Hypercohomology Spectra and
Thomason's Descent Theorem
Stephen A. Mitchell*
Revised October 1996
Dedicated to the memory of Bob Thomason
The celebrated LichtenbaumQuillen conjectures predict that for a suffi
ciently nice scheme and given prime `, the `adic algebraic Kgroups of X are
closely related to the `adicetale cohomology groups of X. More precisely,
one version of the conjectures asserts that there is a descent spectral sequen*
*ce
of AtiyahHirzebruch type
q ^
Hpet(X; Z`(__)) ) ssqp(KX )
2 `
but with the convergence only valid in sufficiently high degrees. Here the coe*
*f
ficient sheaves are Tate twists of the `adic integers, and are to be interpre*
*ted
as zero if q is odd. Throughout this paper,etale cohomology is continuous
etale cohomology [19], and the indicated abutment of the spectral sequence
consists of the homotopy groups of the Bousfield `adic completion of the
spectrum KX, not the naive `adic completion of the Kgroups.
In a remarkable paper [42], Thomason proved the LichtenbaumQuillen
conjectures for a certain localized form of Ktheory  socalled "Bottperiodi*
*c"
Ktheory. The first step was the development of an elaborate theory of
hypercohomology spectra H .(X; E) associated toetale presheaves of spectra
E  or more generally, to presheaves of spectra on a Grothendieck site.
These hypercohomology spectra are by their very construction naturally
______________________________
*Supported by a grant from the National Science Foundation
1
equipped with a suitable descent spectral sequence, and there is a natu
ral map E(X) ! H.(X; E). In particular, the Ktheory of a scheme X maps
to its associated hypercohomology spectrum H.et(X; K).
Fix a prime `, which we will assume is odd in order to simplify the
discussion.Let L() denote Bousfield localization with respect to complex
Ktheory, and let ^L() denote its `adic completion. The main theorem of
[42] can be stated as follows.
Theorem 0.1 Let X be a separated noetherian regular scheme of finite Krull
dimension, with sufficiently nice residue fields of characteristic different fr*
*om
`. Then the natural map KX ! H.et(X; K) induces a weak equivalence
^LK(X) ~=!^LH.et(X; K)
and there is a convergent descent spectral sequence
q
Hpet(X; Z`(__)) ) ssqp^LKX
2
For a precise statement see Theorem 4.13. Several remarks are in order:
o Usually the theorem is stated in terms of Bottperiodic Ktheory. There
are nonnilpotent elements
fin 2 ss2(`1)`n1KZ ^ MZ =`n
that ultimately arise from a primitive `nth root of unity (MZ =`n is
the mod `nMoore spectrum). Using the KZ algebra structure on KX,
we can form the mapping telescope fi1nKX ^ MZ =`n, and it is known
that this mapping telescope is weak equivalent to ^LKX ^MZ =`n. Thus
one gets a spectral sequence going from mod `n cohomology to mod `n
Bottperiodic Ktheory. In this paper, however, the emphasis will be
on Bousfield localization, with the Bott elements entering only as tools
in the proofs. For more information on Bott elements, see [31] and the
references cited there, as well as the recent paper by B. Kahn [24].
o The localization functor L^ is only the first of an infinite sequence
of localization functors L^n defined on spectra. Thus from a global
2
homotopytheoretic perspective, the localization imposed by Thoma
son is a priori rather drastic. However the author showed in [30] that
Theorem 0.1 remains true with ^Lreplaced by ^Lnfor any n, including
the "harmonic localization" ^L1, so this is not such a drastic localization
after all. For further discussion of this point, see [31].
o The second statement of the theorem is not merely a corollary of the
first. The difficulty is with the identification of the E2term; nowadays
this depends on the Gabber rigidity theorem for hensel local rings and
Suslin's theorem on algebraically closed fields. When Thomason first
proved Theorem 0.1, however, these results were not available and he
had to prove them in localized form.
o As a corollary to another major work [44], Thomason showed that
if Ktheory is replaced by Bass Ktheory, the assumption that X is
separated and regular can be dropped. This more general result will
not be considered here.
Thomason's theorem has many interesting applications. For example, let
F be a totally real number field with ring of integers O F. Let LKn() =
ssnLK(). Then up to powers of 2 the value of the Dedekind zeta function
at the odd negative integers is given by the amazing formula
 LK4n2O F 
iF (1  2n) = _____________
 LK4n1O F 
The point is that the descent spectral sequence collapses, and the theorem
of Wiles [46] proving Iwasawa's "Main Conjecture" computes the given zeta
values in terms ofetale cohomology.
Another application, due to Bill Dwyer and the author [9], uses Thoma
son's theorem to compute the `adic complex Ktheory of the spectrum
KO F [_1`] in terms of the Iwasawa theory of the cyclotomic Z`extension (F an
arbitary number field). In the case of abelian fields, this leads to a computa
tion of the complex Ktheory of the space BGLO F[_1`]+ . This work also yields
an explicit conjectural description of the Hopf algebra H*(BGLO F[_1`]+ ; Z=`),
and leads to some partial computations in [32] and [33].
Further applications and motivation can be found in [42], [43], and [31].
The purpose of the present paper, however, is to explain the proof of Thoma
son's Theorem, and the vast array of machinery underlying it. It took the
3
author rather a long time to progress from "thrill seeker" to "reckless cheat"
to "honest man" (Thomason's amusing terminology; see [42], introduction).
Our hope is that this essay will smooth the way for others. At the same
time, we will survey the work of Jardine [20] [21] [22] that recasts the entire
theory in terms of closed model categories. In this approach, the role of the
hypercohomology presheaf H .(; E) is played by a globally fibrant model for
E. Here again, the technical prerequisites are formidable; our goal is to help
make these works more accessible.
The paper is organized as follows. Each of the five sections has its own
introduction. The first two sections are intended for readers who may not be
familiar with schemes and sites. Section 1 is an introduction to Thomason's
theorem for fields, eschewing all schemetheoretic language to minimize the
prerequisites. Section 2 provides a short introduction to sites and cohomol
ogy, and includes a discussion of the very useful Nisnevich site.
The experts can proceed directly to the third section. Here we introduce
presheaves of spectra on a Grothendieck site, and the associated hyperco
homology spectrum. The latter construction is a direct generalization to
spectra of the Godement resolution of an abelian sheaf. Then we consider
Jardine's closed model category structure for presheaves of spectra on a site.
Section 4 begins by discussing the descent theorems of BrownGersten
and Nisnevich. Then the Nisnevich descent theorem is used to deduce The
orem 0.1 for schemes from Theorem 0.1 for fields. Section 5 sketches the
ingenious proof for fields.
Acknowledgements: I would especially like to thank Rick Jardine for
conversations and correspondence, particularly concerning his book [22], on
which much of this paper is based. Thanks are also due to Paul Goerss, for
sharing his expertise on closed model categories and related matters, and to
Bill Dwyer for helpful conversations on the material of Section 5.
Notation, terminology, and a confession of sins. The mod n Moore
spectrum is denoted MZ =n. For a spectrum E, the `adic completion (=Bous
field localization with respect to MZ =`) is denoted E^. We write LE for Bous
field localization with respect to complex Ktheory, and abbreviate (LE )^ as
^LE. Recall also that ^L() is the same thing as Bousfield localization with
respect to mod ` complex Ktheory. Bear in mind that L commutes with
pseudofiltered (e.g. direct) colimits, but ^L does not. Readers unfamiliar
4
with Bousfield localization might consult [31].
If S is a topological space, possibly discrete, Map(S; X) denotes the space
(spectrum) of pointed maps to the pointed space (spectrum) X.
If A is a presheaf of sets or abelian groups, we usually write aA for the
sheafification of A, but when A = ssqE is a presheaf of homotopy groups
attached to a presheaf of spectra, we follow a common practice and write
"ssqEfor assqE.
The Ktheory spectrum of the category of locally free sheaves on a scheme
X is denoted KX. The Ktheory spectrum attached to the coherent sheaves
is denoted K0X, since the alternate notation GX would just about exhaust
the already overworked letter G.
Now, as to our sins: Among the many technical problems that arise in
the theory discussed here, there are several that we have chosen to ignore.
Let us at least be explicit about what they are. First, there are the usual
settheoretic difficulties  for example, many of our categorical definitions
only make sense if the category in question is small. Occasionally hypotheses
of this sort are made explicitly, but more often it is left to the reader to
make the necessary adjustments. Second, there is the problem that some of
the "functors" lurking in the background are in fact only pseudofunctors 
for example, the "functor" that assigns to each ring its category of finitely
generated projective modules. In addition, the passage from schemes to
Ktheory spectra is left inside a black box  particularly when it comes to
questions involving the multiplicative structure. For a careful and rigorous
treatment of all these matters, see [22].
Contents
1. Thomason's theorem for fields. Homotopy fixed point sets. Galois
cohomology. A provisional definition of hypercohomology spectra for fields.
Statement of Thomason's descent theorem for fields.
2. Schemes, sites and cohomology. Schemes,etale morphisms, and
hensel local rings. Grothendieck sites: theetale, Nisnevich and Zariski sites.
Presheaves and sheaves. Points and stalks. Sheaf cohomology and cohomo
logical dimension. C ech cohomology. Noetherian sites.
5
3. Hypercohomology spectra. Spectral sequences, closed model cate
gories, filtered colimits. Hypercohomology: Thomason's Godement construc
tion for spectra. Jardine's closed model category structure for presheaves of
spectra. C ech hypercohomology spectra.
4. Descent theorems in Ktheory. The BrownGersten theorem (Zariski
descent for Ktheory). Nisnevich descent. Gabber rigidity and the Ktheory
sheaf on theetale site. Thomason's descent theorem and the proof for
schemes.
5. The proof of Thomason's theorem for fields. Hyperhomology spec
tra. Norm and hypernorm. The analogue for spectra of a theorem of Tate.
Transfer and hypertransfer. The proof, at last.
