Generalised sheaf cohomology theories
J.F. Jardine*
October 14, 2002
Introduction
This paper is an expanded version of notes for a set of lectures given at the I*
*saac
Newton Institute for Mathematical Sciences during a NATO ASI Workshop
entitled öH motopy Theory of Geometric Categoriesö n September 23 and 24,
2002. This workshop was part of a program entitled New Contexts in Stable
Homotopy Theory that was held at the Institute during the fall of 2002.
The idea was to present some of the basic features of the homotopy theory of
simplicial presheaves and the stable homotopy theory of presheaves of spectra,
and then display their use in applications. A general outline of these theories
forms the subject of Sections 1 and 2 of this paper.
There has been some renewed interest in equivariant stable categories for
profinite groups of late, and the main features of that theory have been de
scibed here in Sections 3 and 4. I wanted to stress the calculational aspects
of that theory as well as display its main features. This is done in the course
of presenting an outline of the proof of Thomason's descent theorem for Bott
periodic algebraic Ktheory, which appears in Section 5.
The outline of the Thomason result which is presented here is a stripped
down version of the proof appearing in [19], with all of the hard bits (ie. the
coherence issues) carefully swept under the rug. Also, the proof works as stated
only for good schemes and at good primes. The other cases, which are much
more complicated to discuss, have been treated in detail elsewhere, particularly
in Thomason's original paper [38] and the ThomasonTrobaugh paper [39]. One
should also look at the commentary given by Mitchell in [30].
I would like to thank the überorganizers of the Workshop and the New
Contexts program at the Institute for giving me the opportunity to speak. I
would also like to take this opportunity to thank the Newton Institute for its
hospitality and support.
____________________________*
This research was partially supported by NSERC.
1
Contents
1 Simplicial presheaves 2
2 Presheaves of spectra 13
3 Profinite groups 20
4 Generalised Galois cohomology theory 23
5 Thomason's descent theorem 30
1 Simplicial presheaves
In all that follows, C will be a small Grothendieck site, meaning a category
having a set of objects and equipped with some notion of covering. Standard
examples include the following:
1)The category opX of open subsets of a topological space X has as objects
all inclusions U X of X, with coverings given by open coverings in the
traditional sense.
2)The category underlying the Zariski site ZarS of a scheme S has objects
consisting of all open subschemes U S of the scheme S. A covering
family is a collection of open subschemes Ui U such that [Ui = U.
This site is really just the site of open subsets of the topological space
underlying the scheme S, but we give it a special name.
3)The category underlying the 'etale site etS of a scheme S consists of all
'etale maps U ! X, and has covering families consisting of sets of mor
phisms OEi: Ui! U over X such that [OEi(Ui) = U.
4)The Nisnevich site NisS of a scheme S has the same underlying category
as the 'etale site, but a covering of U=S is an 'etale covering family OEi*
* :
Ui ! U such that every scheme morphism Sp(F ) ! U defined on some
field F lifts to the total space of the scheme homomorphism t Ui ! U
which is defined by all of the maps OEi. There are fewer Nisnevich covers
than 'etale covers. Note, however, that that every Zariski cover Ui U is
a Nisnevich cover, so that the Nisnevich topology is finer than the Zariski
topology, but coarser than the 'etale topology.
5)All of the algebraic geometric sites described above have "big" broth
ers, namely (SchS)Zar (SchS)etand (SchS)Nis respectively. These are
the socalled big sites for the respective topologies. The underlying cate
gories consist, in all cases of Sschemes Y ! S which are locally of finite
type, and secretly have some fixed infinite cardinal bounding their sets of
points. The covering families have the same definitions as for the corre
sponding versions above. One can impose extra structure on the schemes
2
over S: most commonly one decorates the category SmS of smooth S
schemes with any of the above topologies, so that one has the the smooth
sites (SmS)Zar, (SmS)et and (SmS)Nis. The smooth Nisnevich site
(SmS)Nis is the basis for the standard description of motivic homotopy
theory over S.
6)Any small category I gives rise to a Grothendieck site, with the socalled
"chaotic" topology, which really is no topology at all: the covering famil*
*ies
are precisely the identity maps a ! a in the category I.
Most of the examples of Grothendieck sites listed above arise in algebraic
geometry. The displayed list is is by no means complete: there are, for example,
the various flavours of flat topologies, Voevodsky's h topologies, and so on.
Grothendieck sites abound in nature.
A presheaf on a site C is a contravariant functor F : Cop ! Set taking values
in the set category. Contravariant functors on C taking values in categories D
are said to be presheaves in D, or presheaves of objects in D. For example, a
presheaf of abelian groups is a contravariant functor A : Cop ! Ab taking values
in the category of abelian groups. A presheaf of simplicial sets, or a simplici*
*al
presheaf, is a contravariant functor X : Cop ! S taking values in the category
S of simplicial sets.
The presheaves on C taking values in D are the objects of a category, for
which the morphisms are the natural transformations. From this point of view,
a simplicial presheaf is a simplicial object in the category of presheaves of s*
*ets.
A morphism f : X ! Y of simplicial presheaves consists of simplicial set maps
f : X(U) ! Y (U), one for each object U of C, which are natural with respect
to the morphisms of C in the obvious sense. I write s Pre(C) for the category of
simplicial presheaves on the site C.
Here are some basic examples of simplicial presheaves:
1)Every simplicial set Y determines a constant simplicial presheaf *Y de
fined by *Y (U) = Y , with all morphisms of C sent to the identity on
Y . One often dispenses with the * and just writes Y for the constant
simplicial presheaf associated to Y .
2)Any object A 2 C represents a presheaf U 7! hom (U, A) = A(U). Again,
objects in C and the presheaves that they represent are often confused
notationally. Any simplicial object X in C represents a simplicial presheaf
(of the same name), by exactly the same process.
2a)A group object G of C represents a presheaf of groups G(U) = hom (U, G).
The associated classifying simplicial presheaf BG is defined in sections by
BG(U) = B hom(U, G).
2b)Suppose that W ! V is a covering map of V in C. The corresponding ~Cech
object is the simplicial presheaf arising from the iterated fibre products
W, W xV W, W xV W xV W, . . .
3
in the presheaf category, and all of the associated projections and diagon*
*als
relating them.
It's more conceptually satisfying to think of this object as the nerve of a
groupoid: if f : X ! Y is a plain old function, then there is a groupoid
whose objects are the elements of X, and we say that there is a (unique)
morphism x ! y if f(x) = f(y) in Y . The nerve of this groupoid has X
as its set of vertices and has all iterated cartesian products X xY . .x.YX
as simplices. Apply this construction to all functions W (U) ! V (U) in
all sections and you get the ~Cech object associated to the covering.
2c)Suppose that L is a finite Galois extension of a field k with Galois group
G. Then the ~Cech object associated to the 'etale covering Sp(L) ! Sp(k)
on any good site of kschemes consists of the iterated pullbacks
Sp(L), Sp(L) xSp(k)Sp(L), . . .
which can be identified withQthe Borel construction EG xG Sp(L), just by
the identification L k L ~= G L given by Galois theory. The notation
EG xG Sp(L) also stands for the nerve of the translation category for
the action of G on Sp(L), and the isomorphism between the ~Cech object
associated to the cover Sp(L) ! Sp(k) and the Borel construction EGxG
Sp(L) is induced by an isomorphism of presheaves of groupoids.
3)Suppose_that A is a presheaf of abelian groups, and write K(A, n) =
W (A[n]). Here, A[n] means the presheaf of chain complexes which
consists of a copy of A concentrated in degree_n, and then we obtain
K(A, n) by applying the EilenbergMac Lane W construction to obtain
a simplicial abelian group in each section. The EilenbergMac Lane con
struction is natural so this works, and K(A, n)(U) = K(A(U), n) is the
standard construction of an EilenbergMac Lane space. The simplicial
presheaf K(A, n) is usually called an EilenbergMac Lane object associ
ated to the abelian presheaf A. Of course, there is one of these for each
n 0.
Of course there are such things as sheaves, which are presheaves which satis*
*fy
a patching property defined by the topology on C. Most often one is dealing with
covering families Ui! U in sites (as in algebraic geometry) where pullbacks are
defined. In such a context, one says that a presheaf F is a sheaf if the diagra*
*ms
Y Y
F (U) ! F (Ui) ' F (UixU Uj)
i i,j
are equalizers for all covering families Ui! U of all objects U of C. The categ*
*ory
Shv (C) is the full subcategory of the category of presheaves whose objects are
the sheaves; in other words one constructs the sheaf category by taking all
natural transformations between sheaves. The inclusion of the sheaf category
in the category of presheaves has a left adjoint F 7! ~F, called the associated
4
sheaf functor. The canonical map j : F ! F~arising from the adjunction is
called the associated sheaf map. The associated sheaf is constructed by formally
adjoining (twice) all solutions of patching problems in F via certain filtered
colimit diagrams; most people find this construction obtuse, and it will not be
repeated here. The important point to remember is that, since ~Fis constructed
from F by a filtered colimit construction, the functor F 7! ~Fis exact in the
sense that it preserves all finite limits up to isomorphism.
I generally like to look at [33] for the basics about sheaves on Grothendieck
sites, but your mileage may vary; the presentation in [26] is not quite as seve*
*re.
The basic idea behind the homotopy theory of simplicial presheaves on a
site C is that it is determined by the topology of C. Simplicial presheaves
X : Cop ! S are just diagrams of simplicial sets, and the option of choosing the
chaotic topology (or rather, no topology at all) produces one of the standard
diagramtheoretic homotopy theories. All other topologies on C are finer than
the chaotic topology, and all produce different homotopy theories.
