotient Z[S]=Z[*] (where * ! S is the base point). Now a morphism
of sheaves of sets Gm =2 ! M is the same as a morphism of sheaves of
abelian groups Z[Gm ] = Z Z(Gm ) ! M. By the Theorem 2.37 a morphism
Z(Gm ) ! M is the same as a morphism K_MW1 ! M.
Thus to give a morphism of sheaves of sets Gm =2 ! M is the same as
to give a morphism of sheaves of abelian groups Z K_MW1 ! M together
with extra conditions. One of this conditions is clearly that the composition
[2] MW MW [*] MW
Z K_MW1 ! Z K_1 ! M is equal to`Z K_1 '! Z K_1 ! M.
IdZ 0
Here [*] is represented by the matrix and [2] by the matrix
` ' 0 0
IdZ 0 MW MW
. The morphism [2]1 : K_1 ! K_1 is the one induced by
0 [2]1
the square map on Gm . From Lemma 2.14, we know that this map is the
multiplication by 2ffl= h. recall that we set K_W1 := K_MW1 =h. Thus any
morphism of sheaves of sets Gm =2 ! M determines a canonical morphism
Z K_W1! M. Moreover the morphism Z[Gm ] ! Z K_W1factors through
Z[Gm ] ! Z[Gm =2]; this morphism is induced by the map U 7! (1, < U >).
We have thus proven that given any morphism OE : Z[Gm =2] ! M,
there exists a unique morphism Z K_W1 ! M such that the composition
Z[Gm =2] ! Z K_W1! M is OE. As Z K_W1is a strongly A1-invariant sheaf
of abelian groups, it is the free one on Gm =2.
Our claim is now that the canonical morphism i : Z KW1 ! K_MW0 is an
isomorphism.
We know proceed closely to proof of Theorem 2.37. We first observe
that for any F 2 Fk, the canonical map Z[F x=2] ! Z KW1(F ) fac-
tors through Z[F x=2] i KMW0 (F ). This is indeed very simple to check
using the presentation of KMW0 (F ) given in Lemma 2.9. We denote by
83
j(F ) : KMW0 (F ) ! Z KW1(F ) the morphism so obtained.
Using Theorem 2.22 and the same argument as in the end of the proof
of Theorem 2.37 we see that the j(F )'s actually come from a morphism of
sheaves j : K_MW0 ! Z K_W1. It is easy to check on F 2 Fk that i and j are
inverse morphisms to each other.
The following corollary is immediate from the Theorem and its proof:
Corollary 2.47 The canonical morphism
KW1(F ) ! I(F )
is an isomorphism.
3 A1-homotopy sheaves and A1-homology sheaves
In this section we assume the reader is conformable with [39]. We will freely
use the basic notions and some of the results.
1
3.1 Strongly A1-invariance of the sheaves ssAn , n 1
Our aim in this section is to prove:
Theorem 3.1 For any pointed space B, its A1-fundamental sheaf of groups
1 1
ssA1(B) is strongly A -invariant.
To prove this theorem, we will "directly" observe that the sheaf G :=
1
ssA1(B) is unramified and satisfies the assumption of Theorem 1.22 of [?].
The previous Theorem is equivalent to the following:
Theorem 3.2 Let B be a pointed simplicial presheaf of sets on Smk which
satisfies the B.G. property in the Nisnevich topology and the A1-invariance
property (see [39]). Then the associated sheaf of groups to the presheaf U 7!
ss1(B(U)) is strongly A1-invariant.
84
Proof. The simplicial fibrant resolution LA1(X ) of the A1-localization
of a pointed space X satisfies the assumptions of Theorem 3.2. This proves
one implication. By the results of [39] any pointed simplicial presheaf of sets
B satisfying the assumptions of Theorem 3.2 is simplicially equivalent to the
fibrant resolution of the A1-localization of the associated sheaf to B.
We observe the following immediate corollary:
Corollary 3.3 For any pointed space B, and any integer n 1, the A1-
1 1
homotopy sheaf of groups ssAn(B) is strongly A -invariant.
Proof. Apply the Theorem to the (n - 1)-th iterated simplicial loop
space (n-1)s(B) of B, which is still A1-local.
We now start the proof of Theorem 3.1 with some remarks and prelim-
inaries. We observe first that we may assume B is A1-local and, by the
following lemma, we may assume further that B is 0-connected:
Lemma 3.4 Given a pointed A1-local space B, the connected component of
the base point B(0)is also A1-local and the morphism
1 (0) A1
ssA1(B ) ! ss1 (B)
is an isomorphism.
Proof. Indeed, by [39] the A1-localization of a 0-connected space is still
0-connected; thus the obvious morphism LA1(B(0)) ! B induced by B(0)! B
and the fact that B is A1-connected, induces LA1(B(0)) ! B(0), providing a
left inverse to B(0)! LA1(B(0)). Thus B(0)is a retract in H(k) of the A1-local
space LA1(B(0)) so is also A1-local.
>From now on, B is a fixed A1-connected and A1-local space. For an open
immersion U X and any n 0 we set
n(X, U) := [Sn ^ (X=U), B]Ho(k)= ssn(B(X=U))
where Sn denotes the simplicial n-sphere. For n = 0 these are just pointed
sets, for n = 1 these are groups and for i 2 these are abelian groups. In
fact in the proof below we will only use the case n = 0 and n = 1. We may
85
extend these definitions to an open immersion U X between essentially
smooth k-schemes, by passing to the (co)limit.
The following is our main technical Lemma, and will be proven following
the lines of [9, Key Lemma], using Gabber's presentation Lemma:
Lemma 3.5 Assume k is infinite. Let X be a smooth k-scheme, S X
be a finite set of points and Z X be a closed subscheme of codimension
d > 0. Then there exists an open subscheme X containing S and a
closed subscheme Z0 , of codimension d - 1, containing Z := Z \ and
such that the map of pointed sheaves
=( - Z0) ! =( - Z )
is the trivial map in Ho(k).
Proof. By Gabber's geometric presentation Lemma of loc. cit. there
exists an open neighborhood of S, and an 'etale morphism OE : ! A1Vwith
V some open subset in some affine space over k such that Z := Z \ ! A1V
is a closed immersion, OE-1(Z ) = Z and Z ! V is a finite morphism. Let
F denotes the image of Z in V . Then set Z0 := OE-1(A1F). Observe that
dim(F ) = dim(Z) thus codim(Z0) = d-1. Because we work in the Nisnevich
topology, the morphism of sheaves
=( - Z ) ! A1V=(A1V- Z )
is an isomorphism. The commutative square
=( - Z0) ! =( - Z )
# # o
A1V=(A1V- A1F) ! A1V=(A1V- Z )
implies that it suffices to show that the map of pointed sheaves
A1V=(A1V- A1F) ! A1V=(A1V- Z )
is the trivial map in Ho(k). Now because Z ! F is finite, the composition
Z ! A1F P1Fis still a closed immersion, which has thus empty intersection
with the section at infinity s1 : V ! P1V. By the Mayer-Vietoris property
86
the morphism A1V=(A1V- Z ) ! P1V=(P1V- Z ) is an isomorphism of pointed
sheaves. It suffices thus to check that
A1V=(A1V- A1F) ! P1V=(P1V- Z )
is the trivial map in Ho(k). But clearly the morphism s0 : V=(V - F ) !
A1V=(A1V- A1F) induced by the zero section is an A1-weak equivalence. As
the composition s0 : V=(V - F ) ! A1V=(A1V- A1F) ! P1V=(P1V- Z ) is A1-
homotopic (by the obvious A1-homotopy which relates the zero section to the
section at infinity) to the section at infinity s1 : V=(V -F ) ! P1V=(P1V-Z )
we get the result because as noted previously s1 is disjoint from Z and thus
s1 : V=(V - F ) ! P1V=(P1V- Z ) is equal to the point. .
Corollary 3.6 Assume k is infinite. Let X be a smooth (or essentially
smooth) k-scheme, S 2 X be a finite set of points and Z X be a closed
subscheme of codimension d > 0. Then there exists an open subscheme
X containing S and a closed subscheme Z0 , of codimension d - 1,
containing Z := Z \ and such that for any n 2 N the map
n( , - Z ) ! n( , - Z0)
is the trivial map.
In particular, observe that if Z has codimension 1 and X is irreducible,
Z0 must be . Thus for any n 2 N the map
n( , - Z ) ! n( )
is the trivial map.
Proof. For X smooth this is an immediate consequence of the Lemma.
In case X is an essentially smooth k-scheme, we get the result by an obvious
passage to the colimit, using standard results on limit of schemes [16].
Fix an essentially smooth k-scheme X. For any flag of open subschemes
of the form V U X one has the following homotopy exact sequence
(which could be continued on the left):
. .!. 1(X, U) ! 1(X, V ) ! 1(U, V ) !
(3.1)
0(X, U) ! 0(X, V ) ! 0(U, V )
87
where the exactness at 0(X, V ) is the exactness in the sense of pointed sets,
and at 0(X, U) we observe that there is an action of the group 1(X, U)
on the set 0(X, U) and the exactness is in the usual sense. The exactness
everywhere else is as diagram of groups.
We now assume that X is the localization of a smooth k-scheme at a
point x. We still denote by x the close point in X. For any flag F: Z2
Z1 X of closed reduced subschemes, with Zi of codimension at least i,
we set Ui = X - Zi so that we get a corresponding flag of open subschemes
U1 U2 X. The set F of such flags is ordered by increasing inclusion (of
closed subschemes). Given a flag as above and applying the above observation
with U = U1 and V = ; we get an exact sequence:
. .!. 1(X, U1) ! 1(X) ! 1(U1) ! 0(X, U1) ! 0(X) ! 0(U1)
By the corollary above, applied to X, to S = {x}, and to the closed subset
Z1, we see that must be X itself and thus that the maps (for any n)
n(X, U1) ! n(X)
are trivial. We thus get a short exact sequence
1 ! 1(X) ! 1(U1) ! 0(X, U1) ! * (3.2)
and a map of pointed sets 0(X) ! 0(U1) which has trivial kernel.
Passing to the right filtering colimit on flags we get a short exact sequence
1 ! 1(X) ! 1(F ) ! colimF 0(X, U1) ! * (3.3)
and a pointed map with trivial kernel 0(X) ! 0(F ), where we denote by
F the field of functions of X. But now we observe that B being 0-connected
we have 0(F ) = *, and thus 0(X) = *.
To understand a bit further the short exact sequence (3.3) we now consider
for each flag F as above the part of the exact sequence obtained above for
the flag of open subschemes U1 U2 X:
! 0(X, U2) ! 0(X, U1) ! 0(U2, U1) (3.4)
88
By the Corollary 3.6 applied to X, S = {x} and to the closed subset Z2 X,
we see that must be X and that there exists Z0 X of codimension 1,
containing Z such that
0(X, U2) ! 0(X, X - Z0)
is the trivial map. Define the flag F0 : Z02 Z01 X by setting Z02= Z2
and Z01= Z1 [ Z0 we see that the map
colimF 0(X, U2) ! colimF 0(X, U1)
is trivial. Thus we conclude that
colimF 0(X, U1) ! colimF 0(U2, U1) (3.5)
has trivial kernel. However using now the exact sequence involving the flags
of open subsets of the form ; U1 U2 we see that there is a natural
action of 1(F ) on colimF 0(U2, U1) which makes the map (3.5) 1(F )-
equivariant. As the source colimF 0(X, U1) is one orbit under 1(F ) by
(3.3), the equivariant map (3.5) which has trivial kernel must be injective.
We thus have proven that if k is an infinite field and X is a smooth local
k-scheme with function field F . The natural sequence:
1 ! 1(X) ! 1(F ) ) colimF 0(U2, U1)
(the double arrow refereing to an action) is exact.
An interesting example is the case where X is the localization at a point
x of codimension 1. The set colimF 0(U2, U1) reduces to the 1(F )-set
0(X, X - {x}) because there is only one non-empty closed subset of codi-
mension > 0, the closed point itself. Moreover by the exact sequence (3.2)
shows that the action of 1(F ) on 0(Y, U - {y}) is transitive and the latter
set can be identified with the quotient 1(F )= 1(X); in that case we simply
denote this set by H1y(X; 1).
We observe that any 'etale morphism X0 ! X between smooth local
k-schemes induces a morphism of corresponding associated exact sequences
1 ! 1(X) ! 1(F ) ) colimF 0(U2, U1)
# # #
1 ! 1(X0) ! 1(F 0) ) colimF0 0(U02, U01)
89
When X0 ! X runs over the set of localizations at points of codimension
one in X we get a 1(F )-equivariant map
colimF 0(U2, U1) ! y2X(1)H1y( 1)
Lemma 3.7 (compare [9, Lemma 1.2.1]) The above map is injective and its
image is the weak product, yielding a bijection:
colimF 0(U2, U1) ~= 0y2X(1)H1y( 1)
Corollary 3.8 Assume k is infinite.
1) let X be a smooth local k-scheme with function field F . Then the
natural sequence:
1 ! 1(X) ! 1(F ) ) 0y2X(1)H1y(X; 1)
is exact.
2) the Zariski sheaf associated with X 7! 1(X) is a sheaf in the Nisnevich
1
topology and coincides with ssA1(B)(F ), which is thus unramified.
Proof. 1) is clear. Let's prove the 2). Let's denote by G the sheaf
( 1)Zar. Observe that for X local G(X) = 1(X). 1) implies that for any
k-smooth X irreducible with function field F the natural sequence:
1 ! G(X) ! G(F ) ) 0y2X(1)H1y(X; G)
is exact.
For X of dimension 1 with closed point y, the exact sequence 3.3 yields
a bijection H1y(X; 1) = H1y(X; G) = H1Nis(X, X - {y}; ss1(B)).
If V ! X is an 'etale morphism between local k-smooth schemes of di-
mension 1, with closed points y0 and y respectfully, and with same residue
fields ~(y) = ~(y0), the map
H1Nis(X, X - {y}; ss1(B)) ! H1Nis(V, V - {y0}; ss1(B)) (3.6)
is thus bijective.
It follows that the correspondence X 7! 0y2X(1)H1y(X; G) is a sheaf in the
Nisnevich topology on S"mk.
90
Using our above exact sequence this implies easily that X 7! G(X) is
a sheaf in the Nisnevich topology. The same exact sequence applied to the
henselization X of a k-smooth local scheme implies that the obvious mor-
1 A1
phism G(X) ! ssA1(B)(X) is a bijection. Thus the morphism G ! ss1 (B) is
an isomorphism of sheaves of groups in the Nisnevich topology.
The sheaf ss1(B) is strongly A1-invariant.
We now want to use the results of Section 1.2 to prove that G = ss1(B) is
strongly A1-invariant.
We still denote by G the Nisnevich sheaf ss1(B). By the previous corol-
1
lary, for any smooth local k-scheme X, one has G(X) = ssA1(B)(X) = 1(X).
In view of Theorem 1.26 the following result implies Theorem 3.1 over an
infinite field. Theorem A.7 deduce Theorem 3.1 over any finite field.