1 Thomason's Theorem for Fields
Let F be a field, ` a prime not equal to the characteristic of F. One of the
fundamental problems of algebraic Ktheory is to compute the `adic K
theory of F, where by "`adic Ktheory" we mean the homotopy groups of
the `adic completion KF ^. The following beautiful theorem was proved by
Suslin in 1984 [38] [39].
Theorem 1.1 Let F be a separably closed field, ` a prime not equal to the
characteristic of F. Then
ss2nKF ^~= Z`(n)
for n 0, and
ssoddKF ^= 0. In fact there is a weak equivalence KF ^~= bu^, where bu is
the 1connected cover of the complex Ktheory spectrum.
Here Z`(n) denotes the nth Tate twist of the `adic integers. In other
words,
Z`(n) = nZ` (1)
where Z`(1) denotes the inverse limit of the groups ` (F ).
6
Theorem 1.1 was originally stated for algebraically closed fields, but the
extension to the separably closed case is immediate because `adic Ktheory
(with `6= char F ) is invariant under purely inseparable extension ([35],7,4.8).
We also note that there is some choice involved in the weak equivalence
KF ^~= bu^. It is defined by a diagram
KF ^ KE^ ! bu^
Here E is the algebraic closure of the prime subfield, and the first map is just
the canonical inclusion. This map is a weak equivalence [38]. If Char F=0,
the second map arises from a choice of embedding of E into the complex
numbers. If Char F=p, it arises in effect from a choice of Brauer lift (or
equivalently, by a choice of embedding of the Witt vectors of E into C).
Given Suslin's theorem, it is natural to try to compute the Ktheory of
F by "descent" from its separable closure. In other words, one wants some
kind of spectral sequence whose input is ss*KF ^ as Galois module and whose
output is ss*KF ^. The way to approach such a problem is to cook up an
auxiliary spectrum Y which by its very construction admits such a spectral
sequence, and which is equipped with a natural map KF ^! Y . Then one
tries to show this map is a weak equivalence. In fact, however, life is not
so simple; it turns out that the best one can hope for is a weak equivalence
on some connected cover. All this will be discussed further once we have
Thomason's descent machinery in place. We will first try to motivate this
machinery via the prototypical example of homotopy fixed point sets.
1.1 Homotopy fixed point sets
Let G be a discrete group, X a Gspectrum. Here we simply mean that
each space Xn of the spectrum is a pointed Gspace, and the structure maps
Xn ! Xn+1 are Gmaps (with G acting trivially on the loop coordinate).
The homotopy fixedpoint spectrum is
XhG = MapG (EG+ ; X)
Here the nth space of the spectrum on the right is the function space of
equivariant pointed maps MapG (EG+ ; Xn). The structure maps are the
usual, obvious ones. Filtering EG by skeleta exhibits XhG as the inverse
limit of a tower of fibrations
7
:::  MapG ((EG)n1+; X)  MapG ((EG)n+; X)  :::
Q n
in which the fibres have the form X. This yields a "conditionally conver
gent" right halfplane cohomology spectral sequence  the "descent spectral
sequence"
Ep;q2= Hp(G; ssqX) =) ssqpX
See section 3.1 for a discussion of convergence. Here it will suffice to note
that any of the following conditions will guarantee convergence:
o G has finite cohomological dimension  or more generally, there exists
a positive integer d such that Hp(G; ssqX) = 0 for all p > d and all q.
In this case the spectral sequence collapses at Ed+1.
o The spectrum X is bounded above  that is, there exists an integer M
such that ssqX = 0 for all q > M. In this case the spectral sequence is
formally the same as a first quadrant spectral sequence.
o The groups Hp(G; ssqX) are all finite.
There is a natural map XG ! XhG defined as the composite
XG = MapG (S0; X) ! MapG (EG+ ; X)
induced by the augmentation EG+ ! S0. If this map is a weak equivalence,
then the descent spectral sequence converges to the homotopy of the actual
fixed point set.
There is another way to describe XhG that will be essential later. Recall
that EG+ is the geometric realization of a certain simplicial pointed Gset
E.G+ whose nth level is Gn+1+. Thinking of E.G+ as a simplicial pointed
discrete space, we can apply the functor MapG (; X) to obtain a cosimplicial
spectrum that we will denote S.(G; X). Then
XhG = T ot S.(G; X)
where "Tot" is the BousfieldKan total space construction [6] applied level
wise. It is shown in [6] that this Tot is the inverse limit of a tower of fibra*
*tions
8
 and indeed it is just the same tower constructed above. Moreover there is
a weak equivalence
XhG ~= holim S.(G; X)
and it is this holim construction that we will actually use.
Roughly speaking, we want to apply this machinery to the case X = KF ,
where F is the separable closure of F, with G = GF the absolute Galois group.
However we need to modify the homotopy fixed point construction in two
ways. First, G is a profinite group and we want to take into account this
profinite structure. In particular we want a spectral sequence involving Galois
cohomology.1 Secondly, we will change our point of view by replacing the
GF spectrum KF by the "presheaf of spectra" L 7! KL, L a finite separable
extension of F (for an elegant treatment of homotopy fixed point spaces for
Galois groups, see [15]). We first digress to discuss Galois cohomology.
1.2 Galois cohomology
Let G be a profinite group.2 This is equivalent to saying that G is a compact
and totally disconnected topological group, but in more practical terms it
means that G is the inverse limit of its finite quotients G=U, with U varying
over the closed normal subgroups of finite index (note that the closed sub
groups of finite index are precisely the open subgroups). A discrete Gmodule
M is a Gmodule such that M is the union of the various fixed submodules
MU . The cohomology groups H*(G; M) can be defined in two equivalent
ways.
o "Sheaf": The category of discrete Gmodules is abelian and has enough
injectives. The functors H*(G; M) are the derived functors of the left
exact functor M! MG .
______________________________
1If GF is finite no modifications are needed. However fields with nontrivial*
* but finite
absolute Galois group are extremely rare  the example of the real numbers notw*
*ithstand
ing! See [3],X, Theorem 2.1 for a discussion of this claim.
2Logically, the cohomology to be discussed here should be called "profinite *
*group co
homology". However it turns out that every profinite group is the Galois group *
*of some
Galois extension [45], so the theories of Galois extensions and profinite group*
*s are ab
stractly equivalent.
9
o "Cech": H*(G; M) = colimU H*(G=U; MU ). Here the terms in the
colimit are just ordinary group cohomology. The maps in the colimit
combine the inclusions MU MV with pullback along the quotients
G=V ! G=U in the obvious way.
The reason for the slogans "sheaf" and "Cech" may not be apparent, but
should become clear in the next section.
Here is one way to see that the two definitions agree. We include this
here because it helps to illuminate some of the spectrumlevel constructions
that come later:
Form the simplicial Gset E.G as usual. In this case it is a simplicial
space with the profinite topology on the terms Gn+1. Apply the functor
Map(; M)  continuous Mvalued functions  to obtain a cosimplicial G
module Map(E.G; M). Here the Gaction is the usual left action whose
fixed points yield the group of equivariant maps. The natural augmentation
M! Map(G; M) makes the associated cochain complex into a resolution of
M by acyclic Gmodules. Applying the Ginvariants functor then yields a
cosimplicial abelian group T .(G; M). Of course, this is nothing but the usual
bartype complex (a.k.a. the Godement complex, cf. 2.6). The associated
cohomology groups are then precisely the groups H*(G; M) according to the
"sheaf" definition.
On the other hand, continuity in this context just means that
Map(Gn; M) = colimU Map((G=U)n; M)
It follows easily that
T .(G; M) = colimU T .(G=U; MU )
so this is the same as "Cech" cohomology.
There is yet another description that will be useful (compare section 3.1).
Proposition 1.2 For a discrete Gmodule M, Hp(G; M) = limp T .(G; M).
We will also need to consider cohomology with coefficients in nondiscrete
modules such as Z`(n). The category of inverse systems {Mn} of discrete
Gmodules is also abelian with enough injectives, and the composite func
tor {Mn} 7! lim0nMGn is left exact. Its derived functors will be denoted
10
H*(G; {Mn}). This is a special case of "continuous" cohomology in the sense
of Janssen [19] and will arise in its more general form in 2.7. Often it is
written with a subscript viz. H*cont, but we can dispense with this because
as far as we are concerned continuous cohomology is the only cohomology.
Note that there are short exact sequences
0 ! lim1nHp1(G; Mn) ! Hp(G; {Mn}) ! lim0nHp(G; Mn) ! 0
If M is given as an inverse limit of discrete modules  M = lim0nMn  we
will write simply H*(G; M) for H*(G; {Mn}). The main case of interest is
Z`(n) = lim0m(Z=`m (n)).
If L=F is a Galois extension, we will usually write H*(L=F ; M) for
H*(G(L=F ); M) and simply H*(F ; M) for H*(F =F ; M).
We say that cd`G = d if d is maximal such that there exists a discrete
`torsion module M with Hd(G; M) 6= 0. An argument using "Shapiro's
lemma" shows that if H is any closed subgroup of G, cd`H cd`G. Trans
lating into field notation, this implies that if E is any separable algebraic
extension of F, cd`E cd`F .
Some examples:
o If F is separably closed, obviously cd`F=0 for all `. In fact it is not
hard to show that cd`F=0 if and only if F has no finite separable `
extensions.
o If F is finite, cd`F=1.
o Tsen's theorem: If F is a function field of dimension one over a sep
arably closed field k (i.e., F is a finite separable extension of k(T )),
cd`F 1.
o If F is a number field and either `is odd or F is pure imaginary, cd`F=2.
Otherwise cd2F = 1.
Proposition 1.3 If F has transcendence degree n over a subfield F0, then
cd`F cd`F0 + n. (Usually equality holds; see [3],X,2.1.)
The proof involves inducting over a "TateTsen" filtration; compare the
proof of Theorem 5.1 below.
One similarly defines cohomological dimension for inverse systems of `
torsion modules.