Another observation is that, while it is true that different topologies dete*
*r
mine different homotopy theories for simplicial presheaves on a fixed category
C, the sheaves and simplicial sheaves on C are somehow beside the point, except
that they give the means of specifying weak equivalences.
The slick way of defining weak equivalences (which as far as I know is due
to Joyal [25]) starts with looking at weak equivalences of simplicial sets a li*
*ttle
differently. A simplicial set X has homotopy groups ßn(X, x), one for each
vertex x 2 X0. Collecting all of the terminal maps ßn(X, x) ! * taking values
in the one point set together (one for each x 2 X0) produces a function
G G
ßn(X, x) ! *
x2X0 x2X0
produces a group object ßnX ! X0 over X0 in the category of sets, for every
n 1. If f : X ! Y is a simplicial set map, then the induced group maps
f* : ßn(X, x) ! ßn(Y, f(x)) can be bundled together to form a morphism of
group objects which is compatible with the vertex function f : X0 ! Y0 in the
sense that the following diagram of functions commutes:
f*
ßnX _____//ßnY (1)
 
 
fflffl fflffl
X0 ___f___//Y0
The one sees that a map f : X ! Y is a weak equivalence of simplicial sets in
the usual sense if and only if
1)the induced function f* : ß0X ! ß0Y is a bijection, and
2)the square (1) is a pullback for all n 1.
5
Here of course, ß0X is the set of path components of X, which can be defined
to be the coequalizer
X1 ' X0 ! ß0X
of the face maps d0, d1 : X1 ! X0.
These constructions are completely natural, so it makes sense to talk about
the homotopy group objects ßnX ! X0 over the presheaf of vertices X0 of a
simplicial presheaf X. The presheaf of path components ß0X is also defined
by a coequalizer of the two face maps d0, d1 : X1 ! X0, but this time in the
presheaf category. Both constructions specialize to the corresponding thing for
simplicial sets in all sections.
Definition 1. A morphism f : X ! Y of simplicial presheaves on a Grothen
dieck site C is said to be a (local) weak equivalence if the following hold:
1)the induced map of sheaves f* : ~ß0X ! ~ß0Y is an isomorphism of sheaves,
and
2)the diagram of sheaf morphisms
f*
~ßnX ____//_~ßnY
 
 
fflffl fflffl
~X0__f___//_~Y
* 0
is a pullback in the sheaf category for all n 1.
Here, ~ßnX is the sheaf associated to the presheaf ßnX, for all n 0.
In the presence of stalks, this definition is equivalent (this is an exercis*
*e)
to saying that a map f : X ! Y is a local weak equivalence if and only if
all induced maps f* : Xx ! Yx of stalks are weak equivalences of ordinary
simplicial sets. There is another more exotic definition which involve Boolean
localization, which amounts to taking a fat öp int" in some category of diagrams
[18]. Alternatively, one can say that a map of simplicial presheaves is a local
weak equivalence if it induces an isomorphism in all possible sheaves of homoto*
*py
groups, for all local choices of base points: see the definition of öc mbinator*
*ial"
weak equivalence in [13, p.48] _ it's the same thing.
Example 2. The associated sheaf map j : X ! ~Xis a local weak equivalence
of simplicial sheaves, since it is locally the identity map. This is why one te*
*nds
not to single out simplicial sheaves as separate objects of study even thought
there is a perfectly good historical reason for doing so in Joyal's work [25].
Definition 3. A map i : A ! B of simplicial presheaves is said to be a cofibra
tion if the induced functions i : An(U) ! Bn(U) are one to one, for all n 0
and objects U of C.
Equivalently, a map i : A ! B is a cofibration if it is a monomorphism in
the category of simplicial presheaves on C.
6
Definition 4. A map p : X ! Y of simplicial presheaves is said to be a global
fibration if it has the right lifting property with respect to all maps of simp*
*licial
presheaves which are simultaneously cofibrations and local weak equivalences.
A simplicial presheaf X is said to be globally fibrant if the unique map X ! *
to the terminal simplicial presheaf is a global fibration.
The definition of global fibration means that if p : X ! Y sits inside a sol*
*id
arrow commutative diagram of simplicial presheaf morphisms
A _____//X>>___
____
j _____p__
fflffl_fflffl____
B _____//Y
where j is a cofibration and a local weak equivalence, then the dotted arrow
exists making the diagram commute.
These days (this first appeared in [1]), people sometimes use the term "in
jective fibration" for global fibration. The use of the term "global fibration"*
* in
[13] follows the usage of Brown and Gersten [3].
Now here's the result [13], [18] that gives the homotopy theory of simplicial
presheaves:
Theorem 5. Suppose that C is a small Grothendieck site. Then with the defi
nitions of local weak equivalences, cofibrations and global fibrations given ab*
*ove,
the category s Pre(C) satisfies the axioms for a proper closed simplicial model
category.
The original result of this type is the corresponding statement about sim
plicial sheaves, which is due to Joyal [25]. Joyal's result is a consequence of
Theorem 5 (see [13]), essentially on account of the fact that the canonical map
j : X ! ~Xrelating a simplicial presheaf X and its associated simplicial sheaf
is a local weak equivalence.
The adjective "simplicial" in the statement of the theorem means that there
is a wellbehaved notion of function space hom (X, Y ) for simplicial presheaves
X and Y . Explicitly, the nsimplices of simplicial set hom (X, Y ) are the sim*
*pli
cial presheaf maps X x n ! Y , where we have followed practice of identifying
the standard nsimplex n with its associated constant simplicial presheaf. The
term "proper" means that local weak equivalences are preserved by pullback
along global fibrations and pushout along cofibrations; this is a property that
is inherited from simplicial sets, in that it can be checked stalkwise or with a
Boolean localization argument.
As usual, the function space construction only has homotopical content when
the simplicial presheaf Y is globally fibrant; this is enough, since all simpli*
*cial
presheaves are cofibrant. When Y is globally fibrant, the usual closed simplici*
*al
model category tricks imply that there is a natural bijection
ß0hom (X, Y ) ~=[X, Y ]
7
relating the set of path components of the simplicial set hom (X, Y ) with the
set of morphisms [X, Y ] in the homotopy category Ho(s Pre(C)) associated to
the category of simplicial presheaves on C.
Suppose that K is a simplicial set, and that Y is a globally fibrant simplic*
*ial
presheaf. Then there is a natural isomorphism
hom ( *K, Y ) ~=hom (K, *Y ),
where the functor *Y (called global sections of Y ) is the right adjoint of the
constant presheaf functor, and is thus defined by
*Y = limY (U).
U2C
It follows that there are induced bijections
[ *K, Y ]s Pre(C)~=[K, *Y ]S
relating morphisms in the homotopy categories (suitably labelled) if Y is globa*
*lly
fibrant.
If the site C happens to have a terminal object t (as is almost always the
case with the sites arising in algebraic geometry), then *Y = Y (t) by formal
nonsense. In that case, one uses the homotopy category of pointed simplicial
presheaves (which exists formally, in the presence of Theorem 5) to see that
there is an isomorphism
[Sn, Y ]s Pre(C)*~=ßnY (t).
In other words, the homotopy groups of global sections of globally fibrant obje*
*cts
are isomorphic to groups of morphisms in the homotopy category of simplicial
presheaves, and are thus determined by the topology on the site C.
Suppose that A is an abelian sheaf. Then there is a sequence of natural
isomorphisms
[*, K(A, n)]~=[~Z*, K(A, n)] (simplicial abelian sheaves)
~=[~Z[0], A[n]] (sheaves of chain complexes)
(2)
~=ExtnC(~Z, A)
~=Hn(C, A).
Some comments:
1)The first of these isomorphisms relates morphisms [*, K(A, n)] in the ho
motopy category of simplicial presheaves with morphisms in the corre
sponding homotopy category of simplicial abelian presheaves (or sheaves)
(see [12],[24]); it is an adjointness relation that follows from the fact *
*that
the free simplicial abelian presheaf functor X ! ZX preserves local equiv
alences. This is trivial to verify in the presence of stalks, but more gen*
*er
ally used to be known as the Illusie conjecture; the result has been known
for a long time, and was one of the early applications of the Boolean
localization [41].
8
2)The second isomorphism in the list is a consequence of the DoldKan
correspondence, which in this case identifies morphisms in the homotopy
category of simplicial abelian presheaves with morphisms in the derived
category.
3)The identification of Extn(~Z, A) with morphisms [~Z[0], A[n]] is often
taken to be a definition of the Ext group these days. Actually proving
that it coincides with a more traditional description of Ext seems to re
quire the use of hypercovers [12].
We have seen, in other words that the standard description of the nth co
homology group Hn(C, A) of the site C with coefficients in the abelian sheaf
A coincides with a group of morphisms in the homotopy category of simplicial
presheaves. Thus, for example, if S is a scheme and ~m is the 'etale sheaf of m*
*th
roots of unity on S, then there is an isomorphism
[*, K(~m , n)]etS~=Hnet(S, ~m )
relating the 'etale cohomology of S with coefficients in ~m to morphisms in
the homotopy category of simplicial presheaves on the 'etale site etS for S.
The identification of sheaf cohomology with morphisms in simplicial presheaf
homotopy categories is wildly general: it applies universally.
There is also a convenient way, with these techniques, to capture sheaf co
homology within standard homotopy theory. We need a definition first:
Definition 6. Suppose that X is a simplicial presheaf. A globally fibrant model
for X is a local weak equivalence j : X ! Z where Z is globally fibrant.
Here's an easily proved lemma, with farreaching consequences:
Lemma 7. Suppose that f : X ! Y is a local weak equivalence of globally
fibrant simplicial presheaves. Then f is a simplicial homotopy equivalence, and
all induced maps in sections f : X(U) ! Y (U), U 2 C, are simplicial homotopy
equivalences.