Theorem 3.9 Assume k is infinite. The unramified sheaf of groups G sat-
isfies the Axioms (A2'), (A5) and (A6) of Theorem 1.26. In particular G
is strongly A1-invariant.
Proof. We first prove Axiom (A5). Axiom (A5) (i) follows at once
from the fact proven above that (3.6) is a bijection. From that fact we see
that
1 ! G(X) ! G(F ) ) 0y2X(1)H1y(X; G)
defines on the category of smooth k-schemes of dimension 1 a short exact
sequence of Zariski and Nisnevich sheaves. As the right hand side is flasque
in the Nisnevich topology, we get for any smooth k-scheme V of dimension
1 a bijection
H1Zar(V ; G) = H1Nis(V ; G) = G(F)\ 0y2X(1)H1y(X; G)
For X a smooth local k-scheme of dimension 2 with closed point z and
V = X - {z} (which is of dimension 1), we get H1Nis(V ; G) = H2z(X; G).
Proceeding as in the proof of Lemma 1.24 we get Axiom (A5) (ii).
91
Now we prove Axiom (A2'). We recall from Lemma 3.7 that the map
colimF 0(U2, U1) ~= 0y2X(1)H1y(X; G)
is a bijection for any smooth k-scheme X.
Let z 2 X(2). Denote by Xz the localization of X at z and by Vz = Xz -
{z}. We have just proven that H1Zar(Vz; G) = H1Nis(Vz; G) = H2z(X; G). The
middle term is also equal to 0(Vz) = [(Vz)+ , B]Ho(k)because B is connected
with ss1(B) = G and Vz is smooth of dimension 1.
Now for a fixed flag F in X, by definition, the composition 0(U2, U1) !
H2z(X; G) is trivial if z 2 U2 and is the composition of the map 0(U2, U1) !
0(U2) and of the map 0(U2) ! 0(Vz) = H2z(X; G). Thus given an element
of 0y2X(1)H1y(X; G) which comes from 0(U2, U1), its boundary to H2z(X; G)
at points z of codimension 2 are trivial except maybe for those z not in U2:
but there are only finitely many of those, which establishes Axiom (A2').
We now prove Axiom (A6). Using the Lemma 3.10 below, we see by that
for any field F 2 Fk, the map [ ((A1F)+ ), B]Ho(k)! [ ((A1F)+ ), B(G))]Ho(k)=
G(A1F) is onto. As B is A1-local, [ ((A1F)+ ), B]Ho(k)= [ (SpecF+ ), B]Ho(k)=
G(F ) and this shows that the map G(F ) ! G(A1F) is onto. Thus it is an
isomorphism as any F -rational point of A1Fprovides a left inverse. By part
2) of Lemma 1.16 this implies that G is A1-invariant.
By 2) of Lemma 3.10 we see that for any (essentially) smooth k-scheme
X of dimension 1, the map [(A1X)+, B]Ho(k) ! [(A1X)+ , B(G)]Ho(k) =
H1Nis(A1X; G) is onto. As B is 0-connected and A1-local, this shows that
if moreover X is a local scheme H1Nis(A1X; G) = *.
As we know that G satisfies (A5), Lemma 1.24 implies that H1Zar(A1X; G) =
*. By Remark 1.22 we conclude that C*(A1X; G) is exact, the axiom (A6) is
proven, and the Theorem as well.
Lemma 3.10 1) For any smooth k-scheme X of dimension 1 the map
HomHs,o(k)( (X+ ), B) ! HomHs,o(k)( (X+ ), B(G)) = G(X)
is surjective.
2) For any smooth k-scheme X of dimension 2 the map
HomHs,o(k)(X+ , B) ! HomHs,o(k)(X+ , B(G)) = H1Nis(X; G)
92
is surjective and injective if dim(X) 1.
Proof. This is proven using the Postnikov tower {P n(B)}n2N of B, see
[39] for instance, together with standard obstruction theory, see [36, Ap-
pendix B].
Gm -loop spaces
Theorem 3.11 For any pointed A1-local space B which is 0-connected, so is
the function space RHom o(Gm , B) and for any integer n > 0, the canonical
morphism 1 1
ssAn(RHom o(Gm , B)) ! (ssAn(B))-1
is an isomorphism.
In particular, by induction on i 0, one gets an isomorphism for any
n > 0 1
[Sn ^ (Gm )^i, B]Ho(k)~= ssAn(B)-i(k)
Proof. The fact that RHom o(Gm , B) is A1-connected is proven as fol-
lows. We know from [32] that to show that a space Z is A1-connected, it
suffices to show that the sets [(Spec(F )+ ; Z) are trivial for any F 2 Fk. Base
an easy base change argument we may reduce to F = k. Gm having dimen-
sion one, we conclude from the Lemma 3.12 below and an obvious obstruction
theory argument using Lemma 3.10.
The canonical morphism of the statement is induced by the natural trans-
formation of presheaves of groups "evaluation on the n-th homotopy sheaves"
[Sn ^ Gm ^ (U+ ), B]Ho(k)! ssn(B)-1(U)
Observe that the associated sheaf to the presheaf on the left is exactly
1
ssAn(RHom o(Gm , B)).
Now by Lemma 1.31 and Corollary 3.3 both sheaves involved in the mor-
phism are strongly A1-invariant. To check it is an isomorphism it is sufficient
to check that it is an isomorphism on each F 2 Fk.
It is also clear that the morphism in degree n applied to R 1s(B) is the
morphism in degree n + 1 corresponding to B. Thus by induction, it is
sufficient to treat the case n = 1.
By an easy base change argument we may assume F = k is the base field.
Using again Lemma 3.10 we easily get the result from Lemma 3.12.
93
Lemma 3.12 Let G be a strongly A1-invariant sheaf of groups. Then H1(Gm ; G)
is trivial.
Proof. For k infinite, we use the results of section 1.2. For k finite we use
the results of the Appendix. We know from there that H1 is always computed
using the explicit complex C*(-; G). Thus we reduce to proving the fact
that the action of G(k(T )) on 0y2(Gm )(1)H1y(Gm ; G) is transitive. But this *
*fol-
lows at once from the fact that the action of G(k(T )) on 0y2(A1)(1)H1y(A1; G)
is transitive (because H1(A1; G) is trivial) and the fact that the epimor-
phism 0y2(Gm )(1)H1y(Gm ; G) is an obvious quotient of 0y2(A1)(1)H1y(A1; G) as
a G(k(T ))-set.
3.2 A1-derived category and Eilenberg-MacLane spaces
The derived category. Let us denote by Ab(k) the abelian category of
sheaves of abelian groups on SmS in the Nisnevich topology. Let C*(Ab(k))
be the category of chain complexes5 in Ab(k).
The derived category of Ab(k) is the category D(Ab(k)) obtained from
C*(Ab(k)) by inverting the class Qis of quasi-isomorphisms between chain
complexes. There are several ways to describe this category. The closest to
the intuition coming from standard homological algebra [14] is the following.
Definition 3.13 1) A morphism of chain complexes C* ! D* in C*(Ab(k))
is said to be a cofibration if it is a monomorphism. It is called a trivial
cofibration if it is furthermore a quasi-isomorphism.
2) A chain complex K* is said to be fibrant if for any trivial cofibration
i : C* ! D* and any morphism f : C* ! K*, there exists a morphism
g : D* ! K* such that g O i = f.
The following "fundamental lemma of homological algebra" seems to
be due to Joyal [20] in the more general context of chain complexes in a
Grothendieck abelian category [14]. One can find a proof in the case of
abelian category of sheaves in [18]. In fact in both cases one endows the
______________________________
5with differential of degree -1
94
category C*(Ab(k)) with a structure of model category and apply the homo-
topical of Quillen [43].
Lemma 3.14 1) For any chain complex D* 2 C*(Ab(k)) there exists a func-
torial trivial cofibration D* ! Df*to a fibrant complex.
2) A quasi-isomorphism between fibrant complexes is a homotopy equiva-
lence.
3) If D* is a fibrant chain complex, then for any chain complex C* the
natural map
ss(C*, D*) ! HomD(Ab(k))(C*, D*)
is an isomorphism.
Here we denote by ss(C*, D*) the group of homotopy classes of mor-
phisms of chain complexes in the usual sense. Thus to compute the group
HomD(Ab(k))(C*, D*) for any chain complexes C* and D*, one just chooses
a quasi-isomorphism D* ! Df*to a fibrant complex (also called a fibrant
resolution) and then one uses the chain of isomorphisms
ss(C*, Df*) ~=HomD(Ab(k))(C*, Df*) ~=HomD(Ab(k))(C*, D*)
The main use we will make of this property is a "concrete" description of in-
ternal derived Hom-complex RHom__(C*, D*): it is given by the naive internal
Hom-complex Hom__(C*, Df*), for C* a chain complex which sections on any
smooth k-scheme are torsion free abelian groups (to simplify). Indeed, it is
clear that Hom__(C*, Df*) is fibrant; using part 2 of the above Lemma and and
obvious adjunction formula for homotopies of morphisms of chain complexes
we get that this functor D(Ab(k)) ! D(Ab(k)), D* 7! Hom__(C*, Df*) is the
right adjoint to the functor D(Ab(k)) ! D(Ab(k)), B* 7! B* C*.
The A1-derived category. The following definition was mentioned in
[32, Remark 9] and is directly inspired from [39, 49]:
Definition 3.15 1) A chain complex D* 2 C*(Ab(k)) is called A1-local if
and only if for any C* 2 C*(Ab(k)), the projection C* Z(A1) ! C* induces
a bijection :
HomD(Ab(k))(C*, D*) ! HomD(Ab(k))(C* Z(A1), D*)
95
We will denote by DA1-loc(Ab(k)) D(Ab(k)) the full subcategory consisting
of A1-local complexes.
2) A morphism f : C* ! D* in C*(Ab(k)) is called an A1-quasi isomor-
phism if and only if for any A1-local chain complex E*, the morphism :
HomD(Ab(k))(D*, E*) ! HomD(Ab(k))(C*, E*)
is bijective. We will denote by A1 -Qis the class of A1-quasi isomorphisms.
3) The A1-derived category DA1(Ab(k)) is the category obtained by invert-
ing the all the A1-quasi isomorphisms.
All the relevant properties we need are consequences of the following:
Lemma 3.16 [39, 32] There exists a functor LA1 : C*(Ab(k)) ! C*(Ab(k)),
called the A1-localization functor, together with a natural transformation
` : Id ! LA1
such that for any chain complex C*, `C* : C* ! LA1(C*) is an A1-quasi
isomorphism whose target is an A1-local fibrant chain complex.
It is standard to deduce:
Corollary 3.17 The functor LA1 : C*(Ab(k)) ! C*(Ab(k)) induces a func-
tor
D(Ab(k)) ! DA1-loc(Ab(k))
which is left adjoint to the inclusion DA1-loc(Ab(k)) D(Ab(k)), and which
induces an equivalence of categories
DA1(Ab(k)) ! DA1-loc(Ab(k))
Proof of Lemma 3.16. We proceed as in [32]. We fix once for all a
functorial fibrant resolution C* ! Cf*. Let C* be a chain complex. We let
L(1)A1(C*) be the cone in C*(Ab(k)) of the obvious morphism
ev1 : Hom__(Zo(A1 ), Cf*) ! Cf*
96
We let C* ! L(1)A1(C*) denote the obvious morphism. Define by induction
on n 0, L(n)A1:= L(1)A1O L(n-1)A1. We have natural morphisms, for any chain
complex C*, L(n-1)A1(C*) ! L(n)A1(C*) and we set
L1A1(C*) = colimn2NL(n)A1(C*)
As in [32, Theorem 4.2.1] we have:
Proposition 3.18 For any chain complex C* the complex L1A1(C*) is A1 -
local and the morphism
C* ! L1A1(C*)
is an A1-quasi isomorphism.
This proves Lemma 3.16. In the sequel we set LA1(C*) := L1A1(C*)f: this
is the A1-localization of C*.
Remark 3.19 It should be noted that we have used implicitly the fact that
we are working with the Nisnevich topology, as well as the B.G.-property from
[39]: for a general topology on a site together with an interval in the sense
of [39], the analogue localization functor would require more "iterations",
indexed by some well chosen big enough ordinal number.
The (analogue of the) stable A1-connectivity theorem of [32] in D(Ab(k))
is the following:
Theorem 3.20 Let C* be a (-1)-connected chain complex. Then its A1-
localization LA1(C*) is still (-1)-connected.
The proof is exactly the same as the case of S1-spectra treated in [32].
Following the same procedure as in loc. cit., this implies that for an A1-local
chain complex C* each of its truncations o n (C*) is still A1-local and thus
each of its homology sheaves are automatically strictly A1-invariant. This
endows the triangulated category D(Ab(k)) with a natural non degenerated t-
structure [6] analogous to the homotopy t-structure of Voevodsky on DM(k).
The heart of that t-structure on D(Ab(k)) is precisely the category AbA1(k)
of strictly A1-invariant sheaves.
An easy consequence is:
97
Corollary 3.21 The category AbA1(k) of strictly A1-invariant sheaves is
abelian, and the inclusion functor AbA1(k) Ab(k) is exact.
Chain complexes and Eilenberg-MacLane spaces. Recall from [39],
that for any simplicial sheaf of sets X we denote by C*(X ) the (normalized)
chain complex in C*(Ab(k)) associated to the free simplicial sheaf of abelian
groups Z(X ) on X . This construction defines a functor
C* : opShvNis(Smk) ! C*(Ab(k))
which is well known (see [39, 25] for instance) to have a right adjoint
K : C*(Ab(k)) ! opShvNis(Smk)
called the Eilenberg-MacLane space functor.
For an abelian sheaf M 2 Ab(k) and an integer n we define the pointed
simplicial sheaf K(M, n) (see [39, page 56]) by applying K to the shifted
complex M[n], of the complex M placed in degree 0. If n < 0, the space
K(M, n) is a point. If n 0 then K(M, n) has only one non-trivial homotopy
sheaf which is the n-th and which is canonically isomorphic to M. More
generally, for a chain complex C*, the space KC* has for n-th homotopy
sheaf 0 for n < 0, and the n-th homology sheaf Hn(C*) for n 0.
It is clear that C* : opShvNis(Smk) ! C*(Ab(k)) sends simplicial weak
equivalences to quasi-isomorphisms and K : C*(Ab(k)) ! opShvNis(Smk)
maps quasi-isomorphisms to simplicial weak equivalences. If C* is fibrant, it
follows that K(C*) is simplicially fibrant. Thus the two functors induce a
pair of adjoint functors
C* : Hs(k) ! D(Ab(k))
and
K : D(Ab(k)) ! Hs(k)
As a consequence it is clear that if C* is an A1-local complex, the space
K(C*) is an A1-local space. Thus C* : Hs(k) ! D(Ab(k)) maps A1-weak
equivalences to A1-quasi isomorphisms and induces a functor
1
CA* : H(k) ! DA1(Ab(k))
which in concrete terms, maps a space X to the A1-localization of C*(X ).
1 1
We denote the latter by CA*(X ) and call it the A -chain complex of X .