11
1.3 Hypercohomology spectra for fields
A presheaf of spectra on F is a covariant functor E from the category of
finite separable extensions L of F (which we assume to be subfields of a fixed
separable closure F ). Later this definition will be modified and put in its
proper context of "presheaves of spectra on theetale site of F", but for now
we want to keep things as simple as possible. The main example we have in
mind if of course the presheaf L 7! KL.
The stalk of a presheaf E is the colimit over finite separable extensions
L of the spectra E(L). In practice, the functor E will be defined on infinite
extensions as well and will commute with colimits (up to weak equivalence),
so we will write E F for the stalk of E . The stalk provides the connection
between the presheaf point of view and the GF spectrum point of view: each
presheaf has its stalk, and conversely given a GF spectrum X we get a
presheaf E with E(L) = XGL .
We will give two equivalent versions of the hypercohomology spectrum
H.(F,E ). The first is to be taken as the definition. For any profinite group G
and spectrum X, Map(G; X) will by definition mean colimU Map(G=U; X)
(where as usual U ranges over open normal subgroups). Passing to equivari
ant maps, this means that
MapG (G; X) = colimU MapG=U (G=U; XU )
o "Sheaf": Apply the functor MapGF (; EF) (continuous maps, as al
ways) to E.G, obtaining a cosimplicial spectrum T .(F; E). Define
H .(F; E) = holim T .(F; E)
o "Cech": Let S.(F; E) denote the cosimplicial spectrum colimLS.(G(L=F ); E(*
*L)).
Define
H .(F; E) = holim S.(F; E)
In fact these two spectra are naturally weak equivalent. The point is that
the obvious map of cosimplicial spectra
S.(F; E) ! T .(F; E)
is a weak equivalence at each cosimplicial level and hence induces a weak
equivalence on holim .
12
It is important to note that the two limit operations in the definition of
the Cech hypercohomology spectrum do not, in general, commute up to weak
equivalence (but see Example 3.22 below).
Theorem 1.4 Let F be any field, E a presheaf of spectra on F. There is a
conditionally convergent right halfplane cohomology spectral sequence
Ep;q2= Hp(F; ssqEF ) ) ssqpH .(F; E)
Our main example is the case En = K () ^ MZ =`n, where K is the K
theory presheaf. Then Suslin's theorem yields
Corollary 1.5 For any field F and prime `, there is a conditionally conver
gent fourth quadrant cohomology spectral sequence
q . ^
Ep;q2= Hp(F ; Z`(__)) ) ssqpH (F ; K)
2
(Ep;q2is zero for q odd or q < 0.)
Note that Suslin's theorem is needed to identify the E2term. Also, we
have implicitly used the fact that smashing with a finite spectrum commutes
with hypercohomology, in order to make the identification
holimnH .(F; K ^ MZ =`n) = holimnH .(F; K) ^ MZ =`n
This will be justified in a more general context in section 3. An explanation
of why we get continuous cohomology can be found in section 4.11.
Note that the spectral sequence of the corollary  the descent spectral
sequence for Ktheory  will converge if, for example, cd`F < 1. When
F = R and `=2, we have cd2R = 1 but one can show the spectral sequence
converges nevertheless.
Note also that when cd`F = d < 1, we conclude that H .(F; K)^ is (d
1)connected, and
ssdH .(F; K)^ = Hd(F ; Z`(0)
This group can easily be nonzero  for example, if F is any finite field, ` any
prime, we have ss1 = Z`.
13
For any presheaf E, there is a natural augmentation map E(F )! H .(F ; E)
(analogous to XG ! XhG ). In this context, the LichtenbaumQuillen conjec
ture asserts:
Conjecture 1.6 If cd`F < 1, the augmentation KF ^! H .(F; K)^ is a
weak equivalence on some connected cover.
There are some explicit conjectures as to what the isomorphism range
is. Clearly one must at least pass to the (1)connected cover of the hyper
cohomology spectrum  but this is not enough. Suppose for example that
cd`F 2. Then the spectral sequence collapses and
ss0H .(F; K)^ ~=Z` H2(F ; Z`(1))
The term H2(F ; Z`(1)) is closely related to the Brauer group and is typ
ically nonzero. Hence in this case, there is at best an isomorphism above
dimension zero.
1.4 Statement of Thomason's Descent Theorem for
Fields
Definition 1.7 A field F is `good if char F 6= ` and F has finite transcen
dence degree over a subfield E satisfying cd`E1 1, where E1 is obtained
frompE_by adjoining all the `power roots of unity. If ` = 2 we also require
1 2 E.
p ___
The condition 1 2 F when ` = 2 is needed for two reasons. First of
all, we will need to know that cd`F < 1. For that we could getpby_with
the weaker assumption cd`(E1 =E) 1. However one also needs 1 2
F for the construction of Bott elements. Examples of fields E as above
includepseparably_closed fields, finite fields, local fields, and number fields
(with 1 2 F if ` = 2). Note that an `good field has finite cd`.
Theorem 1.8 Let F be an `good field. Then the natural augmentation
KF ! H .(F; K) induces a weak equivalence
^LKF ~= ^LH.(F; K)
14
and there is a convergent spectral sequence
q
Ep;q2= Hp(F ; Z`(__)) ) ssqp^LKF
2
One can show that the functor L commutes with hypercohomology (see
Corollary 3.12), so ^LH.(F; K) = (H .(F; LK ))^. Then the spectral sequence is
obtained as before.
The theorem can be reformulated in terms of "Bottperiodic" Ktheory.
Let kF = KF ^ MZ =`. Then the theorem is equivalent to the assertion
that LkF ! H .(F ; Lk) is a weak equivalence , and moreover an easy transfer
argument reduces to the case when F contains `th roots of unity. In that
case we fix once and for all a Bott element fi 2 ss2kF . Then by a theorem of
Snaith and Dwyer, LkF ~= fi1kF . Theorem 1.8 is then equivalent to:
Theorem 1.9 Let F be an `good field containing `th roots of unity. Then
the natural augmentation kF ! H .(F; k) induces a weak equivalence
fi1kF ~= fi1H .(F ; k)
The proof of this theorem will be sketched in the final section.
2 Schemes, Sites and Cohomology
This section is directed strictly from one nonexpert to another. Its pur
pose is to provide, in condensed form, the technical background on sites and
cohomology necessary to penetrate [42] and [22].
Just keeping track of the schemetheoretic hypotheses in [42] is already
a nontrivial task. The most basic assumption will usually be that "X is a
separated noetherian regular scheme, of finite Krull dimension", so we briefly
review these notions in the first subsection. In the next two subsections we
recall, for the reader's convenience, the definitions ofetale morphisms and
hensel local rings.
In the fourth subsection we introduce Grothendieck sites. A site is a
beautiful and farreaching generalization of a topological space. It is an
extremely general object, which at the same time is far simpler than any of
its special cases. The examples of most importance for this paper are the
15
etale, Nisnevich and Zariski sites of a scheme. The Nisnevich site lies between
theetale and Zariski sites, and serves to bridge the gap between fields and
general schemes. Next we introduce presheaves and sheaves, and discuss the
various direct and inverse image functors associated to a site morphism.
The topos associated to a site is its category of sheaves of sets. According
to [3] (IV, introduction) or [23] (introduction), it is the topos that is of
paramount importance  different sites can yield the same topos, and a choice
of site is just a particular way to construct the topos. Be that as it may, it*
* is
not the point of view that will be taken here. Our ultimate goal is to study
the homotopy theory of presheaves of spectra on a site, and it would seem
to be highly desirable  even essential  to work on an actual site rather than
the disembodied topos. Thus topos theory per se will make only a minor
appearance.
One place where the topos does enter is in the next section, on points
and stalks. There is a beautiful theory ([3], IV.6) extending the concept of
stalk at a point, familiar from classical sheaf theory, to general sites. The
idea is that it is the inverse image or stalk functor on sheaves  associated
to inclusion of a point in the usual sense  which carries all the essential
information. Formulated in this way, the classical theory of points and stalks
carries over to general sites to a remarkable degree. The most noteworthy
difference is that a given site may or may not have "enough points", meaning
that stalks detect isomorphisms in the usual way. Theetale, Nisnevich and
Zariski sites have enough points.
The next two subsections discuss sheaf cohomology, including some re
sults onetale and Nisnevich cohomological dimension that are crucial for the
applications to spectra. Then we introduce C ech cohomology and compare
it with sheaf cohomology.
In the final section on noetherian sites we discuss two crucial results on
commuting sheaf cohomology with filtered colimits. The first of these involves
colimits in the sheaf variable, and is straightforward. The second result  the
socalled "Continuity Theorem" has to do with cofiltered inverse limits in
the scheme variable, with theetale topology (or more generally inverse limits
of sites, but discretion seems the better part of valor here). This theorem
is somewhat more difficult to state, and much more difficult to prove. In
the affine case it says roughly thatetale cohomology commutes with filtered
colimits of rings.
For a quick introduction to sites andetale cohomology, see [41] or [1].
16
The definitive treatment is the original source [3]. For the Nisnevich site see
[34], [25], Appendix E in [44] and Chapter 7 of [22].
2.1 Schemes
We recall that a scheme X is separated if the diagonal X is closed in X x X.
For example any affine scheme or algebraic variety is separated, whereas e.g.
affine nspace "with a point doubled" is not. More generally a morphism
X ! Y is separated if X is closed in X xY X. Separation hypotheses enter
in various ways. For example, in general the intersection of two affine open
subsets need not be affine (consider the example above with n 2), but
this is easily checked to be true for separated schemes. Another point is
that valuations behave well on separated schemes ([18],II.6, remark at the
bottom of p.130). General considerations like these lie behind the separated
hypothesis in the resolution theorem below.
A scheme X is noetherian if it has a finite open affine cover
X = [Spec Ai
with Ai a noetherian ring. This implies that the underlying space of X is
noetherian and in particular quasicompact ([18], II, Ex. 2.13). The noethe
rian hypothesis is convenient for mostly obvious reasons. In some cases it
could be replaced by "locally noetherian"  or even omitted entirely, provided
that "finitetype" assumptions are replaced by their "finitelypresented" ana
logues. We will usually assume that schemes are at least locally noetherian.