Proof. The map f is a weak equivalence of objects which are both cofibrant
and fibrant, so it is a homotopy equivalence according to standard closed model
category tricks. The simplicial presheaf X x 1 is a cylinder object for a simpl*
*i
cial presheaf X, so we can assume that the homotopy equivalence is simplicial.
If the homotopy equivalence is globally simplicial then it is simplicial_in each
section. __
Corollary 8. Suppose that f : X ! Z and f0 : X ! Z0 are globally fibrant
models for a fixed simplicial presheaf X. Then Z and Z0 are simplicially homo
topy equivalent.
In other words, any two choices of globally fibrant models for a fixed sim
plicial presheaf X are homotopy equivalent in all sections. One often write
j : X ! GX for a choice of globally fibrant model for X: such things always
9
exist by the closed model axioms, and j could be a trivial cofibration (ie. a c*
*ofi
bration and a local weak equivalence) if one likes. Some culture: the "G" stands
for "global", but it can also stand for öG dement", in cases where Godement
resolution theory works [13], [38], [32].
It is now a consequence of the identifications (2) that if A is an abelian s*
*heaf
on a site C having terminal object t, and GK(A, n) is a globally fibrant model
for the EilenbergMac Lane object K(A, n) then there are isomorphisms
( ni
ßiGK(A, n)(t) ~= H (C, A) if 0 i n, and
0 otherwise.
The moral is that sheaf cohomology with coefficients in A can be recovered
from the spaces of global sections GK(A, n)(t) of the globally fibrant objects
GK(A, n).
Remark 9. It causes no pain at all to see that if B is an abelian presheaf and
GK(B, n) is a globally fibrant model for K(B, n), then there are isomorphisms
( ni
~B) if 0 i n, and
ßiGK(B, n)(t) ~= H (C,
0 otherwise.
The point is that there is a local weak equivalence K(B, n) ! K(B~, n) which
induces a homotopy equivalence GK(B, n) ! GK(B~, n), so that the map
GK(B, n)(t) ! GK(B~, n)(t) in global sections is also a homotopy equivalence.
Suppose that G is a sheaf of groups on the site C, and let BG be its associa*
*ted
classifying simplicial sheaf. There is a nonabelian analogue of the cohomology
identifications (2), in that there is a bijection
[*, BG] ~={isomorphism classes of Gtorsor over the point}*.(3)
The thing on the right is one of the standard descriptions of the classical non
abelian invariant H1(C, G). The result itself is a hypercover argument (a hyper
cover is a map which is both a öl cal fibrationä nd a local weak equivalences *
*in
the modern world [12] _ this concept will not be explained here) which makes
use of the fact that the fundamental groupoid functor preserves local weak equi*
*v
alences [16]. Insofar as the category of Gtorsors "is" the stack associated to*
* the
sheaf of groups G, this result gave the first indication that simplicial preshe*
*af
homotopy theory had something to do with stacks [22].
Here is how it came up in a first application: with the identification (3)
in hand, it is clear that if k is a field of characteristic not 2 and On is the
algebraic group of automorphisms for the trivial form of rank n, then there is
an identification of the set [*, BOn] of the set of morphisms in the homotopy
category of simpicial presheaves for the 'etale topology on Sp(k) with the set *
*of
isomorphism classes of nondegenerate symmetric bilinear forms of rank n over
k. This identification gives rise to a theory of characteristic classes for qua*
*dratic
forms over k in the mod 2 Galois cohomology of k as follows:
10
1)there is a ring isomorphism
H*et(BOn, Z=2) ~=A[HW1, . .,.HW2]
where A = H*et(k, Z=2) is the mod 2 Galois cohomology of k and the
degree of the polynomial generator HWi is i.
2)Every form ff of rank n determines a morphism [ff] : * ! BOn in the ho
motopy category of simplicial presheaves for the 'etale topology on Sp(k),
and hence determines a map
ff* : H*et(BOn, Z=2) ! H*et(k, Z=2)
taking values in the mod 2 Galois cohomology of k. The generators HWi
get mapped to elements HWi(ff) 2 Hiet(k, Z=2) which are called the higher
HasseWitt invariants of the form ff.
It is easy to see that HW1(ff) is induced by the determinant On ! Z=2, and
that HW2(ff) coincides with the classical HasseWitt invariant. The higher
HasseWitt invariants were originally defined by Delzant, but not in this form.
The foregoing will make more sense in the presence of the definition of the
cohomology of a simplicial presheaf. Explicitly, if X is a simplicial presheaf *
*on
C and A is an abelian presheaf on that site, then one defines the cohomology
group Hn(X, A) by setting
Hn(X, A) = [X, K(A, n)].
This definition specializes to all geometric cohomology theories of schemes (use
the constant simplicial presheaf associated to a scheme) and simplicial schemes
(use the simplicial presheaf represented by the simplicial scheme) [13]. In par
ticular, in the example above,
Hnet(BOn, Z=2) ~=[BOn, K(Z=2)]
where the morphisms are in the homotopy category of simplicial presheaves on
the 'etale site for k.
The definition of the cohomology groups must be held in stark contrast to
the homology sheaves of a simplicial presheaf X: if A is a presheaf of abelian
groups on C, then one defines the nth homology sheaf Hn(X, A) by
Hn(X, A) = ~Hn(ZX A),
where the object on the right is the sheaf associated to the nth homology sheaf
of the simplicial abelian presheaf ZX A.
Homology sheaves and cohomology groups are related by a universal coeffi
cients spectral sequence
Ep,q2= Extp(Hq(X, Z), A) ) Hp+q(X, A).
11
It follows from a standard comparison argument that any map f : X ! Y
which induces an isomorphism in all homology sheaves must also induce an
isomorphism cohomology groups. Similar statements apply to `torsion coeffi
cients, where ` is a prime: if f induces an isomorphism of sheaves H*(X, Z=`) ~=
H*(Y, Z=`) then f induces an isomorphism in all cohomology groups with `
torsion coefficients.
Example 10. Suppose that k is an algebraically closed field of characteristic
not equal to `, where ` is some prime number. The general linear presheaf of
groups Gl is defined by Gl = lim!Gln in the presheaf category on the smooth
'etale site (Smk)et. One interpretation of the Gabber rigidity theorem says th*
*at
the adjunction map ffl : *BGl(k) ! BGl of simplicial presheaves induces an
isomorphism
H*( *BGl(k), Z=`) ~=H*(BGl, Z=`)
in all mod ` homology sheaves for the 'etale topology. It follows that the map *
*ffl
induces cohomology isomorphisms
H*et(BGl, Z=`) ~=H*et( *BGl(k), Z=`).
On the other hand, the algebraically closed field k has trivial 'etale cohomolo*
*gy
groups, so that there are isomorphisms
H*et( *BGl(k), Z=`) ~=H*(BGl(k), Z=`).
We have therefore identified the mod ` cohomology of the discrete group Gl(k)
with 'etale cohomology H*et(BGl, Z=`), which is well known to be a polynomial
ring Z=`[c1, c2, . .].in Chern classes which is invariant of the underlying alg*
*e
braically closed field. Suslin's calculation
(
Kn(k, Z=`) ~= Z=` if n = 2i, i 0,
0 if n = 2i + 1, i 0,
follow pretty quickly.
This last example was one of the early calculational successes of the homo
topy theory of simplicial presheaves. It also illustrates an idea, namely rigid*
*ity,
which has proven to be quite robust:
1)There are comparison maps ffl : *BG(k) ! BG associated to any reduc
tive algebraic group over G, and the generalized isomorphism conjecture of
Friedlander and Milnor says that the discrete and 'etale cohomology with
mod ` coefficients for the classifying spaces of such groups G should be an
isomorphism. The rigidity program for proving the conjecture is staring
at you: show that the map *BG(k) ! BG induces an isomorphism in
mod ` 'etale homology sheaves (it's just too bad that it hasn't worked yet
outside of stable cases).
2)A rigidity argument like the one displayed above is at the heart of the
proof of the SuslinVoevodsky theorem [37] which asserts that the singular
cohomology of a scheme coincides with its 'etale or qfh cohomology if the
coefficient sheaves are finite and constant.
12
2 Presheaves of spectra
Suppose that A is a presheaf of abelian groups on C. The EilenbergMac Lane
objects K(A, n) naturally organize themselves into a presheaf of spectra, which
will be called H(A). Specifically, this structure includes the full sequence of
pointed simplicial presheaves
K(A, 0), K(A, 1), K(A, 2), . . .
and maps of pointed simplicial presheaves S1 ^ K(A, n) ! K(A, n + 1) which
are induced by shifting the underlying chain complexes, as usual. Here, S1 is
the simplicial circle 1=@ 1, identified with a constant simplicial presheaf _
Voevodsky denotes this object by S1s[40].
More generally, a presheaf of spectra X on the site C consists of pointed
simplicial presheaves Xn, n 0, and maps of pointed simplicial presheaves
oe : S1 ^ Xn ! Xn+1 which are sometimes called bonding maps. Really what
we're doing here by initiating the study of these objects is taking the Bousfie*
*ld
Friedlander model for the stable category [2], and adapting it to the presheaf
context: a spectrum, for us, will be just a sequence of pointed simplicial sets
and bonding maps displayed according to the recipe above, and the category
of (BousfieldFriedlander) spectra will be denoted by Spt. From this point of
view, a presheaf of spectra X on the site C is a functor X : Cop ! Spt.
A map f : X ! Y of presheaves of spectra is the obvious thing; it consists
of maps f : Xn ! Y n, n 0, of pointed simplicial presheaves which respect
structure in the obvious sense that the diagrams
_oe_//_n+1
S1 ^ Xn X
S1^oe f
fflffl fflffl
S1 ^ Y n_oe_//Y n+1
commute. Alternatively, f is just a natural transformation between the functors
X, Y : Cop ! Spt. I shall write Spt(C) for the category of presheaves of spectra
on the site C.