98
1
The functor CA* : H(k) ! DA1(Ab(k)) admits as right adjoint the functor
1
KA : DA1(Ab(k)) ! H(k) induced by C* 7! K(LA1(C*)). We observe that
for an A1-local complex C*, the space K(C*) is automatically A1-local and
1
thus simplicially equivalent to the space KA (C*).
We will need the following:
Proposition 3.22 Let C* be a 0-connected chain complex in C*(Ab(k)).
Then the following conditions are equivalent:
(i) the space K(C*) is A1-local.
(ii) the chain complex C* is A1-local.
1)
Proof. For each complex C* we simply denote by (C*)(A the function
1)
complex Hom__(Zo(A1), Cf*). And we let (C*)(A 0denote the non negative part
1) 1 (A1) 1
of (C*)(A . It is clear that the tautological A -homotopy (C*) Z(A ) !
1) 1
(C*)(A between the Identity and the 0-morphism, induces an A -homotopy
1) (A1) (A1)
(C*)(A 0 Z(A1) ! (C*) 0 as well. Thus (C*) 0 is A1-contractible. We
1)
consider the morphism of "evaluation at one" (C*)(A 0! C*. And we set
1) (n)
UA1(C*) := cone((C*)(A 0! C*). For each n > 0 we let UA1 denote the
n-iteration of that functor. We then denote by U1A1(C*) the colimit of the
diagram
C* ! UA1(C*) ! . .!.U(n)A1(C*) ! . . .
Some observations:
(1) By the very construction, for any n 1, there is a canonical morphism
U(n)A1(C*) ! L(n)A1(C*)
which induces an isomorphism on each homology sheaves in dimension 1.
When C* is 0-connected, it is exactly the truncation L(n)A1(C*) 1: this is one
the main point here!
(2) each morphism U(n)A1(C*) ! U(n+1)A1(C*) is an A1-quasi-isomorphism
1)
because (C*)(A 0was shown above to be A1-contractible. Moreover:
Lemma 3.23 For any C* the morphism of simplicial sheaves
K(U(n)A1(C*)) ! K(U(n+1)A1(C*))
99
is an A1-weak equivalence of spaces.
As a consequence
K(C*) ! K(U(1)A1(C*))
is an A1-weak equivalence of spaces.
1)
Proof. Indeed, this is a principal K((C*)(A 0)-principal fibration by
construction. Thus K(U(n+1)A1(C*)) is simplicially weakly equivalent to the
Borel construction of K(U(n)A1(C*)) with respect to the action of the group
1)
K((C*)(A 0). But now the Borel construction
1) (n)
E(K((C*)(A 0)) xK((C*)(A1)K(UA1 (C*))
0 )
1) (n)
is filtered by the skeleton of E(K((C*)(A . The first filtration is K(UA1 (C*))
1) (n)
and the others are of the form (K((C*)(A 0))^i ^ Si ^ (K(UA1 (C*))+ ) with
i > 0 which is thus A1-weakly contractible.
If C* is 0-connected, by property (1) the colimit U(1)A1(C*) ! LA1(C*)
of the morphisms U(n)A1(C*) ! L(n)A1(C*) is isomorphic in D(Ab(k)) to the
truncation
LA1(C*) 1 ! LA1(C*)
By the connectivity Theorem 3.20, the previous morphism is a quasi-isomorphism.
Recall that the space K(LA1(C*)) is A1-local. The following obvious corollary
of what we have done easily implies the Proposition 3.22.
Corollary 3.24 For any 0-connected C*, the morphism of simplicial sheaves
K(C*) ! K(UA1(C*)) ~=K(LA1(C*) 1) ~=K(LA1(C*))
is an A1-weak equivalence of 0-connected spaces to an A1-local space. It is
thus isomorphic to the A1-localization of the source.
We now deduce the following important property, which is one of the
main tool in this paper:
Theorem 3.25 A strongly A1-invariant sheaf of abelian groups is strictly
A1-invariant.
100
Proof. Indeed, M is strongly A1-invariant means that K(M, 1) is A1-
local. By the Proposition 3.22, it follows that the chain complex M[1] is
A1-local, which means that M is strictly A1-invariant.
Remark 3.26 The previous result can be used to simplify some proofs in
[48]. We may also suppress in most results the assumption of perfectness of
the base field.
Remark 3.27 It could be possible to prove the Theorem without mention-
ing spaces and working only with complexes by introducing the A1-homotopy
category of non-negative chain complexes which is equivalent to that of sim-
plicial abelian sheaves by the Dold-Kan correspondence [25]. The proof would
be exactly along the same lines. However, our method of proof yields slightly
more: for instance Corollary 3.24 will be used below in a non trivial way.
The following consequence is one of our main structural result:
Theorem 3.28 Let X be a pointed 0-connected space. Then X is A1-local if
and only if ss1(X ) is strongly A1-invariant and if, for n 2, ssn(X ) is stri*
*ctly
A1-invariant.
This clearly follows from Corollary 3.3 and Theorem 3.25.
3.3 The Hurewicz theorem and some of its consequences
The following definition was made in [32]:
1
Definition 3.29 Let X be a space and n 2 Z be an integer. We let HAn(X )
1
denote the n-th homology sheaf of the A1-chain complex CA*(X ) of X , and
call it the n-th homology sheaf of X .
1 A1 A1
If X is pointed, we set H"An(X ) = Ker(Hn (X ) ! Hn (Spec(k)) and call
1
it the n-th reduced homology sheaf of X . As HAn(Spec(k)) = 0 for n 6= 0 and
Z for n = 0, this means that as graded abelian sheaves
1 A1
HA*(X ) = Z "H* (X )
Remark 3.30 We observe that the A1-localization functor commutes to the
suspension in D(Ab(k)). As an immediate consequence, we see that there
exists a canonical suspension isomorphism for any pointed space X and any
integer n 2 Z:
"HA1n(X ) ~=H"A1n+1( (X ))
101
Using the A1-connectivity Theorem 3.20 and its consequences, we get
1
Corollary 3.31 The A1-homology sheaves HAn(X ) of a space X vanish for
n < 0 and are are strictly A1-invariant sheaves for n 0.
Remark 3.32 We conjectured in [32] that this result should still hold over
a general base; J. Ayoub produced in [1] a counter-example over a base of
dimension 2. The case of a base dimension 1 is still open.
Remark 3.33 In classical topology, one easily computes the whole homol-
ogy of the sphere Sn: Hi(Sn) = 0 for i > n. In the A1-homotopy world,
the analogue of this vanishing in big dimensions is unfortunately highly non-
trivial and unknown. It is natural to make the:
Conjecture 3.34 Let X be a smooth quasi-projective k-scheme of dimension
1
d. Then HAn(X) = 0 for n > 2d and in fact if moreover X is affine then
1
HAn(X) = 0 for n > d.
That would implies that the A1-homology of (P1)^n vanishes in degrees
> 2n. This is in fact a stronger version of the vanishing conjecture of
Beilinson-Soul'e. It was also formulated in [32].
Computations of higher A1-homotopy or A1-homology sheaves seem rather
difficult in general. In fact, given a space, we now "understand" its first non-
trivial A1-homotopy sheaf, but we do not know at the moment any "non-
trivial" example where one can compute the next non-trivial A1-homotopy
sheaf without using deep results like Milnor or Bloch-Kato conjectures.
Using the adjunction between the functors C* and K it is clear that for
a fixed pointed space X the adjunction morphism
X ! K(C*(X ))
induces a morphism, for each n 2 Z
1 A1
ssAn(X ) ! Hn (X )
which we call the Hurewicz morphism.
The following two theorems form the weak form of our Hurewicz theorem:
102
Theorem 3.35 Let X be a pointed 0-A1-connected space. Then the Hurewicz
morphism 1 1
ssA1(X ) ! HA1(X )
1 1
is the initial morphism from the sheaf of groups ssA1(X ) to a strictly A -
invariant sheaf (of abelian groups). This means that given a strictly A1-
invariant sheaf M and a morphism of sheaves of groups
1
ssA1(X ) ! M
1 A1
it factors uniquely through ssA1(X ) ! H1 (X ).
Proof. Let M be a strictly A1-invariant sheaf. The group of mor-
1
phisms of sheaves HomGr(ssA1(X ), M) is equal to the group of simplicial
homotopy classes HomHs(k)(LA1(X ), K(M, 1)) which, because K(M, 1) is
A1-local, is also HomH(k)(X , K(M, 1)); by our above adjunction, this is also
1 A1
HomDA1(Ab(k))(CA*(X ), K(M, 1)), and the latter is exactly HomAb(k)(H1 (X ), M)
1
because CA*(X ) is 0-connected.
Remark 3.36 It is not yet known, though expected, that the Hurewicz
morphism is an epimorphism in degree one and that its kernel is always the
commutator subgroup.
Theorem 3.37 Let n > 1 be an integer and let X be a pointed (n - 1)-A1-
connected space. Then 1
HAi(X ) = 0
for each i 2 {0, . .,.n - 1} and the Hurewicz morphism
1 A1
ssAn(X ) ! Hn (X )
is an isomorphism between strictly A1-invariant sheaves.
Proof. Apply the same argument as in the previous theorem, using
K(M, n), and the fact from Theorem 3.28 that the A1-homotopy sheaves
1 1
ssAn(X ) are strictly A -invariant.
The following immediate consequence is the unstable A1-connectivity the-
orem:
103
Theorem 3.38 Let n > 0 be an integer and let X be a pointed (n - 1)-
connected space. Then its A1-localization is still (n - 1)-connected.
For any sheaf of sets F on Smk, let us denote by ZA1(F ) the strictly
A1-invariant sheaf 1
ZA1(F ) := HA0(F )
where F is considered as a space in the right hand side. This strictly A1-
invariant sheaf is the free strictly A1-invariant sheaf generated by in the
following sense: for any strictly A1-invariant sheaf M the natural map
HomAb(k)(ZA1(F ), M) ! HomShv(Smk)(F, M)
is clearly a bijection.
1
If F is pointed, we denote by ZA1,o(F ) the reduced homology sheaf "HA0(F ).
Our previous results and proofs immediately yield:
Corollary 3.39 For any integer n 2 and any pointed sheaf of sets F the
canonical morphism
1 n A1 n
ssAn( (F )) ! Hn ( (F )) ~=ZA1,o(F )
is an isomorphism.
The last isomorphism is the suspension isomorphism from Remark 3.30.
Now by Theorem 3.25, the free strictly A1-invariant sheaf generated by a
(pointed) sheaf F is the same sheaf as the free strongly A1-invariant sheaf of
abelian groups generated by the same (pointed) sheaf. Our main computa-
tion in Theorem 2.37 thus yields the following analogue of Theorem 1 which
was announced as Theorem 19 in the introduction:
Theorem 3.40 For n 2 one has canonical isomorphisms of strictly A1-
invariant sheaves
1 n A1 1 ^n ^n MW
ssAn-1(A - {0}) ~=ssn ((P ) ) ~=ZA1,o((Gm ) ) ~=K_n
104
1 2
Remark 3.41 Observe that the previous computation of ssA1(A - {0}) re-
quires a slightly more subtle argument, as it concerns the A1-fundamental
group. The morphism SL2 ! A2 - {0} being an A1-weak equivalence, we
1 2 1
know a priori that ssA1(A - {0}) is a strongly A -invariant sheaf of abelian
groups, as is the A1-fundamental group of any group (or h-group) as usual.
The free strongly A1-invariant sheaf of groups on Gm ^ Gm is commutative
and it is thus K_MW2.
1 1
We postpone the computation of ssA1(P ) to section ??.
Remark 3.42 For any n 0 we let Sn denote (S1)^n. We observe that
An - {0} is canonically isomorphic in Ho(k) to Sn-1 ^ (Gm )^n and (P1)^n is
canonically isomorphic to Sn ^ (Gm )^n, see [39, x Spheres, suspensions and
Thom spaces p. 110]. It is thus natural for any n 0 and any i 0 to study
the "sphere" of the form Sn ^ (Gm )^i.
The Hurewicz Theorem implies that it is at least n - 1 connected and if
n 2, provides a canonical isomorphism
1 n ^i MW
ssAn(S ^ (Gm ) ) ~=K_i
1 n
for i 1 and ssAn(S ) = Z for i = 0 (and n 1).
In case n = 0 our sphere is just a smash-power of Gm which is itself
A1-invariant.
For n = 1 the question is harder and we only get, by the Hurewicz Theo-
1 1 ^i MW
rem, a canonical epimorphism ssA1(S ^ (Gm ) ) i K_i . This epimorphism
1 1
has a non trivial kernel for i = 1 (see the computation of ssA1(P ) in Section
??). We have just observed in the previous remark that this epimorphism is
1 1 ^i
an isomorphism for i = 2. We don't know ssA1(S ^ (Gm ) ) for i > 2.
Corollary 3.43 Let (n, i) 2 N2 and (m, j) 2 N2 be pairs of integers. For
n 2 we have a canonical isomorphism:
8
>> 0 ifm < n
< KMW (k) ifm = n and i > 0
HomHo(k)(Sm ^ (Gm )^j, Sn ^ (Gm )^i) ~= i-j
>> 0 ifm = n ,j > 0 and i = 0
: Z ifm = n and j = i = 0
Proof. This follows immediately from our previous computation, from
Theorem 3.11 and Remark ?? which clearly implies that the product induces
105
isomorphisms (K_MWn )-1 ~=K_MWn-1.
A1-fibration sequences and applications In this paragraph we give
some natural consequences of the (weak) Hurewicz Theorem and of our struc-
ture result for A1-homotopy sheaves Theorem 3.28.
We first recall some terminology.
Definition 3.44 1) A simplicial fibration sequence between spaces
! X ! Y
with Y pointed, is a diagram such that the composition of the two morphisms
is the trivial one and such that the induced morphism from to the simplicial
homotopy fiber of X ! Y is a simplicial weak equivalence.
2) An A1-fibration sequence between spaces
! X ! Y
with Y pointed, is a diagram such that the composition of the two morphisms
is the trivial one and such that the induced diagram between A1-localizations
LA1( ) ! LA1(X ) ! LA1(Y)
is a simplicial fibration sequence.
A basic problem is that it is not true in general that a simplicial fibration
sequence is an A1-fibration sequence. For instance, let X be a fibrant pointed
space, denote by P(X ) the pointed space Hom__o( 1, X ) of pointed paths
1 ! X in X so that we have a simplicial fibration sequence
1(X ) ! P(X ) ! X
whose fiber 1(X ) := Hom__o(S1, X ) is the simplicial loop space of X (with
S1 = 1=@ 1 is the simplicial circle). The following observation is an im-
mediate consequence of our definitions, the fact that if X is A1-fibrant so is
1(X ), and the fact that an A1-weak equivalence between A1-local space is
a simplicial weak equivalence:
106
Lemma 3.45 Let X be a simplicially fibrant pointed space. The paths sim-
plicial fibration sequence 1(X ) ! P(X ) ! X above is an A1-fibration se-
quence if and only if the canonical morphism
LA1( 1(X )) ! 1(LA1(X ))
is a simplicial weak equivalence.