For a general discussion of finiteness hypotheses, beyond the noetherian con
dition, see [16],6.
The Krull dimension of X is the maximal length n of a descending chain
X0 X1 ::: Xn
of irreducible closed subsets. It's worth pointing out here that although local
noetherian rings always have finite Krull dimension, this isn't true in general
(see [12], exercise 9.6). Hence our typical hypothesis "let X be a separated
noetherian scheme of finite Krull dimension" is not redundant.
X is regular if all of its local rings O xare regular. Here a local ring R w*
*ith
maximal ideal m and residue field k = R=m is regular if it is noetherian and
dimk(m=m2) =Krull dim X. For example, Dedekind domains are regular, and
17
an algebraic variety over a field is regular if and only if it is smooth. The k*
*ey
property of regular schemes that we need is (see for example [18],III,exercise
6.9).
Theorem 2.1 Let X be a separated noetherian regular scheme. Then every
coherent sheaf on X admits a finite resolution by locally free sheaves.
We remind the reader that coherent (resp. locally free) sheaves corre
spond to finitelygenerated (resp. finitelygenerated projective) modules in
the affine case. Theorem 2.1 is the ultimate source of the separated and
regular hypothesis that occurs in several of the main theorems below.
2.2 Etale morphisms
f
A morphism of locally noetherian schemes X ! Y isetale if it is
o flat: for all x 2 X the induced homomorphism O f(x)! O x of local
rings is flat
o unramified: the fibre of f over Spec k = y 2 Y is a finite disjoint union
of finite separable extensions Spec ki of the residue field of y. In other
words, there is a pullback diagram
a
Spec ki ______X
i 
 
 
 
 
 
? ?
Spec k ________Y
o locally of finite type: There is an affine open cover Ui of Y and for each
i an affine open cover Vijof f1 Ui such that the ring homomorphism
corresponding to each Vij! Ui has finite type. (For general schemes
one requires that f is locally of finite presentation; see e.g. [41].)
The "unramified" condition has an alternative formulation. Let R! S be
a map of rings, S=R the "module of relative Kahler differentials" (i.e. the
Rmodule on symbols ds, s 2 S modulo the Leibniz rule  see [18],II.8). Then
18
Spec S ! Spec R is unramified if and only if S=R = 0, with a similar result
holding for arbitrary schemes (cf. [29],I,3.5).
We will invariably assume that ouretale maps actually have finite type;
this just means that in the third condition above we have for each fixed i a
finite such cover Vij(and since our schemes will usually be noetherian, the
cover Ui will also be finite).
The basic examples of anetale morphism that the reader should keep in
mind are the following:
o A smooth unramified covering map between complex algebraic varieties
o The morphism Spec L ! Spec F corresponding to finite separable field
extension L/F
o The morphism Spec S ! Spec R corresponding to an unramified finite
extension of Dedekind domains
In the first and third examples, one should picture more generally a map
which is not onto but rather has image some open subset of the target. For
example in the case of an arbitrary finite extension of Dedekind domains, the
associated map of schemes isetale over the complement of the finite set of
ramified primes.
We note thatetale maps are stable under base change and composition,
any open immersion isetale, and anyetale map is an open map.
2.3 Hensel local rings
Let R be a local ring with maximal ideal m and residue field k. Then R
is a hensel local ring, or is henselian, if every finite Ralgebra S splits as a
finite product of local Ralgebras. This is equivalent to R satisfying "Hensel's
lemma" on lifting factorizations of polynomials from k to R. Any local ring R
admits a henselization R ! Rh which is initial among local homomorphisms
to henselian rings. One can think of henselization as a kind of algebraic
approximation to the analytic process of completion. Every complete local
ring is henselian, and in general Rh sits between R and the completion of R.
If S is a finite Ralgebra and isetale over R, then S splits as a product of
local Ralgebras which are themselves henselian.
19
Theorem 2.2 Let R be a hensel local ring. Then the functor S 7! S R k is
an equivalence between the category of finiteetale Ralgebras and the category
of finiteetale kalgebras.
A strict hensel (or strictly henselian, or "strictly local") ring) is a hens*
*el
local ring R with separably closed residue field. Note that in this case, the
preceeding theorem says that every finiteetale Ralgebra splits as a product
of copies of R.
There is also a strict henselization functor R ! Rsh. For further discus
sion see section 2.6 below.
2.4 Sites
A site = (C ; T) consists of (1) a category C with pullbacks; and (2) a
Grothendieck topology T on C  that is, for each object U of C a collection
of families of morphisms {Ui! U} called covering families, such that every
cover of a cover is a cover, every pullback of a cover is a cover, and every
family consisting of a single isomorphism is a cover. All sites considered in
this paper will also have terminal objects and hence have all finite limits.
The reader can easily guess the precise statements needed in (2). See e.g.
[1], [41], or the original source [3]. We remark that the definition of site gi*
*ven
here is less general than the usual one.
There are several examples that will be important for us. Let X be a
locally noetherian scheme.
o The Zariski site XZar. The underlying category C is the category of
open subsets of X (for the Zariski topology), with morphisms the in
clusions. A family {Ui! U} is a covering family if the Ui cover U in
the usual sense. This site makes sense for any topological space, and
is the motivating example.
o Theetale site Xet. C is the category of schemesetale and of finite type
over X. A family {Ui! U} is a covering family if the images of the Ui
cover U in the usual sense.
o The restrictedetale site Xretof a separated noetherian scheme. This is
defined as in the previous example, except that we require the scheme
over X, U ! X to be separated over X, and the covering families are
20
to have only finitely many elements Ui! U. (It follows automatically
that Ui is separated over U as well.) One advantage of using the re
stricted site is that a covering family {Ui! U}`can be regarded as a
single surjectiveetale map U0! U, where U0 = Ui. For the restricted
etale site of a general scheme, see [41],II,1.5.
o The Nisnevich site XNis [34]. The underlying category is the same as
for theetale site. To describe the covering families, we borrow a term
from number theory. Given anetale h : U0! U, we say that x 2 U is
completely decomposed in U0 if there is a point y 2 h1x such that the
induced map on residue fields k(x) ! k(y) is an isomorphism. A cover
ing family for the Nisnevich topology is a covering family {Ui! U} for
theetale topology such that for all x 2 U, x is completely decomposed
in at least one Ui. This site will be discussed further in section 2.6.
o The site Gfsetsattached to a profinite group G. A Gset is discrete if
all its isotropy groups in G are open. We take for C the discrete finite
Gsets. Covering families are just the surjective families.
o "Big" sites. Theetale site discussed above is analogous to a single
topological space. Theetale morphisms U! X play the role of the
open subsets. But we could also consider some much larger category of
schemes  even all schemes  and still get a Grothendieck site, using the
same covering families. For example we could take all schemes X locally
of finite type over a fixed base scheme S, and equip each X with the
etale topology. This kind of a site (or its Zariski or Nisnevich analogue)
is roughly analogous to the category of all topological spaces, and will
be informally referred to as a "big", or "sufficiently large" site.
Two further examples can be found in the next section.
A morphism of sites f :  ! 0 consists of a functor f1 : C0! C that
preserves pullbacks and preserves covering families in the obvious sense.3 If
the sites have terminal objects, and f1 preserves them, then f1 preserves
all finite limits. This will be the case in the examples considered in this
paper, with one exception discussed in Section 2.5 below. An equivalence
______________________________
3The notation f1 for the functor is meant to be suggestive  perhaps danger*
*ously so,
but we're going to take the risk.
21
of sites is a morphism such that f1 is an equivalence of categories and any
inverse functor is also a site morphism. Exercise: show that if F is a field,
the site (GF )fsetsis equivalent to theetale site of F.
Our convention that the site morphism goes in the opposite direction is
in agreement with some sources but not with others (e.g. [1] or [41], in which
our morphism f would be written as a "morphism of topologies" T 0! T ).
The reason for the convention is just that a morphism of schemes X ! Y
then induces site morphisms in the same direction XZar! YZar, Xet! Yet,
etc., and a continuous homomorphism G ! H of profinite groups induces a
morphism Gfsets! Hfsets. On the other hand it must be admitted that there
are some other cases where our convention looks slightly odd. For example,
it is clear that XZar and XNis are "subsites" of Xet, etc., but in our notation
the evident site morphisms go the other way:
Xet! XNis! XZar
However even in this case, our convention helps the wandering topologist
to keep track of things such as the Leray spectral sequence described below.
Site morphisms associated to inclusion of a subsite as above will be called
restriction morphisms.
2.5 Presheaves and sheaves
A presheaf of sets on a site = (C ; T) is just a contravariant functor from
C to sets. An abelian presheaf is a presheaf of abelian groups. Similarly one
can define presheaves with values in any category (simplicial sets, spectra,
etc.), but for now we will focus on presheaves of sets or abelian groups. A
sheaf is a presheaf E that satisfies the usual type of exactness condition with
respect to covering families.
There is an exact sheafification functor a, left adjoint to the inclusion
functor i of presheaves in sheaves.4 A presheaf of sets E is representable
if E (U) = Hom (U; W ) for some object W. If C is an arbitary category
with pullbacks, the finest topology such that every representable presheaf
is a sheaf is called the canonical topology  see e.g. [41]. On most of the
sites encountered in practice, every representable presheaf is a sheaf  in
______________________________
4Later, when we consider presheaves of homotopy groups ssqE, the associated *
*sheaf will
be denoted "ssqE.
22
other words, the topology in question is no finer than the canonical topology.
In the general case the phrase "sheaf represented by W" secretly means
the sheafification of Hom (; W ). The category C also admits a minimal
topology, called the "chaotic" topology, in which the only covering families
are the isomorphisms. In this topology, every presheaf is a sheaf.
Given a site morphism f :  ! 0, one can define various direct and
inverse image functors on sheaves and presheaves. Since these functors are
extremely important, and keeping them all straight gets a bit confusing, we
will make a few remarks that may be of help.