There are adjoint functors
___*//_
Spt oo___Spt(C)
*
which are induced by the global sections functor * and the constant presheaf
functor *, as the notations suggest. As in the case of simplicial presheaves,
one often confuses a spectrum with its associated constant presheaf of spectra.
Thus, for example, one writes S = *S for the constant object associated to
the sphere spectrum S. This presheaf of spectra, in all sections, consists of t*
*he
pointed simplicial sets
S0, S1, S1 ^ S1, . . .
13
with bonding maps given by identities.
As a matter of convenience, one writes Sn = S1 ^ . .^.S1 (nfold smash); it
is standard to call this object the simplicial nsphere.
The weak equivalences that define the stable homotopy theory of presheaves
are much easier to define than weak equivalences of simplicial presheaves, esse*
*n
tially because there is no ambiguity about the choice of base point. Explicitly,
a presheaf of spectra X has presheaves ßnX, n 2 Z, of stable homotopy groups,
and these presheaves have associated sheaves ~ßnX of stable homotopy groups.
Definition 11. A map f : X ! Y of presheaves of spectra on C is a (local)
stable equivalence if it induces isomorphisms ~ß*X ~=~ßY in all associated shea*
*ves
of stable homotopy groups.
Definition 12. A cofibration is a map i : A ! B of Spt(C) such that
1)the map i : A0 ! B0 is an inclusion (aka. cofibration) of simplicial
presheaves, and
2)all maps
(S1 ^ Bn) [(S1^An)An+1 ! Bn+1, n 0,
are cofibrations of simplicial presheaves.
Definition 13. A global (or stable) fibration is a map p : X ! Y of presheaves
of spectra on C which has the right lifting property with respect to all maps
which are simultaneously cofibrations and local stable equivalences.
Now here's the first main result [14]:
Theorem 14. With the definition of local stable equivalence, cofibration and
global fibration given above, the category Spt (C of presheaves of spectra on a
Grothendieck site C satisfies the axioms for a proper closed simplicial model
category.
The associated homotopy category Ho(Spt (C)) is the stable homotopy category
associated to the site C and its underlying topology.
The statement of the Theorem implies the existence of a wellbehaved func
tion space hom (X, Y ) for presheaves of spectra X and Y . This is the usual
thing: its set hom (X, Y )n of nsimplices is the set of all maps X ^ n+! Y
of presheaves of spectra, where n+is the simplcial set n with a disjoint base
point attached.
There are some things that you just get for free out of this result, along
with some knowledge of the ordinary stable homotopy category, since all maps
X ! Y of presheaves of spectra which induce stable equivalences X(U) ! Y (U)
in each section must also be local stable equivalences. Here are some of the mo*
*re
striking examples:
Example 15. Suppose given a finite list of presheaves of spectra Xi, 1 i n.
Then the canonical map
n` Yn
Xi! Xi
i=1 i=1
14
is a local stable equivalence, because the corresponding property holds for
ordinary spectra. Among other things, this means that the stable category
Ho (Spt (C)) has an additive structure.
Example 16. Fibre and cofibre sequences coincide in the stable category of
presheaves of spectra, just as fibre and cofibre sequences coincide in the ordi*
*nary
stable category. The point is that the proof of the result in the ordinary stab*
*le
category involves natural constructions, and hence passes to the category of
presheaves of spectra.
The proof of Theorem 14 involves a stabilization construction which is very
similar to the one met in real life [2], except that one has to be careful to u*
*se
globally fibrant models on the simplicial presheaf level. More explicitly, if X*
* is a
presheaf of spectra, then one first constructs a level fibrant model j : X ! Xf;
this map consists of inductively constructed globally fibrant models Xn ! Xnf
in all levels. Then there is a presheaf of spectra QXf with a natural map
j : Xf ! QXf such that the simplicial presheaf QXnfis the filtered colimit of
the inductive system
Xnf! Xn+1f! 2Xn+2f!
arising from the adjoints of the bonding maps for X. The indicated loop spaces
make homotopical sense ( Y = hom *(S1, Y ), in general and as usual) because
all of the simplicial presheaves Yfnare globally fibrant. Finally (this is impo*
*r
tant, and is a standard source of errors) a filtered colimit of globally fibrant
objects might not be globally fibrant, so we have to take a level fibrant model
of the presheaf of spectra QXf. The stabilization of X is then the composite
X ! Xf ! QXf ! (QXf)f.
One can assume that this map is natural in X, because the small object con
structions by which we construct globally fibrant models are all natural.
In fact, the composite map X ! (QXf)f is a fibrant model for X: it
induces an isomorphism in stable homotopy groups (this is obvious, because
we're stabilizing almost in a standard way in sections), and the object (QXf)f
is globally fibrant in the sense of Theorem 14.
The observation that this thing is globally fibrant is an outcome of the
proof of Theorem 14, which follows a formal script as outlined by Bousfield and
Friedlander. One of the outcomes of that proof is a formal recognition principle
for global fibrations of presheaves of spectra:
Lemma 17. A map p : X ! Y is a global fibration of presheaves of spectra if
and only if it satisfies the following two properties:
1)all constituent maps p : Xn ! Y n, n 0, are global fibrations of simplic*
*ial
presheaves, and
15
2)all diagrams of simplicial presheaf maps
Xn _____//(QXnf)f

p p*
fflffl fflffln
Y n_____//(QYf )f
are homotopy cartesian diagrams in the category of simplicial presheaves.
It's an easy exercise to draw the following consequence:
Corollary 18. A presheaf of spectra X is globally fibrant if and only if all le*
*vel
objects Xn are globally fibrant simplicial presheaves and all adjoint bonding m*
*aps
Xn ! Xn+1 are local weak equivalences.
In other words, a globally fibrant presheaf of spectra is an spectrum obje*
*ct
for the category of presheaves of spectra on the site C. It is in particular a
presheaf of spectra in the usual sense, but it has more structure arising from
the topology on C. Globally fibrant presheaves of spectra are really the central
mystery of the subject, as they encode the notion of descent.
Definition 19. A presheaf of spectra X on the site C satisfies descent if some
(and hence any) globally fibrant model j : X ! Z induces stable equivalences
j : X(U) ! Z(U), U 2 C, of ordinary spectra in all sections.
This is just an example of a very general notion: we can talk , for example,
about simplicial presheaves that satisfy descent, or presheaves of groupoids G
such that BG satisfies descent. The modern definition of stack is a presheaf of
groupoids that satisfies descent in this sense.
We care about presheaves of spectra X that satisfy descent because we can
often explicitly calculate their presheaves of stable homotopy groups ß*X, mean
ing that we can compute the groups ß*X(U) in all sections. This is usually done
with cohomological techniques that depend on phenomena such as the following
example:
Example 20. Suppose that A is an abelian sheaf on the 'etale site etS for
some scheme S. Let H(A) be the corresponding EilenbergMac Lane presheaf
of spectra, and take a globally fibrant model j : H(A) ! GH(A). In fact, to
make the construction, it is enough to take a level fibrant model H(A)f for
H(A) since we're starting with a presheaf of spectra.
By definition,
ßiGH(A)(S) = lim!ßi+nGK(A, n)(S),
n
and we know from before that there are isomorphisms
( nj
ßjGK(A, n)(S) ~= Het (S, A) if 0 j n,
0 if j > n.
16
Then the object GH(A)(S) is an spectrum by construction, so that there are
isomorphisms (
i(S, A) if i 0,
ßiGH(A)(S) ~= Het
0 if i > 0.
I chose to work on an 'etale site to make the example more real, but this
calculation works in complete generality: if B is an abelian presheaf on a site
C and GH(B) is a globally fibrant model for the corresponding EilenbergMac
Lane presheaf of spectra, then there are isomorphisms
( i
~B) if i 0,
ßi *GH(B) ~= H (C,
0 if i > 0.
The calculations just made are the basis of the construction of the descent
spectral sequence for the stable homotopy groups ß*GF of a globally fibrant
model GF of a presheaf of spectra F .
Again we'll introduce some assumptions to make the construction a little
more real: suppose that F is a presheaf of connective spectra on the 'etale site
etS of a decent scheme S. The connectivity assumption means that ~ßiF = 0
for i < 0, or that the stable homotopy group sheaves of F vanish in negative
degrees. Ordinary spectra E have natural Postnikov towers P*E [19], and so
the presheaf of spectra F has a Postnikov tower
.. .
. ..
 
 
fflffl' fflffl
P2FF_____//GP2FF
flfl 
flfl 
flffflffl'l fflffl
fflfP1F=_____//GP1F=ll
fflfll 
fl  
flfl fflffl fflffl
F _____//P0F__'__//GP0F
The Postnikov tower splits off stable homotopy groups in the usual sense, and
we have taken globally fibrant models PnF ! GPnF of all of the PnF in such
a way that the map GPn+1F ! GPnF are stable fibrations having fibres of the
form GK(ßn+1F, n + 1). It's not hard to see that the inverse limit
limGPnF
n
is globally fibrant. It, however, more difficult to conclude that the map
F ! limGPnF
n
17
is a local stable equivalence; this usually requires an assumption of a uniform
bound on 'etale cohomological dimension in all sections, but this is often met *
*in
decent geometric examples provided that the sheaves of stable homotopy groups
are torsion sheaves of some kind. If that works, the inverse limit of the tower*
* of
fibrations
. .!.GP2F (S) ! GP1F (S) ! GP0F (S)
is stably equivalent to GF (S), and a suitably reindexed BousfieldKan spectral
sequence [38], [19] has the form
Es,t2= Hset(S, ~ßtF ) ) ßtsGF (S). (4)
The convergence has to be taken with a grain of salt: typically (again) a unifo*
*rm
bound on 'etale cohomological dimension with respect to the sheaves ~ß*F is
required to make it work. The spectral sequence (4) is called the descent spect*
*ral
sequence for F , or for GF . One often sees it referred to as the 'etale cohomo*
*logical
descent spectral sequence, or as the topological descent spectral sequence.