We now observe:
Theorem 3.46 Let X be a (simplicially) fibrant pointed space. Then the
canonical morphism
LA1( 1(X )) ! 1(LA1(X ))
1 1
is a simplicial weak equivalence if and only if the sheaf of groups ssA0( (X )*
*) =
ss0(LA1( 1(X ))) is strongly A1-invariant.
Proof. From Theorem 3.1 the condition is clearly necessary. To prove
the converse we may clearly assume X is 0-connected (and fibrant). In that
case the inclusion of X (0) X of the sub-space consisting of "simplices
whose vertices are the base point" is a simplicial weak equivalence: use [25]
and stalks to check it. Using the Kan model G(X (0)) for the simplicial loop
space on a pointed 0-reduced Kan simplicial set (loc. cit. for instance) one
obtain a canonical morphism X (0)! B(G(X (0))) which is clearly also a sim-
plicial weak equivalence (by checking on stalks). Thus this defines in the
simplicial homotopy category Hs,o(k) a canonical pointed isomorphism be-
tween X and B(G(X (0))) and in particular a canonical pointed isomorphism
between 1(X ) and G(X (0)). Now we observe that by Lemma 3.47 below,
we may choose LA1 so that LA1 maps groups to groups. Thus G(X (0)) !
LA1(G(X (0))) is an A1-weak equivalence between simplicial sheaves of groups.
By Lemma 3.48 we see that
X ~= B(G(X (0))) ! B(LA1(G(X (0))))
is always an A1-weak equivalence. Now assuming that ss0(LA1( 1(X ))) ~=
ss1(B(LA1(G(X (0))))) is strongly A1-invariant, and the higher homotopy sheaves
of B(LA1(G(X (0)))) are strictly A1-invariant, we see using Theorem 3.28 that
the space B(LA1(G(X (0)))) is A1-local. It is thus the A1-localization of X .
107
Recall from [39] that an A1-resolution functor is a pair (Ex, `) consisting
of a functor Ex : opShv(Smk) ! opShv(Smk) and a natural transforma-
tion ` : Id ! Ex such that for any space X , Ex(X ) is fibrant and A1-local,
and `(X ) : X ! Ex(X ) is an A1-weak equivalence.
Lemma 3.47 [39] There exists an A1-resolution functor (Ex, `) which com-
mutes to any finite products.
Proof. Combine [39, Theorem 1.66 page 69] with the construction of the
explicit I-resolution functor given page 92 of loc. cit..
Recall that a principal fibration G - X ! Y with simplicial group G is
the same thing as a G-torsor over Y.
Lemma 3.48 Let
G - X ! Y
# # #
G0 - X 0 ! Y0
be a commutative diagram of spaces in which the horizontal lines are principal
fibrations with simplicial groups G and G0. Assume the vertical morphism
(of simplicial groups) G ! G0 and the morphism of spaces X ! X 0are both
A1-weak equivalence. Then
Y ! Y0
is an A1-weak equivalence.
Proof. Given a simplicial sheaf of groups G we use the model E(G) of
simplicially contractible space on which G acts freely given by the diagonal
of the simplicial space n 7! E(Gn) where E(G) for a simplicially constant
sheaf of group is the usual model (see [39, page 128] for instance). We may
as well consider it a the diagonal of the simplicial space m 7! Gm+1 , the
action of G being the diagonal one. For any G-space X we introduce the
Borel construction
EG xG X
where G acts diagonally on E(G) x X . If the action of G is free on X ,
the morphism EG xG X ! G \X is a simplicial weak equivalence. Thus
in the statement we may replace Y by EG xG X and Y0 by EG0xG0 X 0
respectfully. Now from our recollection above, EG xG X is the diagonal
108
space of the simplicial space m 7! Gm+1 xG X ; it thus simplicially equivalent
to its homotopy colimit (see [7] and [39, page 54]). The Lemma thus follows
from Lemma 2.12 page 73 of loc. cit. and the fact that for any m the
morphism
Gm+1 xG X ! (G0)m+1 xG0 X 0
are A1-weak-equivalences. This is easy to prove by observing that the G-
space Gm+1 is functorially G-isomorph to G x (Gm ) with action given on the
left factor only. Thus the spaces Gm+1 xG X are separately (not taking the
simplicial structure into account) isomorph to Gm x X .
Definition 3.49 1) A homotopy principal G-fibration
G - X ! Y
with simplicial group G consists of a G-space X and a G-equivariant mor-
phism X ! Y (with trivial action on Y) such that the obvious morphism
EG xG X ! Y
is a simplicial weak equivalence.
2) Let G - X ! Y be a (homotopy) G-principal fibration with struc-
ture group G. We say that it is an A1-homotopy G-principal fibration if the
diagram
LA1(X ) ! LA1(Y)
is a homotopy principal fibration with structure group LA1(G).
In the previous statement, we used an A1-localization functor which com-
mutes to finite product (such a functor exists by Lemma 3.47).
Theorem 3.50 Let G - X ! Y be a (homotopy) principal fibration with
1 1
structure group G such that ssA0(G) is strongly A -invariant. Then it is an
A1-homotopy G-principal fibration.
Proof. We contemplate the obvious commutative diagram of spaces:
G - X ! Y
|| " o " o
G - E(G) x X ! E(G) xG X
# # #
LA1(G) - E(LA1(G)) x LA1(X ) ! E(LA1(G)) xLA1(G)LA1(X )
109
where the upper vertical arrows are simplicial weak equivalences. By Lemma
3.48 the right bottom vertical arrow is an A1-weak equivalence. By the very
definition, to prove the claim we only have to show now that the obvious
morphism E(LA1(G)) xLA1(G)LA1(X ) ! LA1(Y) is a simplicial weak equiva-
lence.
As E(G) xG X ! E(LA1(G)) xLA1(G)LA1(X ) is an A1-weak equivalence,
we clearly only have to show that the space E(LA1(G)) xLA1(G)LA1(X ) is
A1-local. But it fits, by construction, into a simplicial fibration sequence of
the form
LA1(X ) ! E(LA1(G)) xLA1(G)LA1(X ) ! B(LA1(G))
1 1 1
As ssA0(G) is strongly A -invariant the 0-connected space B(LA1(G)) is A -
local by Theorem 3.28. This easily implies the claim using the Lemma ??
above.
Lemma 3.51 Let ! X ! Y be a simplicial fibration sequence with Y
pointed and 0-connected. If and Y are A1-connected, then so is X .
Proof. We use the commutative diagram of spaces
! X ! Y
# # #
1 A1 A1
A ! X ! Y
where the horizontal rows are both simplicial fibration sequences (we denote
1 1
here by ZA the right simplicially derived functor RHom__(A , Z), see [39]).
We must prove that the middle vertical arrow is a simplicial weak equiva-
lence knowing that both left and right vertical arrows are. But using stalks
we reduce easily to the corresponding case for simplicial sets, which is well-
known.
Example 3.52 1) For instance any SLn-torsors, n 2, satisfy the property
1
of the Theorem because ssA0(SLn) = *: this follows from the fact that over
a field F 2 Fk, any element of SLn(F ) is a product of elementary matrices,
1
which shows that over ssA0(SLn)(F ) = *. From [30] this implies the claim.
110
1
2) Any GLn-torsors, for n 1, satisfy this condition as well as ssA0(GLn) =
Gm is strictly A1-invariant. This equality follows from the previous state-
ment.
3) This is also the case for finite groups or abelian varieties: as these are
flasque as sheaves, the H1Nisis trivial.
4) In fact we do not know any example of smooth algebraic group G over
1 1
k whose ssA0 is strongly A -invariant.
Theorem 3.53 Let ! X ! Y be a simplicial fibration sequence with
1 1
Y pointed and 0-connected. Assume that the sheaf of groups ssA0( (Y)) =
ss0(LA1( 1(Y))) is strongly A1-invariant. Then ! X ! Y is also an
A1-fibration sequence.
Proof. This theorem is an easy reformulation of the previous one (using
a little bit its proof) by considering a simplicial group G with a simplicial
weak equivalence Y ~=B(G).
We observe that the assumptions of the Theorem are fulfilled if Y is
simplicially 1-connected, or if it is 0-connected and if ss1(Y) itself is stron*
*gly
A1-invariant. This follows from the following Lemma applied to 1(Y).
Lemma 3.54 Let X be a space. Assume its sheaf ss0(X ) is A1-invariant.
1
Then the morphism ss0(X ) ! ssA0(X ) = ss0(LA1(X )) is an isomorphism.
1
Proof. This Lemma follows from the fact that ss0(X ) ! ssA0(X ) is al-
ways an epimorphism [39, Corollary 3.22 page 94] and the fact that as a
space the A1-invariant sheaf ss0(X ) is A1-local. This produces a factorization
1
of the identity of ss0(X ) as ss0(X ) ! ssA0(X ) = ss0(LA1(X )) ! ss0(X ) which
clearly implies the result.
The relative A1-connectivity theorem.
Definition 3.55 A morphism of spaces X ! Y is said to be n-connected
for some integer n 0 if each stalk of that morphism (at any point of any
smooth k-scheme) is n-connected in the usual sense.
When the spaces are pointed and Y is 0-connected this is equivalent to the
fact that the simplicial homotopy fiber of the morphism is n-connected.
111
The relative A1-connectivity theorem refers to:
Theorem 3.56 Let f : X ! Y be a morphism with Y pointed and 0-
1 1 1
connected. Assume that the sheaf of groups ssA0( (Y)) = ss0(LA1( (Y)))
is strongly A1-invariant (for instance if Y is simplicially 1-connected, or if
ss1(Y) itself is strongly A1-invariant). Let n 1 be an integer and assume f
is (n - 1)-connected, then so is the morphism
LA1(X ) ! LA1(Y)
Proof. Let ! X be the homotopy fiber. By Theorem 3.53 above
the diagram LA1( ) ! LA1(X ) ! LA1(Y) is a simplicial fibration sequence.
Our connectivity assumption is that ssi( ) = 0 for i 2 {0, . .,.n - 1}. By
the unstable A1-connectivity Theorem 3.38, the space LA1( ) is also (n - 1)-
connected. Thus so is LA1(X ) ! LA1(Y).
The strong form of the Hurewicz theorem. This refers to the fol-
lowing classical improvement of the weak Hurewicz Theorem:
Theorem 3.57 Let n > 1 be an integer and let X be a pointed (n - 1)-A1-
1
connected space. Then HAi(X ) = 0 for each i 2 {0, . .,.n - 1}, the Hurewicz
1 A1
morphism ssAn(X ) ! Hn (X ) is an isomorphism, and moreover the Hurewicz
morphism 1 1
ssAn+1(X ) ! HAn+1(X )
is an epimorphism of sheaves.
Proof. We may assume X fibrant and A1-local. Consider the canonical
morphism X ! K(C*(X )) and let us denote by its simplicial homotopy
fiber. The classical Hurewicz Theorem for simplicial homotopy tells us that
is simplicially n-connected (just compute on the stalks).
Now as K(C*(X )) is 1-connected the Theorem 3.56 above tells us that
the morphism X = LA1(X ) ! LA1(K(C*(X ))) is still n-connected. But
as K(LA1(C*(X ))) ! LA1(K(C*(X ))) is a simplicial weak equivalence by
1
Corollary 3.24 we conclude that X ! K(CA*(X )) is n-connected, which
gives exactly the strong form of Hurewicz Theorem.
1
Remark 3.58 For n = 1 if one assumes that ssA1(X ) is abelian (thus strictly
A1-invariant) the Theorem remains true.
112
A stability result. Recall that for a fibrant space X and an integer n
the space P (n)(X ) denotes the n-th stage of the Postnikov tower for X [39,
page 55]. If X is pointed, we denote by X (n+1)! X the homotopy fiber
at the point of X ! P (n)(X ). The space X (n+1)is of course n-connected.
There exists by functoriality a canonical morphism X (n)! (LA1(X ))(n). As
the target is clearly A1-local, we thus get a canonical morphism of pointed
spaces
LA1(X (n)) ! (LA1(X ))(n)
Theorem 3.59 Let X be a pointed connected space. Assume n > 0 is an
integer such that the sheaf ss1(X ) is strongly A1-invariant and for each 1 <
i n, the sheaf ssi(X ) is strictly A1-invariant. Then for each i n + 1 the
above morphism LA1(X (i)) ! (LA1(X ))(i)is a simplicial weak equivalence.
We obtain immediately the following:
Corollary 3.60 Let X be a pointed connected space. Assume n > 0 is an
integer such that the sheaf ss1(X ) is strongly A1-invariant and for each 1 <
i n, the sheaf ssi(X ) is strictly A1-invariant. Then for i n the morphism
1
ssi(X ) ! ssAi(X ) = ssi(LA1(X ))
is an isomorphism and the morphism
1
ssn+1(X ) ! ssAn+1(X ) = ssn+1(LA1(X ))
is the universal morphism from ssn+1(X ) to a strictly A1-invariant sheaf.
Proof. We proceed by induction on n. Assume the statement of the
Theorem is proven for n - 1. We apply Theorem 3.53 to the simplicial
fibration sequence X (n+1)! X ! P (n)(X ); P (n)(X ) satisfies indeed the
assumptions. Thus we get a simplicial fibration sequence
LA1(X (n+1)) ! LA1(X ) ! LA1(P (n)(X ))
Then we observe that by induction and the Corollary 3.60 above that the
morphism P (n)(X ) ! P (n)(LA1(X )) is a simplicial weak equivalence. Thus
P (n)(X ) ~=LA1(P (n)(X )) ~=P (n)(LA1(X )). These two facts imply the claim.
The A1-simplicial suspension Theorem.
113
Theorem 3.61 Let X be a pointed space and let n 2 be an integer. If X
is (n - 1)-A1-connected space the canonical morphism
LA1(X ) ! 1(LA1( 1(X )))
is 2(n - 1)-(A1)-connected.
Proof. We first observe that the classical suspension Theorem implies
that for any simplicially (n - 1)-connected space Y the canonical morphism
Y ! 1( 1(Y))
is simplicially 2(n - 1)-connected. Thus the theorem follows from: We
apply this to the space Y = LA1(X ) itself, which is simplicially (n - 1)-
connected. Thus the morphism LA1(X ) ! 1( 1(LA1(X ))) is simplicially
2(n - 1)-connected. This implies in particular that the suspension mor-
1 1
phisms ssAi(X ) ! ssi+1( s(LA1(X ))) are isomorphisms for i 2(n - 1) and
an epimorphism for i = 2n - 1.
>From Theorem 3.59 and its corollary, this implies that 1(X ) ! LA1( 1(X ))
induces an isomorphism on ssifor i 2n-1 and that the morphism ss2n( 1(X )) !
ss2n(LA1( 1(X ))) is the universal morphism to a strictly A1-invariant sheaf.
1 1
Thus it follows formally that ssA2n-1(X ) ! ss2n(LA1( s(X ))) is a categorical
epimorphism in the category of strictly A1-invariant sheaves. As by Corollary
3.21 this category is an abelian category for which the inclusion into Ab(k) is
1 1
exact, it follows that the morphism ssA2n-1(X ) ! ss2n(LA1( s(X ))) is actually
an epimorphism of sheaves. Thus the morphism 1(X ) ! LA1( 1(X )) is a
(2n - 1)-simplicial weak-equivalence. The morphism
1( 1(X )) ! 1(LA1( (X )))
is thus a 2(n - 1)-simplicial weak-equivalence. The composition
LA1(X ) ! 1( 1(LA1(X ))) ! 1(LA1( (X )))
is thus also simplicially 2(n - 1)-connected.