Consider a functor g : I ! J between small categories. Let D be another
category, which has all limits and colimits, and consider the restriction func
tor D g : D J! DI on functor categories (or "diagrams"). This functor has
both a left adjoint GL and a right adjoint GR , also know as Kan extensions
([26],X). Bearing in mind that presheaves are contravariant functors, we can
apply this to our site morphism f, with g = f1 . We write f# (direct image)
for the restriction functor, f# (inverse image) for the left adjoint and f! for
the right adjoint. Thus if E is a presheaf on , f# E(U) = E(f1 U). And if
F is a presheaf on 0, f# F (W ) = colimV F(V ), where the colimit is taken
over the opposite of the overcategory (f1 # U). If the sites have terminal
objects and f1 preserves them, (f1 # U)op is filtered. Otherwise it is only
pseudofiltered.
One would like to promote these functors to the sheaf level. There is
no problem with the direct image functor because it preserves sheaves and
so immediately yields a direct image functor on sheaves, denoted f*. On
the other hand the symmetry between left and right adjoints breaks down.
The reason is clear  the sheafification functor a is a left adjoint. Hence,
even though f# may fail to preserve sheaves, we get a left adjoint to f* by
setting f* = af# i. The analogous attempt to extend f! fails simply because
s appears on the wrong side. One nevertheless gets a right adjoint f! on the
sheaf level in certain situations  e.g. inclusion of a closed subscheme in the
Zariski oretale topologies.
To confuse matters still further, there are situations where we have site
morphisms going in both directions. Specifically, consider a fixed object U
of a site that has a terminal object X. Define the local site =U to have
underlying category the category of objects over U and covering families the
same as those in . The simple example to keep in mind is an open subscheme
in the Zariski topology. There is an obvious site morphism j : =U ! with
23
j1V = U xX V . But there is another obvious site morphism k :  ! =U
with k1(V ! U) = V . (Note that k1 does not preserve terminal objects or
products.) Inspection shows that j# = k# . Hence j# preserves sheaves and
has a left adjoint j!(= k# ), called "extension by zero" because in the Zariski
example mentioned above we have j!E(V ) = E(V ) if V U and j!E(V ) = 0
otherwise. In general we have
a
j!E(V ) = E(V )
Hom (V;U)
The category of presheaves on the site is sometimes denoted ^. The
category of sheaves (the "topos") is sometimes denoted ~ .
2.6 Points and stalks
In this section presheaves and sheaves will be setvalued unless otherwise
mentioned.
In classical sheaf theory, on the Zariski site of a scheme X, the notion
of "stalk" at a point x 2 X is extremely useful. For example, stalks detect
isomorphisms in the sense that a map of sheaves is an isomorphism if and
only if it induces an isomorphism on all stalks. It isn't obvious how to define
stalks on an arbitrary site, because it isn't even clear what a "point" should
be. The key is to observe that the category of sheaves on the Zariski site of
x = Spec k is the category of sets, and that the corresponding stalk functor
X~Zar! Sets is a left adjoint that commutes with finite limits. Thus on a
site we define a point p as a pair of adjoint functors
p* : ~ ! Sets; p* : Sets ! ~
where the left adjoint p* is required to commute with finite limits. The stalk
of a sheaf F is then defined by Fp = p*F . In fact the point p is determined
by a functor p* that commutes with finite limits and all colimits, because the
right adjoint exists automatically. Indeed one has the explicit formula for a
set S
p*S(U) = Homsets(p*U"; S)
which is easily verified using the colimit formula below for p* (U" is the sheaf
represented by the object U).
24
A neighborhood of a point p is a pair (U; a) with U an object of the site
and a 2 p*U". Equivalently, this can be thought of as a lift of the point to the
local site =U. The neighborhoods form a category Np in an obvious way,
and this category is cofiltered. For any sheaf F we have
Fp = colimNpF (U)
or more generally for a presheaf F
aFp = colimNpF (U)
A set of points of a site is said to be sufficient if it detects isomorphisms
of sheaves in the sense described above. A site has enough points if it has a
sufficient set of points. Not all sites have enough points.5 However there is
the following elegant and useful criterion ([3],IV,6.5). We say that a set of
points of detects covering families provided that whenever {Ui! U} is a
family of morphisms and {p*Ui! p*U} is a surjective family of sets for all
points p in the set, then {Ui! U} is a covering family. Then the criterion
states that a set of points of is sufficient if and only if it detects covering
families.
In the case of the Zariski site, one can show that the new definitions agree
with the old  although it isn't obvious that every sitetheoretic point comes
from a classical point (see [23],7.24, for a quick proof). Note that in this ca*
*se
the stalk of the structure sheaf is the Zariski local ring Ox.
Now consider theetale site of a scheme X. Given an ordinary point
x ! X, x = Spec k, choose a separable closure x = Spec k. Then the
category of sheaves on xet is the category of sets, and the site morphism
p : xet! X induces a functor p* on sheaves that commutes with finite limits
and all colimits. Thus each such "geometric point" x ! X defines a site
theoretic point of Xet. One can easily check that a neighborhood of x ! X
is the same thing as anetale U ! X together with a lift
______________________________
5See [3],IV,7.4 for an interesting example involving Lebesgue measure on th*
*e unit
interval  of a site with no points.
25
U



?
x ______X
It follows that the stalk at x of the structure sheaf U ! OU is the strict
henselization Oshxof Ox. Thus the rings O shxare the local rings for theetale
topology. Finally, it is easy to see that geometric points detect covering
families, so theetale site has enough points.
Contemplating the vast gap between the Zariski local ring O x and its
strict henselization O shx, one longs for an intermediate world in which the
local rings are the ordinary henselizations O hx. The Nisnevich site is precise*
*ly
such a world.6
Consider first`the Nisnevich site of x = Spec k, k a field. Call a presheaf F
additive if F (U V ) = F (U) x F (V ). Then it is easily seen that the sheaves
on xNis are precisely`the additive presheaves (in effect, any covering family
takes the form U W ! U). Hence sheaves are the same thing as presheaves
on the full subcategory of connectedetale xschemes. In particular, for any
such scheme x0 (i.e. x0= Spec k0, with k0 a finite separable extension of k),
the sections functor F ! F (x0) is exact.
For a general scheme X, consider an ordinary point i : x ! X, and x0! x
as above. Define p* : X~Nis! Sets by p*F = j*NisF (x0), where j : x0! X.
Then, thanks to the exactness of the sections functor, p* defines a point of
XNis. Informally, then, a point of the Nisnevich site is denoted x0! x or
x0! X. A neighborhood of such a point is the same thing as a commutative
diagram
x0 ______U
 
 
 
? ?
x ______X
______________________________
6As will be seen, however, the Nisnevich topology is much closer to the Zari*
*ski topology
than it is to theetale topology.
26
Note that x0 need not be a residue field of U  in general, it is only a
finite extension of such a field. However the neighborhoods for which x0is a
residue field of U form a cofinal subcategory of N p.
It follows that the Nisnevich stalk at x 2 X of the structure sheaf is
Ohx, the henselization of O x. More generally the stalk at x0! x is O hx0, the
hensel extension of O hxcorresponding to x0! x under the equivalence of
Theorem 2.2. We will call this ring the "henselization of O x at x0! x".
Thus the local rings for the Nisnevich topology are the henselizations O hx0.
Finally, it is easy to check that the points described above detect covering
families, and hence the Nisnevich site has enough points.
2.7 Sheaf Cohomology
Let A be an abelian sheaf, U an object of C. The category of sheaves is abelian
with enough injectives, and the sheaf cohomology groups Hp(U; A) are the
pth derived functors of the left exact "sections on U" functor U ! A(U).
For the sites XZar; Xetand so on we will write this as HpZar(U; A), Hpet(U; A)
etc.
When the site has enough points, there is a functorial Godement resolu
tion defined as follows. For each point p, the adjoint functors p*; p* defined a
monad ([26],VI), and any monad defines a cosimplicial object in a standard
way. Here we will take the product over a sufficient set of points, obtaining
a monad on sheaves given by
Y
T (A) = p*p*A
p
The resulting cosimplicial sheaf T .(A) provides a resolution of A by
sheaves which, although not necessarily injective, are "flasque" and in partic
ular acyclic for cohomology. Hence, taking sections over U of the Godement
resolution yields a functorial complex for computing cohomology.
One of the main tools for computing sheaf cohomology is the fantastically
general "Leray" or "changeofsite" spectral sequence:
Theorem 2.3 Let f :  ! 0 be a site morphism, A a sheaf on . Then
for any object U in C there is a first quadrant cohomology spectral sequence
Ep;q2= Hp(U; Rqf*A) ) Hp+q(f1 U; A)
27
Here the Rqf*A are the "higher direct image sheaves" which are the de
rived functors of f*. They can be described as the sheafifications of the
presheaves U ! Hq(f1 U; A). In the case of a morphism of schemes X ! Y ,
we would typically take Y=U, X = f1 U and get the classical Leray spectral
sequence of XZar! YZar, its counterpart for theetale site, etc. For the site
morphism associated to a surjective homomorphism of profinite groups, we
get the HochshildSerre spectral sequence.
Now consider the case of a "restriction" morphism such as Xet! XZar. In
this case the "direct image" functor is just the obvious restriction of sheaves
to the smaller category. Nevertheless one gets a very interesting spectral
sequence . For example in theetaletoZariski case one easily derives the
generalized "Hilbert's Theorem 90" asserting that
H1et(X; Gm ) = H1Zar(X; Gm ) = P ic X
One can also use the spectral sequence to show that a given subsite has
the same cohomology as the ambient site. We'll call the following result the
"weak comparison lemma" ([41], 3.9.3).