Example 21. Suppose that K=` is the mod ` Ktheory presheaf of spectra
on the 'etale site etS, where ` is a prime which is distinct from the residue
characteristics of S. Suppose that S is otherwise well behaved as a geometric
object [38], and in particular has finite Krull dimension d. Take a stably fibr*
*ant
model j : K=` ! GK=`. Then GL=`(S) is a variant of the 'etale Ktheory
spectrum of S, and has descent spectral sequence
Es,t2= Hset(S, ~ßtK=`) ) ßtsGK=`(S).
The 'etale sheaves ~ß*K=` are known, by Suslin's calculation of the mod ` K
theory of algebraically closed fields: the sheaf ~ß2tK=` is the twist ~lt in ev*
*en
positive degrees, and is 0 elsewhere.
The LichtenbaumQuillen Conjecture can be viewed as the assertion that
the canonical homomorphism
ßiK=`(S) ! ßiGK=`(S)
is an isomorphism for i d  1 for such schemes S.
In the case where S is defined over a field F containing a primitive `th of
unity i, "the" Bott element is easily described as an element of ß2K=`(F ) that
maps to i under the surjection
ß2K=`(F ) ! Tor(Z=`, F *)
(The notation is a bit pedantic: the group ß2K=`(F ) is otherwise known as
K2(F, Z=`) and ß1K(F ) = K1(F ) ~=F *.) In more general cases, some care has
to be taken _ see the äG ng of Four" paper [5].
One can show that the groups ßiGK=`(S) are Bott periodic for i 1 [19].
Inverting multiplication by the Bott element fi gives a spectrum GK=`(S)(1=fi),
18
which is stably equivalent to the 'etale Ktheory spectrum of Dwyer and Fried
lander [4]. The canonical map GK=`(S) ! GK=`(S)(1=fi) induces an isomor
phism in stable homotopy groups in degrees i 1, under the standard as
sumptions. One might also want to be careful about the coefficients and insist
that ` 6= 2; these issues have been treated in some detail in the literature [3*
*8].
E'tale Ktheory is an example of a generalized 'etale cohomology theory, or
more broadly, of a a generalized sheaf cohomology theory. Quite generally, if F
is a presheaf of spectra on a site C, take a globally fibrant model j : F ! GF
and define
Hi(C, F ) = ßi *GF, i 2 Z.
All sheaf cohomology theories are examples of generalized sheaf cohomology
theories: in the present notation, we have already seen that there is a natural
isomorphism
Hi(C, H(A)) ~=Hi(C, ~A).
for all abelian presheaves A. This isomorphism also explains the sign change in
the defining index: the idea is to make the notation for the generalized theori*
*es
compatible with the notation for ordinary sheaf cohomology.
Other examples of these theories abound, particularly in algebraic Ktheory,
where there is a flavour of Ktheory for each of the standard geometric topolo
gies, and interesting descent theorems (or conjectures) which relate these theo
ries to algebraic Ktheory itself. In particular, we have
1)Zariski Ktheory arising from the Zariski topology, which coincides with
ordinary Ktheory for regular schemes by the BrownGersten descent the
orem [3], and
2)Nisnevich Ktheory arising from the Nisnevich (or cd) topology, which
coincides with the Ktheory of regular schemes by the Nisnevich descent
theorem [32] (an unstable version of Nisnevich descent is the starting poi*
*nt
for motivic homotopy theory [31]).
Finally, in this language, Thomason's descent theorem [38], [19] asserts that
the Bott periodic Ktheory presheaf of spectra K=`(1=fi) satisfies descent for
the 'etale topology in the same range of examples (including the primes `) for
which the LichtenbaumQuillen conjecture is believed to hold. In other words,
if you take the mod ` Ktheory presheaf of spectra K=` and formally invert
multiplication by the Bott element, you end up constructing a model for 'etale
Ktheory. The overall moral is that descent is everywhere.
I want to close this section by mentioning some other general developments
that could not be treated at length here:
1)There is a homotopy theory (ie. model structure) for the category Spt (C)
of presheaves of symmetric spectra on an arbitrary small site C, such
that the associated homotopy category is equivalent to the stable cate
gory Ho (Spt (C)) of presheaves of spectra on C [20]. The point of the
19
construction, as for the case of ordinary symmetric spectra [11], is that
the category of presheaves of symmetric spectra has an internal symmetric
monoidal smash product.
2)Suppose that S is a scheme of finite Krull dimension, and let (SmS)Nis
denote the site of smooth Sschemes, equipped with the Nisnevich topol
ogy. One can construct the unstable motivic homotopy category [31] from
the category s Pre((SmS)Nis) by formally contracting the affine line A1S
over S [9]. One can then go on to formally invert smashing with an object
T which is compact in a suitable sense (eg. T = S1, Gm , P1) to pro
duce a motivic stable category SptT((SmS)Nis) of T spectra [21]. In the
case T = P1, we obtain the motivic stable category of Morel and Voevod
sky. There is a corresponding category SptT ((SmS)Nis) of symmetric
T spectra which is defined by analogy with the constructions in [11] and
[20]. This category is a model for the motivic stable category of T spect*
*ra
if the cyclic permutation (1, 2, 3) acts trivially on the threefold smash
T ^3; examples include T = S1, P1. Again, this category of symmetric
spectrum objects has an internal symmetric monoidal smash product, so
there is a good theory of smash products for the motivic stable category.
3 Profinite groups
For our purposes, a profinite group G = {Gi} is a finite groupvalued functor
G : I ! Grp with i 7! Gi, which is defined on a small left filtered category
I and such that all morphisms i ! j of I are mapped to surjective group
homomorphisms ß : Gi! Gj.
I'll recall what it means for the index category I to be left filtered. There
are two conditions:
1)any two objects i, i0 of I have a öc mmon lower bound", meaning that
there is an object i00and morphisms
nnni77
nnn
i00OOOO
O''Oi0
2)for any two morphisms ff, ff0: i ! i0 there is a morphism e : i00! i such
that ff . e = ff0. e
As a result (and we will use this all the time) colimits of contravariant funct*
*ors
defined on I are filtered in the usual sense.
Although it's a little crime to do so, I'm going to confuse notations by wri*
*ting
G = limiGi. The main reason for assuming that the all transition homomor
phisms Gi ! Gj in the profinite group G is that all induced homomorphisms
G ! Gi are surjective. I shall write Hi for the kernel of this homomorphism,
20
which can be viewed either as a subgroup of the inverse limit or a profinite
group defined on the left filtered category I # i whose objects are all morphis*
*ms
j ! i in I.
Example 22. The central examples for us will be the Galois group G of Galois
field extensions F=K of a fixed field K. By this we really mean the collection
of all Galois groups G(L=K) of the finite Galois extensions L within F . Keep
in mind that nobody has said anything about F being separably closed.
A discrete Gset X (for a profinite group G) is a set equipped with Gaction
G x X ! X which factors through an action by one of the quotients Gi in the
sense that there is a commutative diagram
G x XG
 GGGG
 GGG
fflffl G##
Gix X _____//X
There is a corresponding category of such gadgets, consists of the discrete G
sets and all Gequivariant maps beteen them; this category will be denoted by
G  Setd.
A finite discrete Gset is a discrete Gset X which (you guessed it) happens
to be finite. We write G  Setdffor the full subcategory of G  Setd on such
finite objects. It is not hard at all to see that a finite discrete Gset X can*
* be
identified up to equivariant isomorphism with a finite disjoint union of the fo*
*rm
G
Gi=Ni.
where Ni is a subgroup (not necessarily normal) of the corresponding group
Gi. Observe that the one point set * with trivial Gaction is a member of this
category, and is terminal.
The category of finite discrete Gsets has the structure of a Grothendieck
site, where the covering families {OEi: Xi! Y } are finite lists of Gequivaria*
*nt
maps OEi: Xi! X such that the induced morphisms t Xi! Y are surjective.
A sheaf F : (G  Setdf)op ! Set is a contravariant functor (or presheaf)
such that the induced map
Y
F (t Gi=Ni) ! F (Gi)Ni
Well, a sheaf had better take finite disjoint unions to finite products, and the
indentification
F (Gi=Ni) ~=F (Gi)Ni
is a consequence of the fact that the covering morphism Gi! Gi=Nidetermines
a coequalizer
Gix Ni~= GixGi=NiGi' Gi! Gi=Ni
in the category of finite discrete Gsets.
21
Write Shv(GSet df) for the category of sheaves on the site of finite discre*
*te
Gmodules. There is an equivalence of categories
__L__//
Shv(G  Setdf)oo___ G  Mod d
R
The notation G  Mod d refers to Serre's category of discrete Gmodules: these
are sets equipped with a Gaction G x X ! X such that X is a filtered colimit
of fixed points in the sense that
X = lim!XHi.
i
Recall that Hi is defined to the kernel of the group homomorphism G ! Gi.
The functors L and R are easy to describe: if F is a sheaf then
LF = lim!F (Gi),
i
while
RX = hom G( , X)
is the functor represented on the category of Gsets by X.
Example 23. Every discrete Gset X is a discrete Gmodule, since it's fixed
by some Hi. The object therefore represents a sheaf X = hom ( , X) on the site
G  Setdfof finite discrete Gmodules. In particular, every object of G  Setdf
represents a sheaf.
