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. 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