Proposition 2.4 Let f :  ! 0be a restriction morphism of sites. Suppose
that for every object U in C 0and every covering family {Vi! U} in the
ambient site , there are covering families {Uij! Vi} with Uij in C 0and
{Uij! U} in T 0. Then for any sheaf A on and object U in 0,
H*0(U; f*A) ~=H*(U; A)
This applies for example to the restriction from theetale site of a sepa
rated noetherian scheme X to its restricted site as above. In fact in this case
a stronger condition holds and one can apply the usual "comparison lemma"
([41], 3.9.1) to conclude the actual sheaf categories are equivalent. Another
illustration is provided by the following important theorem:
Theorem 2.5 Let R be a Hensel local ring with maximal ideal m and residue
field R=m = k. Then for any sheaf A on (Spec R)et, the morphism j :
Spec k ! Spec R induces an isomorphism
j* : H*et(Spec R; A) ~=H*et(Spec k; j*A)
28
The proof of this fact doesn't use the Leray spectral sequence of j. Instead
one constructs a site morphism going the other way, as follows. Let us
abbreviate X = (Spec R) and x = (Spec k). Let denote the subsite
of Xet in which both the objects and covering families consist of a single
map Y ! X corresponding to a finiteetale Ralgebra S. Then we have site
morphisms Xet!  ! xetwith the first a cohomology isomorphism by the
weak comparison lemma and the second an equivalence of sites. Letting f
denote the composite morphism, one has f* = j* and the theorem follows
easily.
Finally, the notion of continuous Galois cohomology discussed in section 1
extends immediately to an arbitrary Grothendieck site. The category of
inverse systems of sheaves has enough injectives, and following [19] we define
continuous cohomology as the derived functors of the composite functor lim O
(global sections).
2.8 Cohomological dimension
We will frequently need to know that a scheme X has finite cohomological
dimension for sheaves (or `torsion sheaves) on theetale, Zariski or Nisnevich
sites. In general, for an object X of a site , the cohomological dimension of
X is the maximal n (possibly infinite) such that there exists a sheaf A with
Hn(X; A) 6= 0. The mod `cohomological dimension is defined similarly, with
the requirement that A is an `torsion sheaf. In the case of a scheme X, and
one of the sites mentioned above, we will write this as cdZar(X), cdet`X, etc.
The following theorem of Grothendieck applies to any noetherian topo
logical space X (see [18], III, 2.7).
Theorem 2.6 Let X be a noetherian scheme of Krull dimension n. Then
cdZar(X) n.
The next result, due to Kato and Saito (see [34]), is another indication
that the Nisnevich topology is not so far from the Zariski topology.
Theorem 2.7 Let X be a noetherian scheme of Krull dimension n. Then
cdNis(X) n.
29
In theetale case we need to pay attention to the cohomological dimension
of the residue fields of X. We say that X is uniformly `bounded, with bound
d, if for all residue fields k(x) we have cd`k(x) d. Here cd` refers to Galois
cohomological dimension, or equivalently cdet`Spec k(x).
Theorem 2.8 Let X be a noetherian scheme of Krull dimension n, and
suppose X is uniformly `bounded with bound d. Then cdet`X n + d. In
particular, cdet`X is finite.
There is an elegant proof of this fact based on the Nisnevich site and the
KatoSeito theorem. We sketch it here as an illustration of how the Nisnevich
site functions as an intermediary between fields and schemes.
Let A be an `torsion sheaf on Xet, and consider the Leray spectral se
quence of the site morphism f : Xet! XNis. We have
Ep;q2= HpNis(X; Rqf*A)
and hence Ep;q2= 0 for p > n by the KatoSaito theorem. On the other
hand we claim that the sheaves Rqf*A on XNis are identically zero for q > d,
whence the result. It is equivalent to show that the stalks are zero. In
the notation of section 2.6, the stalk at a point x is the sheaf on xNis that
assigns to each x0! x the group Hqet(O hx0; g*A), where O hx0is the hensel local
ring attached to x0and g is the natural map to X (here we have made use of
the "continuity theorem" (Theorem 2.11 below)). But this is isomorphic to
Hqet(x0; A), by 2.5. Since k0=k is a finite separable extension, cd`k0 cd`k,
and hence Hqet(x0; A) = 0 for q > d.
For example, suppose X is a variety of dimension n over a field k with
cd`k < 1. Then the residue fields of X have transcendence degree at most
n over k. Hence by 1.3, we get cdet`X 2n + cd`k. Or if X is the ring of
integers in a number field F, with either ` odd or F pure imaginary, then
cdet`X 3.
Finally, we remark that the "uniformly `bounded" hypothesis in the
above theorem is inherited by any object U of Xet(with the same bound),
because the residue fields of U are finite separable extensions of the residue
fields of X. Hence if in addition U has finite type over X, all the hypotheses
of the above theorem are inherited by X, and cdet`U n + d.
30
2.9 C ech cohomology
The usual definition of C ech cohomology for spaces goes through essentially
verbatim for an arbitrary Grothendieck site. Let X be an object of the site
, F an abelian presheaf. Given a covering family U = {Ui! U}, one forms
the associated augmented C ech complex
Y Y
0 ! F (U) ! F (Ui) ! F (UixU Uj) ! :::
i i;j
The cohomology of this complex (after removing F(U)) is denoted H* (U =X; F )
and called C ech cohomology with respect to the cover U . C ech cohomology
proper is defined by passing to a filtered colimit over covers (with the caveat
that the covers themselves do not form a cofiltered category under refine
ment  see for example [41]), and denoted H (X; F ).
It is not hard to see that the groups H* (U =X; F ) are the derived func
tors of H0 (U =X; F ). Since H0 (U =X; F ) = F (X) if F is a sheaf, the global
sections functor for sheaves factors as inclusion into the presheaves followed
by H0 . This leads to a spectral sequence we will call "C ech descent for sheaf
cohomology":
Ep;q2= Hp (U=X; Hq(; A)) ) Hp+q(X; A)
By passing to the limit in the spectral sequence above we get another spectral
sequence
Ep;q2= Hp (X; Hq(; A)) ) Hp+q(X; A)
Exercise: Study the case when X = Spec F , F a field. Show that the
Cech complex obtained from the nerve of a finite Galois extension L/F is
just the usual bar complex, the C ech cohomology groups are exactly the
groups labelled "C ech" in section 1.2, and the C ech descent spectral sequence
associated to L/F is just the HochshildSerre spectral sequence of the group
extension GL! GF ! G(L=F ).
Now consider the question of whether C ech and sheaf cohomology agree.
This certainly isn't true in general, even on the site of a topological space.
However from the C ech descent spectral sequence we see that it would be
31
enough to show that for a sheaf A, Hp (X; Hq(; A)) = 0 for q > 0. The
crudest way to guarantee these groups vanish would be to have some cofinal
system of covers U ! X such that Hq(U nX; A) = 0 for all U, n, and q > 0.
For example, this is how one shows that on the Zariski site of a separated
noetherian scheme, sheaf and C ech cohomology agree for coherent sheaves
([18], III, 4.5). In this case one uses coverings by affine schemes, which are
acyclic for coherent sheaf cohomology by a classical theorem of Serre.
This is far too much to expect in general, however. As a next attempt, one
might start from the observation that on an arbitrary site, the sheafification
of the presheaf Hq(; A) is zero when q > 0. This amounts to the fact
that for any W and ff 2 Hq(W ; A), there is a covering V ! W such that ff
restricts to zero on V . So what one needs to know is:
(*) For any covering U0! X with C ech nerve0U.X and any covering V ! U nX,
there is a covering U  ! U such that (U )nX! U nXfactors through V .
Then we can conclude that every C ech cochain on U =X with coefficients
in Hq(; A) vanishes when we pass to the limit in U . This is exactly what
Artin [2] shows to get the following theorem (see also [42], 1.53).
Theorem 2.9 Let X be any quasicompact scheme such that every finite
subset of X lies in an open affine subscheme  for example, suppose X is
quasiprojective over a noetherian ring. Then for any additive presheaf A on
Xet,
H*et(X; aA) ~=H*et(X; A)
This result is a special result about theetale site, and leaves open the
question of whether there isn't a way to compute cohomology from covers
more generally. An elegant answer to this question was provided by Verdier
([3] ,V,7). The idea is to replace the C ech nerves by a more general simplicial
covering object called a hypercover. The resulting cohomology theory, which
we will call Verdier cohomology, agrees with sheaf cohomology on an arbitrary
site  essentially because the appropriate analogue of (*) holds in general.
2.10 Noetherian sites
A site is said to be noetherian if for every object U and every covering
family {Ui! U}, there is a finite subset {Ui1; :::; Uin} which is also a cover*
*ing
32
family.7 For example, if X is a noetherian scheme then theetale, Nisnevich
and Zariski sites of X are noetherian (although in the case of the Nisnevich
site this isn't obvious  see [44], appendix E, 6a). In fact it is enough to
assume only that X is quasicompact, provided that the definition ofetale
morphism is modified as alluded to earlier (see [41],II,1.5.1).
Proposition 2.10 Let be a noetherian site, Ai a filtered (or even pseudo
filtered) system of sheaves on . Then for all objects U, there is a natural
isomorphism
~= *
colimiH*(U; Ai) ! H (U; colimiAi)
To see why this is true, consider for the moment an arbitrary site, a
presheaf F and a covering family Ui! U. Consider also the associated C ech
complex
Y Y
0 ! F (U) ! F (Ui) ! F (UixU Uj) ! :::
i i;j
Then by definition F is a sheaf if and only if for all such covering families
of all objects U, this complex is exact through the first three arrows, and is
a flasque presheaf if and only if all such sequences are exact from the next
term on (i.e. all the higher C ech cohomology groups vanish). We recall that
injective sheaves are flasque, and that sheaf cohomology can be computed
from flasque resolutions (see for example [41], I, 3.5). Thus the proof of the
theorem boils down to showing that pseudofiltered colimits take sheaves to
sheaves, and flasque presheaves to flasque presheaves. This is clear provided
that pseudofiltered colimits commute with the products in the C ech complex
above. But if we now assume the site is noetherian, then we need only
consider finite coverings, in which case all of these products are finite and
the theorem follows.