Here are some remarks:
1)The topos G  Mod d of discrete Gmodules has enough points: there is
a functor
u* : G  Mod d ! Set
which is defined by forgetting the group structure, so that u*X is the set
underlying X. Colimits and finite limits are formed in the category of
discrete Gmodules as they are in the set category, so it's easy to see th*
*at
the functor u* is faithful and exact. That's fine, but the wierdness here *
*is
that there's only one stalk.
2)For a presheaf X on the site of finite discrete Gmodules, the object LX
is still defined and is a discrete Gmodule, and the canonical map X !
RL(X) can be identified with the associated sheaf map. The map itself is
a little complicated to describe, but reduces in sections corresponding to
Gi=Ni to the composite
1N
X(Gi=Ni) ! X(Gi)Ni ! lim!X(Gj)ß i ~=hom(Gi=Ni, LX).
j!i
22
3)There is an isomorphism
*F ~=LF G
for all sheaves F . Recall that the global sections *F of F is given by
taking the inverse limit of F over the objects of the underlying site. Then
one has
LF G= lim!F (Gi)G = lim!F (Gi)Gi = F (*)
The topos G  Mod d (aka. Shv (G  Setdf) is often called the classifying
topos for the profinite group G.
A well known theorem of Giraud [26], [33] asserts that if a category satisfi*
*es
a certain list of exactness properties, then it must be equivalent to the categ*
*ory
of sheaves on some Grothendieck site. Furthermore, the site itself is defined
on a generating family in a completely explicit way. The category of discrete
Gmodules for a profinite group G, and the identification with sheaves on the
site of finite discrete Gsets is a result of the constructions of Giraud's the*
*orem.
Giraud's theorem is a very useful tool for attaching explicit sites to topos*
*es.
It applies, in particular, to all flavours of categories of sets or spaces admi*
*tting
actions by a group G (topological, discrete, sheaf theoretic, etc.) _ these
categories are the classifying toposes for the corresponding group objects.
4 Generalised Galois cohomology theory
We shall continue to talk about general profinite groups G = {Gi}, as in the
previous section.
The covering map Gi! * determines a ~Cech resolution
oo___ oo___
Gi oo___Gix Gi oo___GixoGixoGi_ . . .


fflffl
*
of the terminal object * in the finite discrete Gmodule set G  Mod df. This
simplicial object of left Gmodules can be identified with the sheaftheoretic
Borel construction Gi~xGiEGi arising from the action of Gi on itself by right
multiplication. The notation Gi~xGiEGi means that this object is the sheaf
associated to the obvious presheaf GixGi EGi _ this distinction can be a very
subtle point.
Suppose that B is an abelian presheaf on G  Setdf. The cochain complex
of presheaf maps hom (GixGi EGi, B) is a cosimplicial abelian group with n
cochains Y
hom ((GixGi EGi)n, B) ~= B(Gi).
*g1*...*gn*
The simplicial presheaf maps Gj xGj EGj ! Gi xGi EGi arising from the
transition homomorphisms Gj ! Gi induce cochain complex maps
hom(GixGi EGi, B) ! hom (Gj xGj EGj, B)
23
and we define the ~Cech cohomology groups ~H*(G, B) of G with coefficients in
the abelian presheaf B by
~H*(G, B) = limH* hom(G x EG , B) ~=limH*(G , B(G ))
!Gi i Gi i !Gi i i
If A is an abelian sheaf on G  Setdf, then there are isomorphisms
Y Y
A(Gi) ~= LAHi,
*g1*...*gn* *g1*...*gn*
where (from the last section) Hiis the kernel of the canonical surjection G ! G*
*i.
It follows there are isomorphisms
lim!H* hom(GixGi EGi, A) ~=lim!H*(Gi, LAHi) ~=H*(G, LA)
i i
In other words, the ~Cech cohomology groups H~*(G, A) of G with coefficients
in the abelian sheaf A coincide up to isomorphism with the traditional Galois
cohomology groups H*(G, LA) of G with coefficients in the discrete Gmodule
LA associated to the sheaf A.
Now here's the central fact about this construction:
Lemma 24. Suppose that B is an abelian presheaf on the site G  Setdf, and
suppose that its associated sheaf ~Bis 0. Then ~H*(G, B) = 0.
Proof. A ncochain in hom (GixGi EGi, B) is a tuple (ffoe) of elements aoe2
B(Gi), where the index oe corresponds to the set of ntuples of elements in
Gi. The associated sheaf B~ is 0, so there is a covering Xoe! Gi such that
ffoe7! 0 2 B(Xoe). It follows that there is a transition homomorphism Gjoe! Gi
in the profinite group G such that ffoe7! 0 2 B(Gjoe). Pick j joefor all oe.
Then the cochain ff maps to 0 in
Y
B(Gj).
(BGopj)n
This is true for all cochains, so the filtered colimit of complexes that defines
H~*(G, B) in all degrees is 0. ___
Here's an easy corollary:
Corollary 25. Suppose that B is an abelian presheaf. Then the associated sheaf
map B ! ~Binduces an isomorphism
~H*(G, B) ~=~H*(G, ~B).
Proof. The kernel and cokernel of the map B ! ~Bare abelian presheaves whose_
associated sheaves are 0. __
24
Write H*(G, A) for the sheaf cohomology of the topos G  Mod d of dis
crete Gmodules with coefficients in an abelian sheaf A. There are, of course,
isomorphisms
H*(G, A) ~=H* *I* ~=H*I*(*)
where A ! I* is an injective resolution of the abelian sheaf A.
The following result says that sheaf and ~Cech cohomology coincide:
Proposition 26. There is an isomorphism
~H*(G, A) ~=H*(G, A)
which is natural in abelian sheaves A.
Proof. Let A ! I* be an injective resolution of A. Then the simplicial presheaf
maps GixGi EGi ! * and the injective resolution together determine natural
isomorphisms
H~*(G, A) (A)~=~H*(G, I*) (B)~=H*(G, A).
The cohomology groups ~H*(G, I*) arise from a filtered colimit of bicomplexes
Y
Ip(Gi)
*g1*...*gn*
in an obvious way. The isomorphism labelled (A) is a consequence of Lemma
24, since the cohomology presheaves H*I* satisfy H~pI* = 0 for p > 0. The
isomorphism (B) is induced by the collection of local weak equivalences
GixGi EGi! *.
One uses the fact that all functors hom ( , Ip) are exact on the sheaf category_
since the abelian sheaves Ip are injective. __
Example 27. Suppose that k is a field, and let G = Gal(ksep=k) be its absolute
Galois group, where ksepdenotes the separable closure of the field k. There is
a "site isomorphism" ~
etk =!G  Setdf
which is defined by taking a finite 'etale map U = t Sp(Li) ! Sp(k) to the
set hom k(Sp(ksep), U). The notation means that U is a finite disjoint union of
spectra of separable extensions Li=k. Any abelian sheaf A for the 'etale topolo*
*gy
on Sp(k) therefore corresponds uniquely to an abelian sheaf on the site GSetdf
of discrete finite Gsets for the Galois group G, and there is an isomorphism
H*et(k, A) ~=~H*(G, A)
which is induced by the site isomorphism and the identification of sheaf coho
mology with ~Cech cohomology given by Proposition 26.
25
These ideas have analogs for presheaves of spectra F on the site G  Setdf.
There is a stable equivalence
hom ((GixGi EGi)+ , F ) ' holimGiF (Gi),
so one is entitled to define the generalised ~Cech cohomology groups ~Hn(G, F )
with coefficients in the presheaf of spectra F by
H~n(G, F ) = limß holim F (G )
!i n  Gi i
for n 2 Z.
Here's a key point: if F is globally fibrant, then the local weak equivalence
GixGi EGi! * induces a stable equivalence
hom*((GixGi EGi)+ , F ) '!hom *(S0, F ) ~=F (*).
In particular, there is a stable equivalence
holimGiF (Gi) ' F (*).
This is the finite descent property for globally fibrant presheaves of spectra *
*on
the siteG  Setdf.
There are two consequences:
1)There is an isomorphism
~H*(G, F ) ~=H*(G, F )
if F is globally fibrant. Recall that Hn(G, F ) ~=ßnF (*) in this case.
2)If E ! F is a globally fibrant model for a presheaf of spectra E then the
induced map ~H*(G, E) ! ~H*(G, F ) can be identified with a morphism
H~*(G, E) ! H*(G, E)
from the generalised ~Cech cohomology theory associated to E to the gen
eralized sheaf cohomology theory associated to E.
Remark 28 (Warning). The map
~H*(G, E) ! H*(G, E)
relating generalised ~Cech and generalised sheaf cohomology is not known to be
an isomorphism in general. We have effectively seen that it is an isomorphism
when E is an EilenbergMac Lane spectrum object K(A, n). Both constructions
see fibre sequences so it follows that the map is an isomorphism if E has only
finitely many nontrivial presheaves of homotopy groups ... but that's it. It is
even not known that ~H*(G, E) = 0 in the presence of a local stable equivalence
E ! *
26
In general, if F is a presheaf of spectra on G  Setdfand Gi is one of the
quotients (components) of the profinite group G, then the right multiplication
maps .g : Gi! Gi by elements g 2 Gi determine an action
Gix F (Gi) ! F (Gi).
Write g = (.g)* : F (Gi) ! F (Gi). The finite collection of maps g : F (Gi) !