1 n A1
4 A1-coverings, ssA1 (P ) and ss1 (SLn )
1
4.1 A1-coverings, universal A1-covering and ssA1
Definition 4.1 1) A simplicial covering Y ! X is a morphism of spaces
which has the unique right lifting property with respect to simplicially trivial
114
cofibrations. This means that given any commutative square of spaces
A ! Y
# #
B ! X
in which A ! B is an simplicially trivial cofibration, there exists one and
exactly one morphism B ! Y which let the whole diagram commutative.
2) An A1-covering Y ! X is a morphism of spaces which has the unique
right lifting property with respect to A1-trivial cofibrations6.
Lemma 4.2 A morphism Y ! X is a simplicial (resp A1-) covering if and
only if it has the unique right lifting property with respect to any simplicial
(resp A1-) weak equivalence.
Proof. It suffices to prove that coverings have the unique lifting property
with respect to weak-equivalences (both in the simplicial and in the A1-
structure). Pick up a commutative square as in the definition with A ! B
a weak-equivalence. Factor it as a trivial cofibration A ! C and a trivial
fibration C ! B. In this way we clearly reduce to the case ss : A ! B is a
trivial fibration. Uniqueness is clear as trivial fibrations are epimorphisms of
spaces. Let's prove the existence statement. For both structures the spaces
are cofibrant thus one gets a section i : B ! A which is of course a trivial
cofibration. Now we claim that f : A ! Y composed with i O ss : A ! A is
equal to f. This follows from the unique lifting property applied to i. Thus
f O i : B ! Y is a solution and we are done.
Remark 4.3 A morphism Y ! X in Smk, with X irreducible, is a sim-
plicial covering if and only if Y is a disjoint union of copies of X mapping
identically to X.
We will see below that Gm -torsor are examples of A1-coverings. It could
be the case that a morphism in Smk is an A1-covering if and only if it has
the right lifting property with respect to only the 0-sections morphisms of
the form U ! A1 x U, for U 2 Smk.
______________________________
6remember [39] that this means both a monomorphism and an A1-weak equivalence
115
The simplicial theory.
Lemma 4.4 If Y ! X is a simplicial covering for each x 2 X 2 Smk the
morphism of simplicial sets Yx ! Xx is a covering of simplicial sets.
Proof. For i 2 {0, . .,.n} we let as usual n,i n be the union of all the
faces of n but the i-th. The inclusion n,i n is then a simplicial equiva-
lence (of simplicial sets). Now for any U 2 Smk and any inclusion n,i n
as above, we apply the definition of simplicial covering to n,ix U n x U.
When U runs over the set of Nisnevich neighborhoods of x 2 X, this eas-
ily implies that Yx ! Xx has the right lifting property with respect to the
n,i n, proving our claim.
For any pointed simplicially connected space Z there exists a canonical
morphism in Hs,o(k) of the form Z ! BG, where G is the fundamental group
sheaf ss1(Z); this relies on the Postnikov tower [39] for instance. Using now
Prop. 1.15 p.130 of loc. cit. one gets a canonical isomorphism class Z"! Z
of G-torsor. Choosing one representative, we may point it by lifting the base
point of Z. Now this pointed G-torsor is canonical up to isomorphism. To
prove this we first observe that Z"is simplicially 1-connected. Now we claim
that any pointed simplicially 1-connected covering Z0 ! Z over Z is canon-
ically isomorphic to this one.
Indeed, one first observe that the composition Z0 ! Z ! BG ! BG
(where BG means a simplicially fibrant resolution of BG) is homotopically
trivial. This follows from the fact that Z0 is 1-connected.
Now let EG ! BG be the universal covering of BG (given by Prop. 1.15
p.130 of loc. cit.). Clearly this is also a simplicial fibration, thus EG is
simplicially fibrant. Thus we get the existence of a lifting Z0 ! EG. Now
clearly the commutative square
Z0 ! EG
# #
Z ! BG
Using the Lemma above, we see that this this square is cartesian on each
stalk (by the classical theory), thus cartesian. This proves precisely that
Z0 as a covering is isomorphic to Z". But then as a pointed covering, it is
116
canonically isomorphic to Z" ! Z because the automorphism group of the
pointed covering Z"! Z is clearly trivial.
Now given any pointed simplicial covering Z0 ! Z one may consider the
connected component of the base point Z0(0)of Z0. Clearly Z0(0)! Z is
still a pointed (simplicial) covering. Now the universal covering (constructed
above) of Z0(0)is clearly also the universal covering of Z. One thus get a
unique isomorphism from the pointed universal covering of Z to that of Z0(0).
The composition "Z! Z0is clearly the unique morphism of pointed coverings
(use stalks). Thus Z"! Z is the universal object in the category of pointed
coverings of Z.
The A1-theory.
We want to prove the analogue statement in the case of A1-coverings. We
observe that as any simplicially trivial cofibration is an A1-trivial cofibrati*
*on,
an A1-covering is in particular a simplicial covering.
Before that we first establish the following Lemma which will provide us
with our two basic examples of A1-coverings.
Lemma 4.5 1) A G-torsor Y ! X with G a strongly A1-invariant sheaf is
an A1-covering.
2) Any G-torsor Y ! X in the 'etale topology, with G a finite 'etale k-
group of order prime to the characteristic, is an A1-covering.
Proof. 1) Recall from [39, Prop. 1.15 p. 130] that the set H1(X ; G) of
isomorphism classes (denoted by P (X ; G) in loc. cit.) of G-torsors over a
space X is in one-to-one correspondence with [X , BG]Hs(k)(observe we used
the simplicial homotopy category). By the assumption on G, BG is A1-local.
Thus we get now a one-to-one correspondence H1(X ; G) ~=[X , BG]H(k). Now
let us choose a commutative square like in the definition, with the right
vertical morphism a G-torsor. This implies that the pull-back of this G-torsor
to B is trivial when restricted to A B. By the property just recalled, we
get that H1(B; G) ! H1(A; G) is a bijection, thus the G-torsor over B itself
is trivial. This fact proves the existence of a section s : B ! Y of Y ! X.
117
The composition s O (A B) : A ! Y may not be equal to the given
top morphism s0 : A ! Y in the square. But then there exists a morphism
g : A ! G with s = g.s0 (by one of the properties of torsors).
But as G is A1-invariant the restriction map G(B) ! G(A) is an isomor-
phism. Let "g: B ! G be the extension of g. Then clearly "g-1.s : B ! Y
is still a section of the torsor, but now moreover its restriction to A B is
equal to s0. We have proven the existence of an s : B ! Y which makes the
diagram commutative. The uniqueness follows from the previous reasoning
as the restriction map G(B) ! G(A) is an isomorphism.
2) Recall from [39, Prop. 3.1 p. 137] that the 'etale classifying space
Bet(G) = Rss*(BG) is A1-local. Here ss : (Smk)et! (Smk)Nis is the canon-
ical morphism of sites. But then for any space X , the set [X , Bet(G)]H(k)~=
[X , Bet(G)]Hs(k) is by adjunction (see loc. cit. x Functoriality p. 61]) in
natural bijection with HomHs(Smk)et(ss*(X ), BG) ~=H1et(X ; G).
This proves also in that case that the restriction map H1et(B; G) ! H1et(A; *
*G)
is a bijection. We know moreover that G is A1-invariant as space, thus
G(B) ! G(A) is also an isomorphism. The same reasoning as previously
yields the result.
Example 4.6 1) Any Gm -torsor Y ! X is an A1-covering. Thus any line
bundle yields a A1-covering. In particular, a connected smooth projective
k-variety of dimension 1 as always non trivial A1-coverings!
2) Any finite 'etale Galois covering Y ! X between smooth k-varieties
whose Galois group has order prime to char(k) is an A1-covering. More
generally, one could show that any finite 'etale covering between smooth k-
varieties which can be covered by a surjective 'etale Galois covering Z ! X
with group a finite 'etale k-group G of order prime to char(k) is an A1-
covering. In characteristic 0, for instance, any finite 'etale covering is an
A1-covering.
Lemma 4.7 1) Any pull-back of an A1-covering is an A1-covering.
2) The composition of two A1-coverings is a A1-covering.
3) Any A1-covering is an A1-fibration in the sense of [39].
118
4) A morphism Y1 ! Y2 of A1-coverings Yi ! X which is an A1-weak
equivalence is an isomorphism.
Proof. Only the last statement requires an argument. It follows from
Lemma 4.2: applying it to Y1 ! Y2 one first get a retraction Y2 ! Y1 and
to check that this retraction composed with Y1 ! Y2 is the identity of Y2
one uses once more the Lemma 4.2.
We now come to the main result of this section:
Theorem 4.8 Any pointed A1-connected space X admits a universal pointed
A1-covering X" ! X in the category of pointed coverings of X . It is (up
to unique isomorphism) the unique pointed A1-covering whose source is A1-
1
simply connected. It is a ssA1(X )-torsor over X and the canonical morphism
1
ssA1(X ) ! AutX (X")
is an isomorphism.
Proof. Let X be a pointed A1-connected space. Let X ! LA1(X )
be its A1-localization. Let X"A1 be the universal covering of LA1(X ) in the
1
simplicial meaning. It is a ssA1(X )-torsor by construction. From Lemma
1 1
4.5 X"A1 ! LA1(X ) is thus also an A1-covering (as ssA1(X ) is strongly A -
1
invariant. Let X"! X be its pull back to X . This is a pointed ssA1(X )-torsor
and still a pointed A1-covering. We claim it is the universal pointed A1-
covering of X .
Next we observe that X" is A1-simply connected. This follows from the
left properness property of the A1-model category structure on the category
of spaces [39] that X" ! X"A1is an A1-weak equivalence.
Now we prove the universal property. Let Y ! X be a pointed A1-
covering. Let Y(0) Y the inverse image of (the image of) the base point in
1 (0)
ssA0(Y). We claim (like in the above simplicial case) that Y ! X is still
an A1-covering. It follows easily from the fact that an A1-trivial cofibration
1
induces an isomorphism on ssA0. In this way we reduce to proving the uni-
versal property for pointed A1-coverings Y ! X with Y also A1-connected.
119
By Lemma 4.9 below there exists a cartesian square of pointed spaces
Y ! Y0
# #
X ! LA1(X )
with Y0 ! LA1(X ) a pointed A1-covering of LA1(X ). By the above theory of
simplicial coverings, there exists a unique morphism of pointed coverings
"XA1 ! Y0
# #
LA1(X ) = LA1(X )
Pulling-back this morphism to X yields a pointed morphism of A1-coverings
X" ! Y
# #
X = X
Now it suffices to check that there is only one such morphism. Let f1 and f2
be morphisms X" ! Y of pointed A1-coverings of X . We want to prove they
are equal. We again apply Lemma 4.9 below to each fi and get a cartesian
square of pointed spaces
"X "fi!"Xi0
# #
Y ! Y0
in which X" ! X"iis an A1-weak equivalence. As a consequence the pointed
A1-coverings X"0i! Y0 to the A1-local space Y0 are simply A1-connected
and are thus both the simplicial universal pointed covering of Y0 (and of
LA1(X )): let OE : X"01~="X20be the canonical isomorphism of pointed coverings.
To check f1 = f2, it clearly suffices to check that OE O "f1= f"2. But there
1 1
exists _ : X"! ssA1(X ) such that "f2= _.(OE O "f1). But as X" is A -connected,
1
_ is constant, i.e. factor as X" ! * ! ssA1(X ). But as all the morphisms
1
are pointed, that constant * ! ssA1(X ) must be the neutral element so that
OE O "f1= "f2.
We observe that if Y ! X is a pointed A1-covering with Y simply A1-
connected, the unique morphism X"! Y is an A1-weak equivalence and thus
120
an isomorphism by Lemma 4.7 4).
Finally it only remains to prove the statement concerning the morphism
1
ssA1(X ) ! AutX (X")
Here the right hand side means the sheaf of groups on Smk which to U as-
sociates the group of automorphisms AutX (X")(U) of the covering X"x U !
X x U. We observe that if two automorphisms OEi 2 AutX (X")(U), i 2 {1, 2},
coincide on the base-point section U ! X" x U then OE1 = OE2. Indeed as
"Xx U ! X x U is a ssA11(X )-torsor, there is ff : X" x U ! ssA11(X ) with
1 A1 A1
OE2 = ff.OE1. But ssA0(X" x U) = ss0 (U) and ff factors through ss0 (U) !
1
ssA1(X ). As the composition of ff with the base-point section U ! X" x U
is the neutral element, we conclude that ff is the neutral element and OE1 = OE*
*2.
This first shows that the above morphism is a monomorphism. Let OE 2
AutX (X")(U). Composing OE with the base-point section U ! X"x U we get
1
_ 2 ssA1(X )(U). But the automorphisms OE and _ coincide by construction
on the base-point section. Thus they are equal and our morphism is also
onto. The Theorem is proven.
Lemma 4.9 Let Y ! X be a pointed A1-covering between pointed A1-
conected spaces. Then for any A1-weak equivalence X ! X 0any there exists
a cartesian square of spaces
Y ! Y0
# #
X ! X 0
in which the right vertical morphism is an A1-covering (and thus the top
horizontal morphism an A1-weak equivalence).
Proof. Let X 0! LA1(X 0) be the A1-localization of X 0. As by construc-
tion, LA1(-) is a functor on spaces we get a commutative square
Y ! LA1(Y)
# #
X ! LA1(X 0)
in which the horizontal arrows are A1-weak equivalences. As the left vertical
arrow is an A1-fibration (by Lemma 4.7) with A1-homotopy fiber equal to
121
the fiber Y, which is an A1-invariant sheaf, thus is A1-local, the A1-
homotopy fiber of the pointed morphism LA1(Y) ! LA1(X 0) is A1-equivalent
to the previous one (because the square is obviously A1-homotopy cartesian).
As both LA1(Y) and LA1(X 0) are A1-fibrant and (simplicially) connected,
this means (using the theory of simplicial coverings for LA1(X 0)) that there
exists a commutative square
LA1(Y) ! Y0
# #
LA1(X 0) = LA1(X 0)
in which Y0 ! LA1(X 0) is an (A1-)covering and LA1(Y) ! Y0 an (A1-)weak
equivalence.
This A1-homotopy cartesian square induces a commutative square
Y ! Y"
# # (4.1)
X = X
in which both vertical morphisms are A1-coverings and the top horizontal
morphism is an A1-weak equivalence (by the properness of the A1-model
structure [39]), where Y" is the fiber product Y0xLA1(X0)X . By Lemma 4.7
Y ! Y" is an isomorphism. This finishes our proof as Y" is clearly the pull-
back of an A1-covering of X 0because X ! LA1(X 0) factor through X ! X 0.