The second important result of this section concerns theetale site of
an inverse limit of schemes. Suppose Ri is a filtered system of rings, and
R = colimiRi. Then Spec R is the (inverse) limit of the Ri in the cate
gory of schemes. More generally, suppose A iis a filtered system of quasi
coherent O Y algebras over a fixed base scheme Y. Then one can define
______________________________
7The reader might be wondering why we don't call this a quasicompact site. N*
*ote, how
ever, that the definition generalizes the notion of a topological space X with *
*the property
that every open subset of X is quasicompact  that is, a noetherian topological*
* space.
33
associated schemes Spec Ai (see [18],II,exercise 5.17), affine8 over Y, and
Spec (colimiA i) is again the limit of the Spec Ai in the category of schemes.
Using this construction, it is not hard to show that if Xi is any cofiltered
system of schemes with affine transition maps Xi! Xj, then the limit of the
Xi exists. However the only case we're really going to use is the affine case
Xi = Spec Ri, so the reader can feel free to picture this case, if desired.
Noetherian hypotheses are a bit awkward in this setting, because ob
viously a filtered colimit of noetherian rings need not be noetherian. For
that reason we will work a bit more generally and assume we are given the
following data:
o a cofiltered system of quasicompact and quasiseparated9 schemes Xi,
with affine transition maps. Such a system will be called admissible.
o an abelian presheaf F on a sufficiently largeetale site (i.e. that includes
all Xi as well as X, etc.). The presheaf F is assumed to be continuous
in the sense that for any admissible system Ui in the site, with limit U,
the natural map colimiF (Ui) ! F (U) is an isomorphism.
For example, any filtered system of rings, or of separated noetherian
schemes with affine transition maps, defines an admissible system. There
is no need to fuss over the precise meaning of "sufficiently large" in the
second condition above, because in practice F will be a presheaf defined on
all schemes e.g. the Ktheory presheaf, which is continuous as discussed in
the next section. The following theorem is the Continuity Theorem.
Theorem 2.11 Let Xi be an admissible system of schemes with limit X, and
let F be a continous presheaf on a sufficiently largeetale site. Then there is
a natural isomorphism
~= *
colimiH*et(Xi; aFi) ! Het(X; aF )
______________________________
8A morphism f : X !Y is affine if Y has an open affine cover Uisuch that ea*
*ch f1Ui
is affine.
9A scheme is quasiseparated if the unique map to Spec Z is quasicompact (i.*
*e. the
inverse image of every quasicompact open set is quasicompact). Every separated *
*scheme
is quasiseparated; see [16],I.6 for this and related issues.
34
The proof of this important theorem is quite pretty, and as we will need
one of the details, we will give a very brief sketch (see [1]). The beautiful f*
*act
is that essentially any morphism of schemes f : U! V over X is pulled back
from a morphism fi : Ui! Vi over some Xi, and moreover any reasonable
property of f can be arranged to hold for fi. Given a scheme U over X, we
call Ui over Xi a source of U if U is isomorphic as Xscheme to UixXi X,
and similarly for maps of Xschemes.
Lemma 2.12 Anyetale f : U ! X has anetale source Ui! Xi, which can
be taken surjective if f is surjective. More generally, any morphism U! V of
schemesetale over X has a source Ui! Vi with the same properties. Sources
are essentially unique, in the sense that any two sources of U! V agree after
base change to some higher Xj.
In fact by theorems of Grothendieck ([17],IV.8 and IV.11), the lemma
holds with "etale" replaced by virtually any property P stable under base
change  separated, finite, affine, projective, etc. The hardest step is obtain
ing a flat source.
Now let G be a presheaf on our bigetale site. Ignoring the fact that G is
already defined on X, we define a presheaf eG on X by
eG = colimiss#iGi
where ss#i is the presheaf inverse image functor and Gi is the restriction of
G to Xi. Recall that the inverse image functor itself involves a gargantuan
filtered colimit, so this definition appears to be a bit complicated. However
it is easy to see on inspection:
Lemma 2.13 Let U beetale over X, and let Ui be any source of U. Then
there is a natural isomorphism
~=
colimjGj(Uj) ! (eG)(U)
where the colimit is over all i! j and Uj is the base change of Ui to Xj.
Note that if G is continuous, this shows that eG is isomorphic to the
restriction of G to Xet. Using this last lemma, and the fact that Xet is a
noetherian site, it is easy to show that the functor e is exact, takes sheaves
35
to sheaves, and takes flasque presheaves to flasque presheaves. The argument
is just as in Theorem 2.10, and the proof of Theorem 2.11 is then completed
the same way.
Here is the sort of application we have in mind. Let X be a scheme, F a
sheaf on Xet.
Corollary 2.14 Consider the presheaf U 7! H*et(U; F ) on the Zariski, Nis
nevich andetale sites of X. Then
(i) The Zariski stalk at x 2 X is H*et(Spec Ox; eF )
(ii) The Nisnevich stalk at x0! x is H*et(Spec Ohx0; eF )
(iii) Theetale stalk at x! x is H*et(Spec Oshx; eF ) (=0 for * > 0).
Note we are not assuming F is continuous, or even defined on the various
local rings, in the above corollary. If it happens to be continuous, that's
fine, but the corollary as stated is needed for example in Theorem 2.8. Its
proof is easy from the continuity theorem and the definition of stalks; recall
that in all three cases these can be computed from a cofiltered limit of affine
schemes. Of course the fact that theetale stalk is zero follows from more
elementary considerations.
3 Hypercohomology spectra on schemes
In this section we introduce and compare the cohomological descent machines
of Thomason ([42], see especially sections 1 and 5) and Jardine ([20], [21],
[22]).
We begin with some homotopytheoretic preliminaries, including a few
words on closed model categories and our conventions on spectral sequences.
It is a basic fact of life that homotopy limits (and various other construc
tions) do not commute with filtered colimits. Nevertheless, there are many
situations where a filtered colimit can be "commuted" with whatever other
construction is at hand. The "colimit lemma" (3.3) is an explicit and prac
tical criterion for deciding the issue; it will be used frequently in the seque*
*l.
Next we introduce Thomason's spectrumlevel Godement construction.
This is the analogue, for presheaves of spectra, of the hypercohomology of
a chain complex of sheaves. (The reader is urged to consult Thomason's
36
"Scholium of Great Enlightenment" at this point10 ([42],5.32).) It can also
be regarded as a generalization of the homotopy fixed point set discussed in
section 1. Virtually any of the standard theorems on abelian sheaf cohomol
ogy can be extended to hypercohomology spectra  although typically some
boundedness or finite cohomological dimension hypotheses must be imposed.
Two important examples are given, including the spectrumlevel continuity
theorem. Further examples can be found in [42],1. We also give conditions
for commuting smash products and Bousfield localization with hypercoho
mology.
It should be noted that the ThomasonGodement construction, as given
here, only applies to sites with enough points, although according to Thoma
son ([42],1.34), there is a modified version that works in the general case.
The next section surveys the elegant work of Jardine [20] [21] that es
tablishes a closed model category structure on presheaves of spectra. In this
approach, Thomason's hypercohomology spectrum H.(; E) is replaced by a
globally fibrant model for E. It is not as obvious in this setting that there is
a descent spectral sequence, so we explain this point in more detail. Finally,
we compare the globally fibrant model with H.(; E) (on a site with enough
points). The conclusion is that, subject to a finite cohomological dimension
assumption, the two constructions agree.
In the last section we give a brief introduction to C ech hypercohomology
spectra ([42],1). Once again, results from abelian sheaf theory generalize
to the spectrum setting without much difficulty  for example, "C ech de
scent for sheaf cohomology", and Artin's theorem [2] relating C ech and sheaf
cohomology on theetale site.
Throughout this work, the adjective pointwise always indicates that a
property of presheaves or morphisms of presheaves holds for the sections
on any object U of the site. For example a map E ! F of presheaves of
spectra is a pointwise weak equivalence if each map E(U) ! F (U) is a weak
equivalence, and so on.
______________________________
10The referee points out that the "scholium" is not as enlightening as it cou*
*ld be, because
the assertions found there are neither proved nor obvious. For proofs, see Chap*
*ter 4 of
[22].
37
3.1 Preliminaries on homotopy theory
3.1.1 Spectral sequences
In this section we explain our conventions on spectral sequences. For a
very complete analysis of convergence see [4], where the term "conditionally
convergent" is coined.
Suppose given a tower of fibrations of spectra
f0 f1 f2
*  X0  X1  :::
and let Wn denote the homotopyfibre of fn. Applying ss* yields an exact
couple and a spectral sequence which we index by setting
Ep;q1= ssqpWp
This is a right halfplane cohomology spectral sequence, with the "Atiyah
Hirzebruch" style of indexing (motivated by the example in which Xn is the
function spectrum F (An; E) for some spectrum E and CWcomplex A with
skeletal filtration An).
Let F p= F pss*X denote the kernel of the projection map ss*X ! ss*Xp1.
The spectral sequence converges if (i) the natural inclusion F p=F p+1 Ep;*1
is an isomorphism for all p; and (ii) lim1nss*Xn = 0. In fact the spectral
sequence is always conditionally convergent  that is, it converges if and only
if lim1rEp;qr= 0. Thus the spectral sequence automatically converges if the
E2term is bounded on the right, bounded below, or is finite in each bidegree.
For example, suppose I is a small category and i ! Xi a functor to spec
tra. Then the methods of [6],XI, lead to a tower of fibrations and a spectral
sequence
Ep;q2= limpissqXi ) ssqpholimiXi
where limp is the pth derived functor of the inverse limit. If I = , so that
X. is a cosimplicial spectrum, we have Ep;q2= ssp(ssqX.), where ssp denotes
the cohomotopy of a cosimplicial abelian group  that is, the cohomology of
the associated chain complex ([6],XI,7.3).
Finally, we remark that we could also consider an extended tower
:::  X1  X0  X1  :::
38
such as the Postnikov tower of a nonconnective spectrum. This leads to a
similar spectral sequence, but now the spectral sequence is no longer con
fined to a halfplane and the convergence question becomes more delicate.
Nevertheless, in some cases it is still conditionally convergent. Suppose, for
example, that sskXn = 0 for k > n. Then it is even possible to reindex
the spectral sequence so that it becomes a right halfplane spectral sequence
 compare [42], p. 542. See Theorem 3.19 below for an example of this
situation.