F (Gi) of spectra can be added up in the stable category to produce the norm
map N : F (Gi) ! F (Gi); this map is defined to be the öc mposite"
Y ' ` r
F (Gi) ! F (Gi)  F (Gi) ! F (Gi).
g2G g2G
in the stable category. The notation here is a little bit strange, although you
find it in the literature [19]: the map is not a diagonal, but is rather defi*
*ned
by the requirement that all diagrams
Q
F (Gi)_____//g2G F (Gi)
LL
LLL prg
gLLLL&&Lfflffl
F (Gi)
commute, and is therefore multiplication by the group elements in the corre
sponding factors.
If g, h are elements of Gi, then there is a commutative diagram
F (Gi)__N__//F (Gi)
g  h
fflffl fflffl
F (Gi)__N__//F (Gi)
in the stable category (actually much more care is required), and so the norm
map has a factorization
F (Gi)_____N_____//FO(Gi)O
 
 
fflffl 
holim!GiF (Gi)N//_holimGiF (Gi)
h
through a map
Nh : holim!GiF (Gi) ! holimGiF (Gi)
called the hypernorm.
Recall that the map
F (*) ! holimGiF (Gi)
27
is a stable equivalence if F is globally fibrant; in that case the composition
F (Gi) ! holim!GiF (Gi) Nh!holimGiF (Gi) ' F (*)
defines an abstract transfer map ø : F (Gi) ! F (*). This transfer map has the
usual list of good properties (including a projection formula) where it is defi*
*ned.
Assumption 29. We're now going to restrict to the cases of presheaves of
spectra F which are bounded below in the sense that the presheaves of stable
homotopy groups ßiF vanish for i below some number N, and such that the
cohomological dimension of the profinite group G with respect to all sheaves
ß~*F is bounded above by some number M. This means that Hj(H, ~ßjF ) = 0
for j > M for all closed subgroups H of G. Saying that H is closed means
in practice that H is the pullback in G of some finite subgroup of some finite
quotient Gi.
Example 30. The examples of such profinite groups to keep in mind arise from
Galois groups of fields k, when the 'etale sheaves ~ß*F are `torsion where ` i*
*s a
prime not equal to either 2 or the characteristic of k, and k has finite transc*
*en
dance degree over some field N containing a primitive `th root of unity i`, and
such that either cd`(N) < 1 (eg. N = Fp(i`) a finite field) or cdl(N(~`1)) < 1
(eg. Q(i`)). These are special examples of fields k for which Thomason's de
scent theorem for Bott periodic Ktheory holds (and is more easily described),
and for which the LichtenbaumQuillen conjecture should hold.
Under these assumptions, all of the descent machinery works:
1)If F is a globally fibrant presheaf of spectra on G  Setdf the Galois
cohomological descent spectral sequence
Es,t2= Hs(G, ~ßtF ) ) ßtsF (*) = Hst(G, F )
converges. If G is the absolute Galois group of one of the fields in the l*
*ist
above, then this spectral sequence has the form
Es,t2= Hset(k, ~ßtF ) ) ßtsF (k) = Hstet(k, F ).
2)Suppose that F ! PnF is a Postnikov section of F , and consider the
composite
F ! PnF j!GPnF
where j is a globally fibrant model. The presheaf of spectra GPnF has
only finitely many nontrivial presheaves of stable homotopy groups on
account of the global bound on cohomological dimension, and the fibre
of the composite map F ! GPnF has presheaves of stable homotopy
groups which are 0 below some fixed integer, in all sections. In fact, one
can show that the presheaf maps ßiF ! ßiGPnF are isomorphisms for
i < n  M. This gives a technique for approximating the presheaves of
stable homotopy groups of F on presheaves of spectra having only finitely
many nontrivial presheaves of stable homotopy groups.
28
3)Suppose that E is a presheaf of spectra whose presheaves of stable ho
motopy groups ß*E satisfy the assumptions above. The nth Postnikov
section E ! PnE induces a comparison diagram
~Hi(G, E)____//~Hi(G, PnE)
 
 '
fflffl fflffl
Hi(G, E)_____//Hi(G, PnE)
We have just seen that the map Hi(G, E) ! Hi(G, PnE) is an isomorphism
for i > M n. It follows that ~Hi(G, PnE) is isomorphic to Hi(G, E) in the
same range. (One should very quickly revert to interpreting this in terms
of actual stable homotopy groups of spectra, because the sign change is
much too confusing).
Now we're in position to believe a pretty good result:
Theorem 31 (Tate Theorem). Suppose that F is a globally fibrant presheaf of
spectra on the site GSet df, and suppose that the presheaves of stable homotopy
groups of F are bounded below and that G has finite cohomological dimension
with respect to all sheaves ~ß*F of F . The the hypernorm map
Nh : holim!GiF (Gi) ! holimGiF (Gi) ' F (*)
is a stable equivalence for all finite quotients Gi of G.
The Tate Theorem is due to Thomason [38], but appeared for the first time
in [19] in its present form. Its proof of is an inductive argument that starts *
*with
the Tate lemma that asserts the norm map induces an isomorphism
H0(Gi, A(Gi)) ~=H0(Gi, A(Gi))
if A is a discrete abelian Gmodule of cohomological dimension 0 [34]. This
gives the case of the Theorem for presheaves of spectra GH(A) associated to
such sheaves A. Both sides of the comparison respect fibre sequences, allowing
one to prove the case corresponding to presheaves of spectra GK(B, n) where
G has bounded cohomological dimension with respect to the abelian sheaf ~B.
A second induction involving fibre sequences allows one to verify the statement
for all globally fibrant models GPnF of finite Postnikov sections, and then one
finishes the proof by using the approximation technique given in item 2) above.
In the cases to which it applies, the Tate Theorem give rise to the Tate
spectral sequence
Ep,q2= Hp(Gi, ßqF (Gi)) ) ßp+qF (*)
Thus, for the sort of base field k listed in Example 30, if L=k is a finite *
*Galois
extension with Galois group G and F is a globally fibrant presheaf of spectra on
29
the 'etale site etk which satisfies the usual assumptions, then the Tate spect*
*ral
sequence has the form
Ep,q2= Hp(G, ßqF (L)) ) ßp+qF (k) (5)
We shall switch to this context for the remainder of this paper.
5 Thomason's descent theorem
The Tate spectral sequence (5) and low dimensional calculations are the basis
for the following result of Thomason. For the statement, we need to know that
if L=k is a finite Galois extension of a field k with Galois group G, then the
Ktheory transfer map i : K=`(L) ! K=`(k) is Gequivariant for the obvious
action of G on K=`(L) and the trivial action on K=`(k). It therefore induces a
map of spectra
ih : holim!GK=`(L) ! K=`(k)
which is called the hypertransfer.
Theorem 32. Suppose that k is a field satisfying the list of assumptions in Ex
ample 30, and suppose that there is a Galois extension N=k such that cd`(N) 1
and cd`(H) 1 where H = Gal(N=k). Suppose that L=k is a finite Galois
subextension of N=k with Galois group G. Then there is an element
Ind(fi) 2 ß2holim!GK=`(L)
such that the homomorphism ih* : ß2holim!GK=`(L) ! ß2K=`(k) induced by
the hypertransfer maps Ind(fi) to the Bott element fi 2 ß2K=`(k).
The element Ind(fi) is called an inductor for the Bott element fi. The punch
line in the proof of Theorem 32 is the Tate Theorem for the globally fibrant
model of the suspended Moore spectrum S[2]=`.
Now, in the same generality as the Theorem 32:
1)The Bott periodic Ktheory presheaf of spectra K=`(1=fi) and its globally
fibrant model GK=`(1=fi) are both modules over the mod ` Ktheory
presheaf K=`. In particular the composite
K=`(1=fi)(L) ^ K=`(L) [!K=`(1=fi)(L) i*!K=`(1=fi)(k)
has an adjoint
c : K=`(1=fi)(L) ! Hom (K=`(L), K=`(1=fi)(k))
30
taking values in a function spectrum. Taking homotopy inverse limits for
the Gaction induces a map c* which fits into the picture
holimGK=`(1=fi)(L)c*//_holimGHom(K=`(L),OK=`(1=fi)(k))O

 ~=
 fflffl
i* Hom (holim!GK=`(L), K=`(1=fi)(k))

 Ind(fi)*
 fflffl
K=`(1=fi)(k)____________//______________________Hom(S[2], K=`(1=fi*
*)(k))
The indicated dotted arrow map is induced by multiplication by the Bott
element, by a projection formula argument. A similar picture exists for
the globally fibrant model GK=`(1=fi), and the two may be compared.
2)Consider the resulting comparison diagram
*
K=`(1=fi)(k)_i___//_holimGK=`(1=fi)(L)_//_ 2K=`(1=fi)(k)
j j* j
fflffl ' fflffl fflffl
GK=`(1=fi)(k)_i*__//holimGGK=`(1=fi)(L)//_ 2GK=`(1=fi)(k)
(6)
The vertical maps arise from the choice j : K=`(1=fi) ! GK=`(1=fi) of
globally fibrant model, the maps labelled i* are canonical maps into the
respective homotopy inverse limits, and both horizontal composites are
induced by multiplication by the Bott element.
Actually, this is all a bit of a lie: all of the objects in the diagram (6) are
supposed to be constructed from finite Postnikov sections of the presheaf of
spectra K=`(1=fi) so that the following "~Cech descentä rguments work _ the
concept of approximation by finite Postnikov sections was invented to handle
exactly this point (and Thomason used to say that this was the hardest thing
in his proof) _ but the display in front of you is already formidable enough.