Remark 4.10 Let us denote by CovA1(X ) the category of A1-coverings of a
fixed pointed A1-connected space X . The fiber x0 of an A1-covering Y ! X
over the base point x0 is clearly an A1-invariant sheaf of sets. One may de-
1
fine a natural right action of ssA1(X ) on x0(Y ! X ) and it can be shown
that the induced functor x0 from CovA1(X ) to the category of A1-invariant
1
sheaves with a right action of ssA1(X ) is an equivalence of categories.
When X is an arbitrary space, this correspondence can be extended to an
equivalence between the category CovA1(X ) and some category of "functor-
sheaves" defined on the fundamental A1-groupoid of X .
We end this section by mentionning the (easy version of the) Van-Kampen
Theorem.
122
Remark 4.11 The trick to deduce this kind of results is to observe that for
any pointed connected space X , the map [X , BG]Hs,o(k)! HomGr(ss1(X ), G)
is a bijection. This follows as usual by considering the functoriality of the
Postnikov tower [39]. But then if G is a strongly A1-invariant sheaf, we get
in the same way:
[X , BG]Ho(k)! HomGrA1(ss1(X ), G)
where GrA1 denotes the category of strongly A1-invariant sheaves of groups.
It follows at once that the inclusion GrA1 Gr admits a left adjoint G 7! GA1,
1
with GA1 := ssA1(BG) = ss1(LA1(BG)). As a consequence, GrA1 admits all
1
colimits. For instance we get the existence of sums denoted by *A in GrA1:
1
if Gi is a family of strongly A1-invariant sheaves, their sum *AiGi is (*iGi)A1
where * means the usual sum in Gr.
Theorem 4.12 Let X be an A1-connected pointed smooth scheme. Let {Ui}i
be a an open covering of X by A1-connected open subschemes which contains
the base point. Assume furthermore that each intersection Ui \ Uj is still
A1-connected. Then for any strongly A1-invariant sheaf of groups G, the
following diagram
1 A1 ! A1 A1 A1
*Ai,jss1 (Ui\ Uj) ! *i ss1 (Ui) ! ss1 (X) ! *
is right exact in GrA1.
Proof. We let ~C(U) the simplicial space associated to the covering Ui of
X (the ~Cech object of the covering). By definition, ~C(U) ! X is a simplicial
weak equivalence. Thus Remark 4.11, it follows that for any G 2 GrA1
HomGrA1(ss1(X), G) = [C~(U), BG]Hs,o(k)
Now the usual skeletal filtration of ~C(U) easily yields the fact that the obvi*
*ous
diagram (of sets)
1 A1 ! A1
HomGr(ssA1(C~(U)), G) ! iHomGr(ss1 (Ui), G) ! i,jHomGr(ss1 (Ui\Uj), G)
is exact. Putting all these together we obtain our claim.
123
1 n A1
4.2 Basic computation: ssA1 (P ) and ss1 (SLn) for n 2
The following is the easiest application of the preceding results:
Theorem 4.13 For n 2 the canonical Gm -torsor
Gm - (An+1 - {0}) ! Pn
is the universal A1-covering of Pn. This defines a canonical isomorphism
1 n
ssA1(P ) ~=Gm .
Proof. For n 2, the pointed space An+1 - {0} is A1-simply connected
by Theorem 3.38. We now conclude by Theorem 4.8.
For n = 1, A2 - {0} is no longer 1-A1-connected. We now compute
1 2 2 1 A1 2
ssA1(A -{0}). As SL2 ! A -{0} is an A -weak equivalence, ss1 (A -{0}) ~=
1 1 A1
ssA1(SL2). Now, the A -fundamental sheaf of groups ss1 (G) of a group-space
G is always a sheaf of abelian groups by the classical argument. Here we mean
by "group-space" a group object in the category of spaces, that is to say a
simplicial sheaf of groups on Smk.
By the Hurewicz Theorem and Theorem 3.25 we get canonical isomor-
1 A1 A1 2 MW
phisms ssA1(SL2) = H1 (SL2) = H1 (A - {0}) = K_2 .
Finally the classical argument also yields:
Lemma 4.14 Let G be a group-space which is A1-connected. Then there
exists a unique group structure on the pointed space G" for which the A1-
covering "G! G is an (epi-)morphism of group-spaces. The kernel is central
1
and canonically isomorphic to ssA1(G").
Altogether we have obtained:
Theorem 4.15 The universal A1-covering of SL2 given by Theorem 4.8 ad-
mits a group structure and we get in this way a central extension of sheaves
of groups
0 ! K_MW2 ! S"L2! SL2 ! 1
Remark 4.16 Over an infinite field, this extension is also a central extensi*
*on
in the Zariski topology by the Theorem 1.26. In fact the results of the
Appendix show that it is always the case.
124
This central extension can be constructed in the following way:
Lemma 4.17 Let B(SL2) denote the simplicial classifying space of SL2.
Then there exists a unique Hs,o(k)-morphism
e2 : B(SL2) ! K(K_MW2 , 2)
which composed with (SL2) B(SL2) gives the canonical cohomology class
(SL2) ~= (A2 - {0}) ! K(K_MW2 , 2).
The central extension of SL2 associated with this element of H2(SL2; K_MW2)
is canonically isomorphic to the central extension of Theorem 4.15.
Proof. We use the skeletal filtration Fs of the classifying space BG; it
has the property that (simplicially) Fs=Fs-1 ~= s(G^s). Clearly now, using
the long exact sequences in cohomology with coefficients in K_MW2 one sees
that the restriction:
H2(B(SL2); K_MW2) ! H2(F1; K_MW2) = H2( (SL2); K_MW2)
is an isomorphism.
Now it is well-known that such an element in H2(B(SL2); K_MW2) cor-
responds to a central extension of sheaves as above: just take the pointed
simplicial homotopy fiber of (a representative of) the previous morphism
B(SL2) ! K(K_MW2 , 2). Using the long exact homotopy sequence of simpli-
cial homotopy sheaves of this fibration yields the required central extension:
0 ! K_MW2 ! ss1( ) ! SL2 ! 0
To check it is the universal A1-covering for SL2, just observe that the map
1 2
B(SL2) ! K(K_MW2 , 2) is onto on ssA2 as the map (SL2) ~= (A - {0}) !
1
K(K_MW2 , 2) is already onto (actually an isomorphism) on ssA2. Now by Theo-
rem 3.50 the simplicial homotopy fiber sequence is also an A1-fiber homotopy
sequence. Then the long exact homotopy sequence in A1-homotopy sheaves
this time shows that is simply 2-A1-connected. Thus the group-object
ss1( ) is simply A1-connected thus is canonically isomorphic to S"L2.
125
Remark 4.18 1) As a K_MW2 -torsor (forgetting the group structure) SL"2
can easily be described as follows. We use the morphism SL2 ! A2 - {0}.
It is thus sufficient to describe a K_MW2 -torsor over A2 - {0}. We use the
open covering of A2 - {0} by the two obvious open subsets Gm x A1 and
A1 x Gm . Their intersection is exactly Gm x Gm . The tautological symbol
Gm x Gm ! K_MW2 (see Section 2.3) defines a 1-cocycle on A2 - {0} with
values in K_MW2 and thus an K_MW2 -torsor. The pull-back of this torsor to
SL2 is S"L2. It suffices to check that it is simply A1-connected. This follows
in the same way as in the previous proof from the fact the Ho(k)-morphism
A2-{0} ! K(K_MW2 , 1) induced by the previous 1-cocycle is an isomorphism
1
on ssA1.
2) For any SL2-torsor , over a smooth scheme X (or equivalently a rank
two vector bundle , over X with a trivialization of 2(,)) the composition
of the Hs,o(k)-morphisms X ! B(SL2) classifying , and of e2 : B(SL2) !
K(K_MW2 , 2) defines an element e2(,) 2 H2(X; K_MW2); this can be shown to
coincide with the Euler class of , defined in [3], see [36].
1
The computation of ssA1(SLn), n 3
We first observe:
Lemma 4.19 1) For n 3, the inclusion SLn SLn+1 induces an isomor-
phism 1 1
ssA1(SLn) ~=ssA1(SLn+1)
2) The inclusion SL2 SL3 induces an epimorphism
1 A1
ssA1(SL2) i ss1 (SL3)
Proof. We denote by SL0n SLn+1 the subgroup formed by the matrices
of the form 0 1
1 0 . .0.
B C
B ? C
B . C
@ .. M A
?
with M 2 SLn. Observe that the group homomorphism SL0n! SLn is an
A1-weak equivalence: indeed the inclusion SLn SL0nshows SL0nis the
126
semi-direct product of SLn and An so that as a space SL0nis the product
An x SLn.
The group SL0nis the isotropy subgroup of (1, 0, . .,.0) under the right
action of SLn+1 on An+1 - {0}. The following diagram
SL0n- SLn+1 ! An+1 - {0} (4.2)
is thus an SL0n-Zariski torsor over An+1 - {0}, where the map SLn+1 !
An+1 - {0} assigns to a matrix its first horizontal line.
By Theorem 3.53, and our computations, the simplicial fibration sequence
(4.2) is still an A1-fibration sequence. The associated long exact sequence of
A1-homotopy sheaves, together with the fact that An+1 - {0} is (n - 1)-A1-
connected and that SLn SL0nan A1-weak equivalence implies the claim.
Now we may state the following result which implies the point 2) of
Theorem 22:
1 MW
Theorem 4.20 The canonical isomorphism ssA1(SL2) ~=K_2 induces through
the inclusions SL2 ! SLn, n 3, an isomorphism
1 A1 A1
K_M2= K_MW2=j ~=ssA1(SLn) = ss1 (SL1 ) = ss1 (GL1 )
Remark 4.21 Let A3 - {0} ! B(SL02) be the morphism in Hs,o(k) which
1
classifies the SL02-torsor (4.2) over A3-{0}. Applying ssA2 yields a morphism:
1 3 A1 0 A1 A1 MW
K_MW3 = ssA2(A -{0}) ! ss2 (B(SL2)) ~=ss2 (B(SL2)) = ss1 (SL2) = K_2
This morphism can be shown in fact to be the Hopf morphism j in Milnor-
Witt K-theory sheaves. The proof we give below only gives that this mor-
phism is j up to multiplication by a unit in W (k).
Remark 4.22 We will use in the proof the "second Chern class morphism",
a canonical Ho(k)-morphism
c2 : B(GL1 ) ! K(K_M2, 2)
more generally the n-th Chern class morphism cn : B(GL1 ) ! K(K_Mn, n) is
defined as follows: in Ho(k), B(GL1 ) is canonically isomorphic to the infinite
grassmanian Gr1 [39]. This space is the filtering colimit of the finite Grass-
manian Grm,i [loc. cit. p. 138]. But clearly, [Grm,i, K(K_Mn, n)]Ho(k)is the
127
cohomology group Hn(Grm,i; K_Mn, n). This group is isomorphic to the n-th
Chow group CHn(Grm,i) by Rost [44], and we let cn 2 [Grm,i, K(KMn, n)]Ho(k)
denote the n-th Chern class of the tautological rank m vector over bundle
on Grm,i[15]. As the Chow groups of the Grassmanians stabilize loc. cit., the
Milnor exact sequence gives a canonical element cn 2 [B(GL1 ), K(K_Mn, n)]Ho(k).
Form this definition it is easy to check that c2 is the unique morphism
B(GL1 ) ! K(K_M2, 2) whose composite with (GL2) ! B(GL2) ! B(GL1 ) !
K(K_M2, 2)) is the canonical composition (GL2) ! (A2-{0}) ! K(K_MW2 , 2)) !
K(K_M2, 2)).
Proof of Theorem 4.20. Lemma 4.19 implies that we only have to
show that the epimorphism
1 A1
ss : K_MW2 = ssA1(SL2) i ss1 (SL3)
has exactly has kernel the image I(j) K_MW2 of j : K_MW3 ! K_MW2.
The long exact sequence of homotopy sheaves of the A1-fibration sequence
4.2 SL02-SL3 ! A3-{0} and the A1-weak equivalence SL02! SL2 provides
an exact sequence
1 3 MW A1 A1
K_MW3 = ssA1(A - {0}) ! K_2 = ss1 (SL2) i ss1 (SL3) ! 0
But from Remark ?? and the fact that K_MWn is the free strongly A1-invariant
sheaf on Gm we get that the obvious morphism
HomAb(k)(K_MW3 , K_MW2) ! K_MW-1(k) = W (k)
is an isomorphism. Thus this means that the connecting homomorphism
K_MW3 ! K_MW2 is a multiple of j. This proves the inclusion Ker(ss) I(j).
1 A1 M
Now the morphism ssA1(SL2) i ss1 (SL3) ! K_2 induced by the second
Chern class (cf remark 4.22) is the obvious projection K_MW2 ! K_MW2 =j =
K_M2. This shows the converse inclusion.
1 1
4.3 The computation of ssA1 (P )
We recall from [39] that there is a canonical Ho(k)-isomorphism P1 ~= (Gm )
induced by the covering of P1 by its two standard A1's intersecting to Gm .
128
1 1 A1
Thus to compute ssA1(P ) is the same thing as to compute ss1 ( (Gm )).
Let us denote by Shvo the category of sheaves of pointed sets on Smk.
1
For any S 2 Shvo, we denote by `S : S ! ssA1( (S)) the canonical Shvo-
1
morphism obtained by composing S ! ss1( (S)) and ss1( (S)) ! ssA1( (S)).
Lemma 4.23 The morphism S induces for any strongly A1-invariant sheaf
1
of groups G a bijection HomGr(ssA1( (S)), G) ~=HomShvo(S, G).
Proof. As the classifying space BG is A1-local the map [ (S), BG]Hs,o(k)!
[ (S), BG]Ho(k)is a bijection.
1
Now the obvious map [ (S), BG]Hs,o(k)! HomGr(ssA1( (S)), G) given
by the functor ss1 is bijective, see Remark 4.11.
The classical adjunction [ (S), BG]Hs,o(k)~=[S, 1(BG)]Hs,o(k)and the
canonical Hs,o(k)-isomorphism G ~= 1(BG) are checked to provide the re-
quired bijection.
The previous result can be expressed by saying that the sheaf of groups
1 1
FA1(S) := ssA1( (S)) is the "free strongly A -invariant" sheaf of groups on
the pointed sheaf of sets S. In the sequel we will simply denote, for n 1,
1 ^n
by FA1(n) the sheaf ssA1( ((Gm ) )).
We have proven in section 4.2 that FA1(2) is abelian and (thus) isomor-
1 1
phic to K_MW2. Our aim is to describe FA1(1) = ssA1(P ).
The Hopf map of a sheaf of group.
Recall that for two pointed spaces X and Y we let X * Y denote the
reduced join of X and Y, that is to say the quotient of 1 x X x Y by the
relations (0, x, y) = (0, x, y0), (1, x, y) = (1, x0, y) and (t, x0, y0) = (t, *
*x0, y0)
where x0 (resp. y0) is the base point of X (resp. Y). It is a homotopy colimit
of the diagram of pointed spaces
X
"
X x Y ! Y
129
Example 4.24 A2- {0} has canonically the A1-homotopy type of Gm * Gm :
use the classical covering of A2-{0} by Gm xA1 and A1xGm with intersection
Gm x Gm .
The join X * (point) of X and the point is called the cone of X and is
denoted by C(X ). It is the smash product 1 ^ X with 1 pointed by 1.
we let X C(X ) denote the canonical inclusion. The quotient is obviously
(X ). The "anticone" C0(X ) is the the smash product 1 ^ X with 1
pointed by 0.
The join obviously contains the wedge C(X )_C0(Y). Clearly the quotient
(X * Y)=(X _ Y) is (X x Y) and the quotient (X * Y)=(C(X ) _ C0(Y)) is
(X ^ Y).
The morphism of pointed spaces X * Y ! (X ^ Y) is thus a simplicial
weak-equivalence. The diagram of pointed spaces
(X x Y) (X x Y)
" #
X * Y ~! (X ^ Y)
defines a canonical Hs,o(k)-morphism
!X,Y : (X x Y) ! (X x Y)
The following result is classical:
Lemma 4.25 The Hs,o(k)-morphism !X,Y is (for the co-h-group structure
on (X x Y) equal in Hs,o(k) to (ss1)-1.Id (XxY).(ss2)-1, where ss1 is the
obvious composition (X x Y) ! (X ) ! (X x Y) and ss2 is defined the
same way using Y.
Proof. To prove this, the idea is to construct an explicit model for the
map (X x Y) ! X * Y. One may use as model for (X x Y) the amalga-
mate sum of C(X x Y), 1X x Y and C0(X x Y) obviously glued together.
Collapsing C(X x Y) _ C0(X x Y) in that space gives exactly (X x Y) thus
! (X xY) is a simplicial weak equivalence. Now there is an obvious map
! X * Y given by the obvious inclusions of the cones and the canonical
projection on the middle. It then remains to understand the composition
! X * Y ! (X x Y). This is easily analyzed and yields the result.
130
Now let G be a sheaf of groups. We consider the pointed map
~0G: G x G ! G , (g, h) 7! g-1.h
This morphism induces a morphism 1 x G x G ! x G which is easily
seen to induce a canonical morphism
jG : G * G ! (G)
which is called the (geometric) Hopf map of G.
We will still denote by jG : (G ^ G) ! (G) the canonical Ho(k)-
morphism obtained as the composition of the geometric Hopf map and the
inverse to G * G ! (G ^ G).
Example 4.26 Example 4.24 implies that the Hopf fibration A2-{0} ! P1
is canonically A1-equivalent to the geometric Hopf map jGm
(Gm ^ Gm ) ! (Gm )
We observe that G acts diagonally on G * G and that the geometric Hopf
map jG : G * G ! (G) is a G-torsor. It is well known that the classifying
map (G) ! BG for this G torsor is the canonical one [27]. By Theorem
1 1
3.53 if ssA0(G) is a strongly A -invariant sheaf, the simplicial fibration
G * G ! (G) ! BG (4.3)
is also an A1-fibration sequence.
Remark 4.27 Examples are given by G = SLn and G = GLn for any
n 1. In fact we do not know any connected smooth algebraic k-group
which doesn't satisfy this assumption.
The following result is an immediate consequence of Lemma 4.25:
Corollary 4.28 For any sheaf of groups G, the composition
jG
(G x G) ! (G ^ G) ! (G)
is equal in [ (G x G), (G)]Hs,o(k)(for the usual group structure) to
( (O1))-1. (~0).( (pr2))-1
pr1 g7!g-1 IdGx*
where O1 is the obvious composition G x G ! G -! G ! G x G and
pr2 *xIdG
pr2 is the composition G x G ! G ! G x G.
131
We now specialize to G = Gm . From what we have just done, the fibration
sequence (4.3) Gm * Gm ! (Gm ) ! BGm is A1-equivalent to
A2 - {0} ! P1 ! P1
As the spaces (Gm ) ~= P1 and B(Gm ) ~= P1 are A1-connected, the long
exact sequence of homotopy sheaves induces at once a short exact sequence
of sheaves of groups
1 ! K_MW2 ! FA1(1) ! Gm ! 1 (4.4)
We simply denote by ` : Gm ! FA1(1) the section `Gm . As the sheaf of
pointed sets FA1(1) is the product K_MW2 x Gm (using `), the following result
entirely describes the group structure on FA1(1) and thus the sheaf of groups
FA1(1):
Theorem 4.29 1) The morphism of sheaves of sets
Gm x Gm ! K_MW2
induced by the morphism (U, V ) 7! `(U-1 )-1`(U-1 V )`(V )-1 is equal to the
symbol (U, V ) 7! [U][V ].
2) The short exact sequence (4.4):
1 ! K_MW2 ! FA1(1) ! Gm ! 1
is a central extension.
Proof. 1) follows directly from the definitions and the Corollary 4.28.
2) For two units U and V in some F 2 Fk the calculation in 1) easily yields
`(U)`(V )-1 = -[U][-V ]`(U-1 V )-1 and `(U)-1`(V ) = [U-1 ][-V ]`(U-1 V ).
Now we want to check that the action by conjugation of Gm on K_MW2
(through `) is trivial. It clearly suffices to check it on fields. For units U,*
* V
and W in some field F 2 Fk, we get (using the previous formulas):
`(W )([U][V ])`(W )-1 = (-[W ][-U-1 ]+[UW ][-U-1 .V ]-[W V ][-V ])`(W -1)-1.`(W*
* )-1
Now applying 1) to U = W = V yields (as ` is pointed) `(W -1)-1.`(W )-1 =
[W ][W ].
132
Now the claim follows from the easily checked equality in K_MW2(F )
-[W ][-U-1 ] + [UW ][-U-1 .V ] - [W V ][-V ] + [W ][W ] = [U][V ]
which finally yields `(W )([U][V ])`(W )-1 = [U][V ].
Remark 4.30 Though it is the more "geometric" way to describe FA1(1) it
is not the most natural. We denote by F (S) the free sheaf of groups on the
pointed sheaf of sets S. This is also the sheaf ss1( (S)). Its stalks are the
free groups generated by the pointed stalks of S.
For a sheaf of group G let cG : F (G) i G be the canonical epimorphism
induced by the identity of G, which admits `G as a section (in Shvo). Con-
sider the Shvo-morphism `(2): G^2 ! F (G) given by (U, V ) 7! `G (U).`G (U).`G *
*(UV )-1
This morphism induces a morphism F (G^2) ! F (G).
A classical result of group theory, a proof of which can be found in [10,
Theorem 4.6] gives that the diagram
1 ! F (G^2) ! F (G) ! G ! 1
is a short exact sequence of sheaves of groups. If G is strongly A1-invariant,
we deduce the compatible short exact sequence of strongly A1-invariant
1 ! FA1(G^2) ! FA1(G) ! G ! 1
But now `G (U).`G (U).`G (UV )-1 is the tautological symbol G2 ! FA1(G^2).
In the case of Gm this implies (in a easier way) that the extension
1 ! K_MW2 = FA1(G^2m) ! FA1(1) ! Gm ! 1
is central. It is of course isomorphic to (4.4) but not equal as an extension!
Indeed, as a consequence of the Theorem, we get for the extension (4.4) the
formula `(U)`(V ) =< -1 > [U][V ]`(UV ), but for the previous extension
one has by construction `(U)`(V ) = [U][V ]`(UV ).
Remark 4.31 As a consequence we also see that the sheaf FA1(1) is never
abelian. Indeed the formula `(U)`(V ) =< -1 > [U][V ]`(UV ) implies
`(U)`(V )`(U)-1 = h([U][V ])`(V )
133
Now given any field k one can show that there always exists such an F and
such units with h([U][V ]) 6= 0 2 K_MW2 (F ). Take F = k(U, V ) to be the
field of rational fraction in U and V over k. The composition of the residues
morphisms @U and partialV : KMW2 (k(U, V ) ! KMW0 (k) commutes to mul-
tiplication by h. As the image of the symbol [U][V ] is one, the claim follows
by observing that h 6= 0 2 kMW0 (k).
1 1
Endomorphisms of FA1(1) = ssA1(P ).
We want to understand the monoid of endomorphisms End(FA1(1)) of the
sheaf of groups FA1(1). As FA1(1) is the free strongly A1-invariant sheaf on the
pointed sheaf Gm , we see that as a set End(FA1(1)) = HomShvo(Gm , FA1(1)).
By definition the latter set is FA1(1)-1(k). As a consequence we observe that
there is a natural group structure on End(FA1(1)).
Remark 4.32 It follows from our results that the obvious map
[P1, P1]Ho(k)! End(FA1(1))
is a bijection. The above group structure comes of course from the natural
group structure on [P1, P1]Ho(k)= [ (Gm ), P1]Ho(k).
The functor G 7! (G)-1 is exact in the following sense:
Lemma 4.33 For any short exact sequence 1 ! K ! G ! H ! 1 of
strongly A1-invariant sheaves yields, the diagram of
1 ! (K)-1 ! (G)-1 ! (H)-1 ! 1
is still a short exact sequence of strongly A1-invariant sheaves.
Proof. We already know from Lemma 1.31 that the sheaves are strongly
A1-invariant sheaves. The only problem is in fact to check that the morphism
G-1 ! H-1 is still an epimorphism. For any x 2 X 2 Smk, let Xx be
the localization of X at x. We claim that the morphism G(Gm x Xx) !
H(Gm x Xx) is an epimorphism of groups. Using the very definition of the
functor G 7! (G)-1 this claim easily implies the result.
Now a element in H(Gm x Xx) comes from an element ff 2 H(Gm x U),
for some open neighborhood U of x. Pulling back the short exact sequence
134
to Gm x U yields a K-torsor on Gm x U. But clearly H1(Gm x Xx; G) is
trivial by our results of section 1.2. This means that up to shrinking a bit U
the K torsor is trivial. But this means exactly that there a fi 2 G(Gm x U)
lifting ff. The Lemma is proven.
Applying this to the short exact sequence (4.4) 1 ! K_MW2 ! FA1(1) !
Gm ! 1, which is a central extension by Theorem 4.29, obviously yields a
central extension as well:
0 ! (K_MW2 )-1 ! (FA1(1))-1 ! (Gm )-1 ! 1
But now observe that Z = (Gm )-1 so that the epimorphism (FA1(1))-1 !
(Gm )-1 admits a canonical section sending 1 to the identity.
Corollary 4.34 The sheaf of groups (FA1(1))-1 is abelian and is canonically
isomorphic to Z K_MW1.
Proof. The only remaining point is to observe from remark ?? that the
products induce an isomorphism K_MW1 ~=(K_MW2 )-1.
We let ae : Gm ! (FA1(1))-1 = Z K_MW1 be the morphism of sheaves
which maps U to (1, [U]). Observe it is not a morphism of sheaves of groups.
Theorem 4.35 Endowed with the previous abelian group structure and the
composition of morphisms End(FA1(1)) ~=[P1, P1]Ho(k)is an associative ring.
ae(k) induces a group homomorphism kx ! End(FA1(1))x to the group of
units and the induced ring homomorphism
(k) : Z[kx ] ! End(FA1(1))
is onto. As a consequence, End(FA1(1)) is a commutative ring.
Proof. Let Z(kx ) be the free abelian group on kx with the relation the
symbol 1 2 kx equals 0. It is clear that Z[kx ] splits as Z Z(kx ) in a com-
patible way to the splitting of Corollary 4.34 so that (k) decomposes as the
identity of Z plus the obvious symbol Z(kx ) ! K_MW1(k). But then we know
from Lemma 2.6 that this is an epimorphism.
135
The canonical morphism
[P1, P1]Ho(k)= End(FA1(1)) ! KMW0 (k) = End(K_MW1 )
given by the "Brouwer degree" (which means evaluation of A1-homology in
degree 1) is thus an epimorphism as Z[kx ] ! KMW0 (k) is onto.
To understand this a bit further, we use Theorem 2.46 and its corollary
which show that K_W1! I_is an isomorphism.
In fact KMW0 (k) splits canonically as Z I(k) as an abelian group, and
moreover this decomposition is compatible through the above epimorphism
to that of Corollary 4.34. This means that the kernel of
[P1, P1]Ho(k)= [ (Gm ), (Gm )]Ho(k)i KMW0 (k)
is isomorphic to the kernel of KMW1 (k) i I(k). As KMW1 (k) !h KMW1 (k) !
I(k) ! 0 is always an exact sequence by Theorem 2.46 and its corollary, and
as KMW1 (k) !h KMW1 (k) factors through KMW1 (k) i KMW1 (k)=j = kx we
get an exact sequence of the form
kx ! KMW1 (k) ! I(k) ! 0
where the map kx ! KMW1 (k) arises from multiplication by h. Clearly
-1 is mapped to 0 in KMW1 (k) because h.[-1] = [-1]+ < -1 > [-1] =
[(-1)(-1)] = [1] = 0. Moreover the composition kx ! KMW1 (k) ! kx is
the squaring map. Thus (kx )=( 1)! KMW1 (k) is injective and equal to the
kernel. We thus get altogether:
Theorem 4.36 The diagram previously constructed
0 ! (kx )=( 1)! [P1, P1]Ho(k)! GW (k) ! 0
is a short exact sequence of abelian groups.
Remark 4.37 J. Lannes has observed that as a ring, G"W (k) := [P1, P1]Ho(k)=
End(FA1(1)) is the Grothendieck ring of isomorphism classes of symmetric
inner product spaces over k with a given diagonal basis, where an isomor-
phism between two such objects is a linear isomorphism preserving the inner
136
product and with determinant 1 in the given basis. It fits in the following
cartesian square of rings
GW" (k) ! Z kx
# #
GW (k) ! Z (kx =2)
The bottom horizontal map is the rank plus the determinant. The group
structure on the right hand side groups is the obvious one. The product
structure is given by (n, U).(m, V ) = (nm, Um .V n).
C. Cazanave in a work in preparation, has attacked a proof of the previous
two results by a different method using the approach of Barge and Lannes
on Bott periodicity for orthogonal algebraic K-theory [2]. This allows him
to construct an invariant [P1, P1]Ho(k)! G"W (k).
Remark 4.38 We may turn (k) into a morphism of sheaves of abelian
groups : Z[Gm ] ! (FA1(1))-1 ~=Z K_MW1 induced by ae. Here Z[S]
means the free sheaf of abelian groups generated by the sheaf of sets S.
: Z[Gm ] ! (FA1(1))-1 ~= Z K_MW1 is then the universal morphism of
sheaves of abelian groups to a strictly A1-invariant sheaf. As a consequence
the target is also a sheaf of commutative rings: it is the A1-group ring on
Gm .
Free homotopy classes [P1, P1]H(k). By Remark 2.44 to understand the
1 1 1 1
set [P1, P1]H(k)we have to understand the action of ssA1(P )(k) on [P , P ]Ho(k)
and to compute the quotient.
Clearly, as [P1, P1]Ho(k)~= [P1, B(FA1(1))]Ho(k)~= End(FA1(1)), this action
is given on the right hand side by the action by conjugation of FA1(1) on the
target. Now the abelian group structure comes from the source and thus this
action is an action of the group FA1(1) on the abelian group End(FA1(1)).