3)We have a diagram (6) for all finite subextensions L=k of N=k, and
the idea is that the map j : K=`(1=fi) ! GK=`(1=fi) should induce
an isomorphism in generalized ~Cech cohomology theory for the Galois
group of N=k (this is the thing that requires a finite Postnikov section
instead). Then any element ff 2 ß*GK=`(1=fi)(k) lifts to an element of
ß*holimGK=`(1=fi)(L) for some extension L, so that fi [ff is in the ima*
*ge
of j* : ß* 2K=`(1=fi)(k) ! ß* 2GK=`(1=fi)(k). Similarly, if j*(fl) = 0 in
ß*GK=`(1=fi)(k) then i*(fl) = 0 in ß2holimGK=`(1=fi)(L) for some L, and
so fi[ff = 0. Multiplication by fi has been inverted on K=`(1=fi) and on i*
*ts
globally fibrant model, so it follows that j : K=`(1=fi)(k) ! GK=`(1=fi)(k)
is a stable equivalence. This is also true if k is replaced by any of its *
*finite
31
extensions inside the field N, because the same assumptions are at work.
One then can replace k by N, by a continuity argument.
4)The previous step effectively finishes the case of relative Galois cohomo
logical dimension 1. One can show that the map j : K=`(1=fi)(k) !
GK=`(1=fi)(k) for all fields satisfying the standard assumption by induct
ing up through a TateTsen filtration [19], [38], meaning via an induction
through steps of relative Galois cohomological dimension one.
These have been, grossly speaking, the steps in proving Thomason's descent
theorem for Bott periodic Ktheory for good fields (and at good primes). The
version of Thomason's theorem that is given in [19, Thm 7.31] asserts the fol
lowing:
Theorem 33. Suppose that ` is a prime such that ` > 3. Suppose that X is
a scheme which is separated, Noetherian and regular, of finite Krull dimension,
and suppose that the ring (X, OX ) of functions on X contains 1=`. Suppose
that each residue field k(x) of X is of finite transcendance degree over some f*
*ield
kx such that cdl(kx) 1 or cd`(kx(~l1) 1. Then the map
K=`(1=fi)(X) ! GK=`(1=fi)(X)
is a stable equivalence.
The assumptions on the residue fields mean that the statement of Thoma
son's theorem holds for those fields: the requirement that the respective fields
contain a primitive `throot of unity disappears by a trick. The important thing
is the regularity assumption, which means that we are in the realm where the
Nisnevich descent theorem holds.
Here's how to finish the proof. Take a globally fibrant model
j : K=`(1=fi) ! GK=`(1=fi)
with respect to the 'etale topology on smooth schemes over X. This is a map of
presheaves of spectra, but now interpret it in the stable category associated to
the Nisnevich topology. The presheaf of spectra GK=`(1=fi) is globally fibrant
with respect to the Nisnevich topology, since direct image functors preserve
global fibrations (this is a ubiquitous fact, which first appeared in [15]). At*
* the
same time, by working in stalks at points x on smooth schemes Y=X, one finds
a commutative diagram
K=`(1=fi)(Ohx,Y)j*_//GK=`(1=fi)(Ohx,Y)
'  '
fflffl ' fflffl
K=`(1=fi)(k(x))_j*_//_gK=`(1=fi)(k(x))
where Ohx,Yis the henselization of the local ring O)x,Y and the vertical maps
are induced by the residue homomorphism Ohx,Y! k(x). The residue maps
32
are both stable equivalences, essentially by Gabber rigidity, and the bottom
horizontal map is a stable equivalence by the Thomason theorem for fields.
It follows that the map j : K=`(1=fi) ! GK=`(1=fi) is a local stable equiv
alence and hence a globally fibrant model for the Nisnevich topology. At the
same time, the Nisnevich descent theorem implies that the presheaf of spectra
K=`(1=fi) satisfies Nisnevich descent, and is therefore sectionwise stably equi*
*v
alent to any globally fibrant model for that topology. The map j is therefore a
stable equivalence in all sections.
References
[1]B. Blander, Local projective model structures on simplicial presheaves, KT*
*heory 24(3)
(2001), 283301.
[2]A.K. Bousfield and E.M. Friedlander, Homotopy theory of spaces, spectra, *
*and bisim
plicial sets, Springer Lecture Notes in Math. 658 (1978), 80150.
[3]K.S. Brown and S.M. Gersten, Algebraic Ktheory as generalized sheaf cohomo*
*logy,
Springer Lecture Notes in Math. 341 SpringerVerlag, BerlinHeidelbergNew Y*
*orkTokyo
(1973), 266292.
[4]W. Dwyer and E. Friedlander, Algebraic and 'etale Ktheory, Trans. AMS 292 *
*(1985),
247280.
[5]W. Dwyer, E. Friedlander, V. Snaith and R. Thomason, Algebraic Ktheory eve*
*ntually
surjects onto topological Ktheory, Invent. math. 66 (1982), 481491.
[6]O. Gabber, Ktheory of Henselian local rings and Henselian pairs, Contemp. *
*Math. 126
(1992), 5970.
[7]H. Gillet and R.W. Thomason, The Ktheory of strict hensel local rings and *
*a theorem
of Suslin, J. Pure Applied Algebra 34 (1984), 241254.
[8]P.G. Goerss, Homotopy fixed points for Galois groups, Contemp. Math. 181 (1*
*995), 187
224.
[9]P.G. Goerss and J.F. Jardine, Localization theories for simplicial presheav*
*es, Can. J.
Math. 50(5) (1998), 10481089.
[10]P.G. Goerss and J.F. Jardine, Simplicial Homotopy Theory, Progress in Mathe*
*matics,
174, Birkhäuser, BaselBostonBerlin (1999).
[11]M. Hovey, B. Shipley and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 1*
*3 (2000),
149208.
[12]J.F. Jardine, Simplicial objects in a Grothendieck topos, Contemp. Math 55 *
*(1986), 193
239.
[13]J.F. Jardine, Simplicial presheaves, J. Pure Applied Algebra 47 (1987), 35*
*87.
[14]J.F. Jardine, Stable homotopy of simplicial presheaves, Can. J. Math. 39 (1*
*987), 733747.
[15]J.F. Jardine, The Leray spectral sequence, J. Pure Applied Algebra 61 (1989*
*), 189196.
[16]J.F. Jardine, Universal HasseWitt classes, Contemporary Math. 83 (1989), 8*
*3100.
[17]J.F. Jardine< The LichtenbaumQuillen conjecture for fields, Canad. Math. B*
*ull. 36
(1993), 426441.
[18]J.F. Jardine, Boolean localization, in practice, Doc. Math. 1 (1996), 2452*
*75.
[19]J.F. Jardine, Generalized Etale Cohomology Theories, Progress in Math. 146,*
* Birkhäuser,
BaselBostonBerlin (1997).
[20]J.F. Jardine, Presheaves of symmetric spectra, J. Pure App. Alg. 150 (2000)*
* 137154.
[21]J.F. Jardine, Motivic symmetric spectra, Doc. Math. 5 (2000), 445552.
33
[22]J.F. Jardine, Stacks and the homotopy theory of simplicial sheaves, Homolog*
*y, Homotopy
and Applications 3(2) (2001), 361384.
[23]J.F. Jardine, The separable transfer, Preprint (2000), to appear in J. Pure*
* Applied Al
gebra.
[24]J.F. Jardine, Presheaves of chain complexes, Preprint (2001).
[25]A. Joyal, Letter to A. Grothendieck (1984).
[26]S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introdu*
*ction to
Topos Theory, SpringerVerlag, BerlinHeidelbergNew York (1992).
[27]A.S. Merkurjev and A.A. Suslin, Kcohomology of SeveriBrauer varieties and*
* the norm
residue homomorphism, Math. USSR Izvestiya 21 (1983), 307340.
[28]A.S. Merkurjev and A.A. Suslin, On the norm residue homomorphism of degree *
*3, Math.
USSR Izvestiya 36 (1991), 349367.
[29]J.S. Milne, 'Etale Cohomology, Princeton University Press, Princeton (1980).
[30]S. Mitchell, Hypercohomology spectra and Thomason's descent theorem, Algebr*
*aic K
theory, Fields Institute Communications 16, AMS (1997), 221278.
[31]F. Morel and V. Voevodsky, A1 homotopy theory of schemes, Publ. Math. IHES *
*90
(1999), 45143.
[32]Y.A. Nisnevich, The completely decomposed topology on schemes and associate*
*d descent
spectral sequences in algebraic Ktheory, Algebraic Ktheory: Connections wi*
*th Geometry
and Topology, NATO ASI Series C 279, Kluwer, Dordrecht (1989), 241342.
[33]H. Schubert, Categories, SpringerVerlag, New YorkHeidelbergBerlin (1972).
[34]JP. Serre, Cohomologie Galoisienne, Springer Lecture Notes in Math. 5 (4th*
* edition),
SpringerVerlag, BerlinHeidelbergNew YorkTokyo (1973).
[35]A.A. Suslin, On the Ktheory of algebraically closed fields, Invent. Math. *
*73 (1983),
241245.
[36]A.A. Suslin, On the Ktheory of local fields, J. Pure Applied Algebra 34 (1*
*984), 301318.
[37]A. Suslin and V. Voevodsky, Singular homology of abstract algebraic varieti*
*es, Invent.
math. 123 (1996), 61103.
[38]R.W. Thomason, Algebraic Ktheory and 'etale cohomology, Ann. Scient. 'Ec. *
*Norm. Sup.
4e s'erie 18 (1985), 437552.
[39]R.W. Thomason and T. Trobaugh, Higher algebraic Ktheory of schemes and of *
*de
rived categories, The Grothendieck Festschrift, Volume III, Progress in Math*
*ematics 88,
Birkhäuser, BostonBaselBerlin (1990), 247436.
[40]V. Voevodsky, A1Homotopy theory, Doc. Math. Extra Vol. ICM I (1998), 5796*
*04.
[41]D.H. Van Osdol, Simplicial homotopy in an exact category, Amer. J. Math. 99*
*(6) (1977),
11931204.
34