As KMW2 (k) FA1(1) is central this action factors through an action of kx
on the abelian group End(FA1(1)).
Lemma 4.39 The action of kx on End(FA1(1)) = KMW1 (k) Z is given as
follows. For any u 2 kx and any (v, n) 2 kx x Z one has in KMW1 (k) Z
cu([v], n) = ([v] - nh[u], n)
137
Proof. To find the action of kx by conjugation on End(FA1(1)) =
(FA1)-1(k) we observe that by Remark 4.31 we understand this action on
FA1(1).
We may explicit this action on FA1(k(T ) and observe that the isomor-
phism End(FA1(1)) = KMW1 (k) Z = (FA1)-1(k) is obtained by cup-product
by T on the left [T ][ : KMW1 (k) Z ! FA1(k(T )). Thus for (v, n) 2 kx x Z
the corresponding element [T ] [ ([v], n) in FA1(k(T )) is [T ][v].`(T )n.
Now by the formula in Remark 4.31 we get for any u 2 kx, and any
(v, n) 2 kx x Z:
cu([T ][v].`(T )n) = [T ][v].(h[u][T ].`(T ))n
= ([T ][v] + nh[u][T ])`(T )n = [T ]([v] - nh[u])`(T )n
This implies the Lemma.
Corollary 4.40 Assume that for each n 1 the map kx ! kx , u 7! un is
onto. Then the surjective map
[P1, P1]H(k)! [P1, P1 ]H(k)= Z
has trivial fibers over any integer n 6= 0 and its fiber over 0 is exactly
KMW1 (k) = [P1, A2 - {0}]H(k).
Proof. First if every unit is a square, by Proposition 2.13, we know that
KMW1 (k) ! kx is an isomorphism. On the set of pairs (v, n) 2 KMW1 (k)xZ =
End(FA1(1)) the action of u 2 kx is thus given as
cu([v], n) = ([vu-2n], n)
But for n 6= 0, any unit w can be written vu-2n for some u and v by assump-
tion on k. Thus the results.
A Unramified and strongly A1-invariant sheaves
over finite fields
Theorem A.1 Any strongly A1-invariant sheaf of groups on Smk is unram-
ified.
138
Proof. Applying 3.8 to B = BG, we already know this result over an
infinite field. It suffices thus to do the case k is perfect which includes fin*
*ite
fields. We now assume k is perfect.
As G is strongly A1-invariant the classifying space BG is A1-local in the
sense of [39], moreover using loc. cit. we know that for any smooth scheme
X, the group of morphism HomHo(k)( (X+ ); BG) is naturally isomorphic to
G(X). Moreover the pointed set HomHo(k)(X+ ; BG) is naturally isomorphic
to H1Nis(X; BG).
The property (0) of unramified sheaves is clear. To prove properties (1)
and (2) we proceed as follows. Let X 2 Smk and let Z X be a closed
subset having everywhere codimension i, and let U X be the open
complement. Endow Z with the reduced induced structure.
Assume first that Z is smooth and has trivial normal bundle in X. The
cofibration sequence X+ ! X=U ! s(U+ ) ! s(X+ ) ! s(X=U), the
result recalled above and the purity Theorem [39] yield an exact sequence
1 ! G(AiZ=(Ai- {0})Z ) ! G(X) ! G(U) ! H1Nis(AiZ=(Ai- {0})Z ); G)
If i 1, G(AiZ=(Ai- {0})Z ) is trivial because G is A1-invariant and G(Z) =
G(AiZ) ! G((Ai- {0})Z ) is injective as the section (1, 0, . .,.0) provides a
left inverse. If i 2, then (AiZ)=(Ai- {0})Z = (Ai=(Ai- {0})) ^ (Z+ ) is A1-
equivalent to a double simplicial suspension, and thus the set H1Nis(AiZ=(Ai-
{0})Z ); G) vanishes.
Because k is perfect, there is an increasing flag of closed subschemes
Zn Zn-1 . . .Z with Zm - Zm+1 smooth over k of codimension i
in X and we may assume further that the normal bundle of Zm - Zm+1 in
X - Zm+1 is trivial. Using the results above, altogether this implies that G
is unramified (see also Remark 1.4).
The following result is useful to deduce results for finite fields from resu*
*lt
for infinite fields. The analogue over an infinite field is trivial (compare [3*
*2,
Lemme 6.4.8]):
Lemma A.2 Let k be a field. Let A1 be a dense open subset. Then
the pointed morphism (A1)+ ! A1= is trivial in Ho(k). As a consequence,
( + ) ! ((A1)+ ) ~= (Spec(k)+ ) admits a section in Ho(k).
139
Proof. By Mayer-Vietoris we get an isomorphism of pointed sheaves
of sets A1= = _xA1=(A1 - {x}), where x runs over the closed points in
A1 - . But in HZar,o(k), A1=(A1 - {x}) is the simplicial suspension of
Spec(~(x)+ ). By Corollary 3.22 p. 95 [39], the A1-localization of a suspension
is 0-connected. Thus _xA1=(A1 - {x}), which is a simplicial suspension is
A1-connected and any morphism in Ho(k) from Spec(k)+ to an A1-connected
pointed space is trivial.
Corollary A.3 Let k be any field. Let B be an A1-local pointed space.
Let f : Spec(k(T )) ! Spec(k) be the canonical morphism. Then for any
X 2 Smk and any n 2 N, the pointed map
HomHo(k)( n(X+ ); B) ! HomHo(k(T))( n((X|k(T))+ ); f*(B)) has trivial ker-
nel. For n 1, being a group homomorphism, it is injective.
Proof. Using the obvious unstable analogues of [32, Corollary 5.2.7] we
see that the map
colim HomHo(k)( n( x X+ ); B) ! HomHo(k(T))( n((X|k(T))+ ); f*(B)) )
is a bijection for any n 2 N, where runs over the ordered set (for inclusion)
of open dense subsets of A1.
Now assume ff 2 HomHo(k)( n(X+ ); B) becomes trivial on k(T ). Then
by the above formula it must be trivial in HomHo(k)( n(( x X)+ ); B) for
some open non-empty in A1. This means, using the cofibration sequence
( + )^(X+ ) ! (A1+)^(X+ ) ! ( =A1)^(X+ ) and observing that ( xX)+ =
( + ) ^ (X+ ), that ff is the restriction of an element fi in HomHo(k)(( =A1) ^
n(X+ ); B). But Lemma A.2 implies that the restriction of this element
to HomHo(k)( n(X+ ); B) = HomHo(k)((A1+) ^ n(X+ ); B), which is ff, is
trivial.
Corollary A.4 Let k be a field. Let B be an A1-local pointed space. For any
field F 2 Fk, the map
HomHo(k)( n(Spec(F )+ ); B) ! HomHo(k)( n((Spec(F (T ))+ ); B) has trivial
kernel. For n 1, being a group homomorphism, it is injective.
Consequently, if Bk(T)is n-A1-connected for some n, B is n-A1-connected.
Proof. By an easy base change argument, the map
HomHo(k)( n(Spec(F )+ ); B) ! HomHo(F)( n(Spec(F )+ ); B|F ) is a bijec-
tion and in the same way,
140
HomHo(k)( n((Spec(F (T ))+ ); B) ~=HomHo(F)( n((Spec(F (T ))+ ); BF ) in the
latter we consider F (T ) as an obvious extension of F .
Now by Corollary A.3, we get the first statement. The last statement is
proven as follows. If Bk(T)is n-A1-connected, then for any F 2 Fk and any i
the same base change argument yields that HomHo(k)( i((Spec(F (T ))+ ); B) ~=
HomHo(k)( i((Spec(F (T ))+ ); B|k(T)) where F (T ) is obviously considered as
a separable extension of k(T ). If i n it follows from that that
HomHo(k)( i((Spec(F (T ))+ ); B) is trivial. By the first statement we deduce
that HomHo(k)( i((Spec(F )+ ); B) is also trivial.
We conclude with [32, Lemma 6.1.3] that B is n-A1-connected.
Corollary A.5 Let M be a strongly A1-invariant sheaf of abelian groups on
a perfect field k. Set for any X 2 Smk and any y 2 X(1)set H1y(X; M) :=
colimy2U[U=(U - __y), BM]Ho(k).
Then for any F 2 Fk, the obvious diagram
0 ! M(F ) ! M(F (T )) ! P2(A1F)(1)H1P(A1F; M) ! 0
is a short exact sequence.
Proof. Using our definitions the construction of that diagram and its
exactness follow from Lemma A.2. Indeed the pointed morphism (A1F)+ !
A1F= is trivial in Ho(F ), with an open subset of A1F. As a consequence,
for any F 2 Fk, the cofibration sequence (A1F= ) ! ( + ) ! ((A1F)+ ) is
split.
Taking morphisms to (the base change to F ) BM and letting decrease
we get a (colimit of split) exact sequence
0 ! M(F ) ! M(F (T )) ! colim [(A1F= ), (BM)|F ]Ho(F) ! 0
But by an easy adjunction, and convenient choices of k-smooth models of
F and we easily obtain an isomorphism colim [(A1F= ), (BM)|F ]Ho(F) !
P2(A1F)(1)H1P(A1F; M).
Corollary A.6 Let k be a field. Let G be sheaf of groups on Smk. Then there
exists a universal morphism G ! GA1 from G to strongly A1-invariant sheaf.
If G is abelian, so is GA1. Moreover, for any field F 2 Fk, the morphism
G|F ! GA1|F is still the universal morphism from G|F to strongly A1-invariant
sheaf.
141
Proof. Let BG ! LA1(BG) be the A1-localization of the pointed space
BG. From [39, Corollary 1.24 p. 104 & Proposition 2.8 p. 108], the in-
verse image functor f*, where f : Spec(k(T )) ! Spec(k) is the structure
morphism, is exact, preserves A1-weak equivalences and A1-local objects.
1 0
Let's denote by G0 the sheaf ssA1(LA1(BG)). By Theorem 1.26 G |k(T ) is
a strongly A1-invariant sheaf and B(f*G) ~= f*(BG) ! f*(LA1(BG)) is the
A1-localization of B(f*G).
1 0
Now let's denote by G" the ssA1(LA1(BG )). One has a canonical morphism
G ! G0 ! G" and any morphism from G to a strongly A1-invariant sheaf H
factors uniquely through G ! G".
It only remains to show G" is strongly A1-invariant. By Corollary A.3
applied to B = LA1(BG0), we see that for any X 2 Smk and any n 2, the
group HomHo(k)( n(X+ ); LA1(BG)) is trivial, as it injects into
HomHo(k(T))( n(Xk(T) +); B(G|k(T))) which is trivial. This shows that the
higher homotopy sheaves of LA1(BG0) are trivial, except ss1 (as we know
from [39] that the A1-localization of a connected space is connected). Thus,
LA1(BG0) is canonically isomorphic to B(G0). Thus G" is strongly A1-invariant.
To prove that if G is abelian so is GA1 we conclude by using the fact
[39] that there exists an A1-localization functor which commutes to finite
products.
The last statement is clear by our construction.
Theorem A.7 Let k be a field and B be a pointed A1-connected space. Then
1 1
ssA1(B) is strongly A -invariant.
Proof. By Theorem 3.9 we know this result for any infinite field. We
might thus assume k is finite but we will only use the fact that for any field
1
F the field F (T ) is infinite! Let us denote by G the sheaf ssA1(B).
We consider the canonical morphism G ! GA1. Theorem 3.9 and Corol-
lary A.6 implies that this morphism induces an isomorphism on any infinite
F 2 Kk.
Now for any F 2 Fk, the morphisms G(F ) ! G(F (T )) and GA1(F ) !
GA1(F (T )) are monomorphism by Corollary A.4.
Thus for any F 2 Fk, the morphism G(F ) ! GA1(F ) is a monomorphism.
We now want to prove now that G(F ) ! GA1(F ) is surjective for any
F 2 Fk.
142
Consider the simplicial homotopy fiber of B ! BGA1. It is an A1-
local space as both B and BGA1 are. But clearly |k(T) is A1-connected.
By Corollary A.4, is A1-connected. The long exact homotopy sequence of
! B ! BGA1 easily implies that G(F ) ! GA1(F is onto for any F 2 Fk
1
and that ssA1( ) = *.
This implies that G ! GA1 is an isomorphism.
We finish the Appendix by proving:
Theorem A.8 Assume k is a finite field. Let M* be a Z-graded A1k-module.
Then for each n, the unramified sheaf Mn obtained by Theorem 1.43 is
strongly A1-invariant.
Proof. From what has been done in Theorem 1.43 and right before,
we know that each Mn defines an unramified sheaf of abelian groups which
satisfies Axioms (A2'), (A6). It also satisfies Axiom (A5) over infinite
fields.
As in the proof of the previous Theorem, we consider the universal mor-
phism Mn ! (Mn)A1 to a strongly A1-invariant sheaf of groups. The latter
is abelian by Corollary A.6.
The axiom (HA)(i) implies that for any F 2 Fk, the morphism M*(F ) !
M*(F (T )) is injective. As in the proof of the previous Theorem, we get that
the morphism Mn ! (Mn)A1 is a monomorphism of sheaves.
Now we claim that as M* satisfies Axiom (HA), there is a canonical
commutative diagram for any F 2 Fk
0 ! Mn(F ) ! M(F (T )) ! P2(A1F)(1)H1P(A1F; Mn) ! 0
# # #
0 ! (Mn)A1(F ) ! (Mn)A1(F (T )) ! P2(A1F)(1)H1P(A1F; (Mn)A1)! 0
The bottom exact sequence being given by Corollary A.5. To check the com-
mutativity, one may extends to an infinite field, where Mn ! (Mn)A1 is an
isomorphism. The point is then to check that the construction from A.5 gives
back the exact sequence of Axiom (HA).
Now the middle vertical arrow is an isomorphism by assumption. Thus it
is clearly sufficient to prove that the morphisms H1P(A1F; Mn) ! H1P(A1F; (Mn)A*
*1)
are monomorphisms. But as P is a tautological uniformizing element of its
associated valuation we get on the left side an isomorphism Mn-1(F [T ]=P ) ~=
143
H1P(A1F; Mn) and using homotopy purity [39], we get on the left side an iso-
morphism ((Mn)A1)-1(F [T ]=P ) ~=H1P(A1F; (Mn)A1). These isomorphisms are
easily checked to be compatible (base change to an infinite field) so that we
reduce to proving that for each P , the induced morphism Mn-1(F [T ]=P ) !
((Mn)A1)-1(F [T ]=P ) is injective. But this follows from the Lemma below.
Lemma A.9 Let M ! N be a be monomorphism of sheaves of abelian
groups on Smk. Then M-1 ! N-1 is still a monomorphism.
Proof. As the sheaves are abelian, the group M(Gm x X) canoni-
cally splits as the direct sum M(X) M-1(X). Thus for any such X, the
monomorphism M(Gm x X) ! N(Gm x X) splits accordingly as a sum of
monomorphisms.
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Fabien Morel
Mathematisches Institut der LMU
Theresienstr. 39
D-80 333 Muenchen
morel@mathematik.uni-muenchen.de
148