3.1.2 Closed model categories
A closed model category is a category C equipped with three distinguished
classes of morphisms, called weak equivalences, fibrations and cofibrations,
that are closed under composition and contain all identities. These classes are
subject to several axioms that codify the abstract properties of a "homotopy
theory".
(CM1): C has all finite limits and colimits.
(CM2): If f = gh is a composite morphism and any two of f; g; h are weak
equivalences, then so is the third.
(CM3): All three classes of morphisms are closed under retracts.
(CM4): Given a commutative square
A ______C
 pp 
i pp p
 pp 
?p ?
B ______D
the dotted arrow exists whenever (i) i is an acyclic cofibration and p is a
fibration, or (ii) i is a cofibration and p is an acyclic fibration ("acyclic" *
*is
shorthand for weak equivalence).
p
(CM5): Any map X! Y can be factored in two ways: (i) as X !i W  ! Y ,
j q
where i is a cofibration and p is an acyclic fibration; and (ii) as X ! Z !
Y , where j is an acylic cofibration and p is a fibration.
39
Let OE and * denote respectively "the" initial and terminal objects of the
category (usually, for us, OE = *). An object X is cofibrant if OE! X is a
cofibration and fibrant if X! * is a fibration. Part of the general philosophy
is that homotopy theory works best with a cofibrant source and a fibrant
target (see [11],3.18 and 3.22). In practice one of these two conditions holds
trivially for all  or almost all  objects of the category, while the other im*
*poses
possibly stringent conditions on an object X. For example, in the category of
presheaves of spectra discussed below, cofibrancy is a harmless condition to
impose (although not all objects are cofibrant), whereas fibrancy or the lack
of it is essentially the descent problem.
The most basic example for present purposes is the category of pointed
simplicial sets. We say that a map of pointed simplicial sets is a weak equiv
alence if the induced map on realizations is a weak equivalence in the usual
topological sense, is a cofibration if it is injective, and is a fibration if i*
*t is a
Kan fibration. With these definitions, pointed simplicial sets form a closed
model category [36]; of course this works in the unpointed category also.
Every object is cofibrant, and the fibrant objects are the Kan complexes.
This closed model category structure was extended to spectra in [5]. Here
a spectrum is defined to be a sequence of pointed simplicial sets Xn, n 0,
equipped with structure maps S1 ^ Xn! Xn+1. The homotopy groups of a
spectrum X are defined as usual by
ssnX = colimk ssn+k  Xn+k 
with the colimit taken in the evident way using the structure maps. We
say that a map of spectra f : X! Y is a weak equivalence if it induces an
isomorphism on homotopy groups, and is a cofibration if for all n 0 the
natural map from the pushout Pn in the diagram
S1 ^ Xn1 ______S1^ Yn1
 
 
 
 
? ?
Xn _____________Pn
to Yn is a cofibration of simplicial sets (in particular f is a spacewise cofib*
*ra
tion). Finally, we say that f is a fibration if it has the right lifting proper*
*ty
40
as in CM4(ii) with respect to all acyclic cofibrations. With these definitions,
the category of spectra is a closed model category [5]. The fibrations are in
particular spacewise fibrations, and can be described explicitly (see [5]). It
turns out that the fibrant spectra are precisely the spectra (in the classical
sense) whose constituent spaces are Kan complexes. The cofibrant spectra
are just the spectra with injective structure maps.
3.1.3 Filtered colimits of spectra
Consider the closed model category of pointed simplicial sets, as defined
above. Then:
Proposition 3.1 Filtered colimits preserve cofibrations and fibrations (in
particular a filtered colimit of Kan complexes is a Kan complex). Filtered
colimits of Kan complexes preserve weak equivalences.
For cofibrations, this is obvious, since a cofibration is just an injective
map. The proof for fibrations may be sketched as follows. Call a simplicial
set K small if it has only finitely many nondegenerate simplices. Then it
is easy to see that the functor Hom(K; ) commutes with filtered colimits
(even when the maps in the colimit are not injective!). But Kan fibrations
are defined in terms of the right lifting property with respect to certain maps
of small simplicial sets (inclusion of the boundary minus one face of n into
n), and the result follows. A similar argument applies to weak equivalences,
provided that the topological homotopy groups occurring in the definition of
weak equivalence can be replaced by simplicial homotopy groups. This is
possible since we have assumed the spaces in question are Kan complexes.
Similar arguments apply to the BousfieldFriedlander closed model cate
gory structure on spectra, although the situation is complicated slightly by
the different notion of fibration. Call a spectrum X a Kan spectrum if each
space Xn is a Kan complex. The result is:
Proposition 3.2 Filtered colimits of spectra preserve cofibrations and fibra
tions, and filtered colimits of Kan spectra preserve weak equivalences.
In fact the assertions for cofibrations and weak equivalences hold for
pseudofiltered colimits. We note that up to functorial weak equivalence,
41
any spectrum can be replaced by a Kan spectrum; this will be done tacitly,
when necessary, in the sequel.
We now turn to the (critical!) question of when homotopy limits and
related constructions commute with filtered colimits. Our goal is to give a
simple criterion that is easily applied in practice  although it will take lon*
*ger
to state this criterion than to prove it. Throughout this section "spectral
sequence" means right halfplane cohomology spectral sequence.
We assume given the following data:
o A spectrum X and a filtered system of spectra Xi, i 2 I.
o A compatible family of maps fi : Xi! X, so there is an induced map
colim Xi! X.
o Spectral sequences E*Xiand E*X converging conditionally to ss*Xiand
ss*X respectively, and that are natural with respect to the maps in the
filtered system and with respect to the fi. (In particular the associated
filtrations on ss*Xi and ss*X are assumed natural in this sense.)
We say that a spectral sequence is bounded on the right if there exists d
such that Ep;*2= 0 for p > d, and is bounded below if there exists m such that
E2*; q= 0 for q < m. Note that either condition implies convergence. If we
are given a family E*(i) of spectral sequences, we say that E*(i) is uniformly
bounded on the right if there is a fixed d that works for all i, etc. The next
proposition will be called the "colimit lemma".
Proposition 3.3 Suppose that (i) either the E2terms E2(Xi) are uniformly
bounded on the right, or else they are uniformly bounded below; and (ii)
the map f* : colimiE2(Xi) ! E2(X) is an isomorphism. Then the map
f : colimiXi! X is a weak equivalence.
Proof: Since the index category I is filtered, the colimit of the spectral se
quences E*(i) is still a spectral sequence. However it need not, in general,
converge to the homotopy of the colimit  the trouble is that nontrivial el
ements of ss*colimiXi = colimiss*Xi can have infinite filtration. But hy
pothesis (i) rules out this possibility; the induced filtration on colimissqXi
is finite for each q, and the colimit of the spectral sequences converges to
the homotopy of the colimit. Then condition (ii) implies that the target
42
spectral sequence is also convergent, the induced map on E1 terms is an
isomorphism, and f is a weak equivalence.
As an illustration, suppose that I and J are small categories with I filtered
and J arbitrary. Let Xi;jbe an (I x J)diagram of spectra, and consider the
natural map
f : colimiholimjXi;j! holimjcolimiXi;j
Here "holim" means the homotopy limit as in [6], extended to spectra as in
[42], section 5. For present purposes, all the reader needs to know is that (1)
holim is a functor from Jdiagrams Yj of spectra to spectra; and (2) there is
a natural conditionally convergent spectral sequence
Ep;q2= limpj(ssqYj) ) ssqpholim Yj
where limp denotes the pth derived functor of the inverse limit. Recall that
a spectrum X is bounded above if its homotopy groups vanish above some
dimension. Then we obtain as an immediate corollary:
Proposition 3.4 Suppose that (i) either there exists d such that for all q; i
the groups limpj(ssqXi;j) vanish for p > d, or else the Xi;jare uniformly
bounded above; and (ii) the natural map
colimilimpjss*Xi;j! limpjcolimiss*Xi;j
is an isomorphism. Then the map
f : colimiholimjXi;j! holimjcolimiXi;j
is a weak equivalence.
For example if J = , so we have a filtered system of cosimplicial spectra,
condition (ii) holds automatically but condition (i) is a nontrivial assumption
about finite cohomological dimension. On the other hand if J is the category
0 1 2 :::, so a Jdiagram is a tower, then condition (i) holds with
d = 1 but condition (ii) can easily fail. Let's call a tower of spectra Yj stab*
*le
if for each fixed q there is an mq such that ssqYj! ssqYj1 is an isomorphism
for all n > mq (for example, a Postnikov tower). We call an Idiagram of
towers Xijas above uniformly stable if each tower Xij(i fixed) is stable and
43
moreover each number mq depends only on q and not on i. In this case
ssqlimjXij= ssqXi;nwith n >> 0 depending only on q. Then (ii) obviously
holds and we get the easy but important corollary:
Corollary 3.5 Let Xijbe a filtered system (in i) of towers (in j), and sup
pose the system is uniformly stable. Then
~=
colimilimjXi;j! limjcolimiXi;j
Remark 3.6 The assumption on E2terms in the colimit lemma can, of
course, be replaced by the same assumption on some other fixed Erterm.
An interesting example is discussed in the introduction to Section 4.
3.2 The Godement construction for spectra
Let be a site with enough points  for example, the Zariski,etale, or
Nisnevich site of a scheme X. Let E be a presheaf of spectra on . Then
the construction of the Godement resolution for ordinary abelian sheaves
carries over almost verbatim to presheaves of spectra. In fact let A be any
category with filtered colimits, products and coproducts, and consider an
Avalued presheaf F. Then we can define the stalk Fp = p*F at a point p
as a filtered colimit over neighborhoods of p as in 2.6. The formula given
there for the right adjoint p* works fine too. The Godement construction
then produces a cosimplicial A valued presheaf T .F , whose sections on X
give a cosimplicial object in A . In particular we get a Godement functor
E! T .Efrom presheaves of spectra to cosimplicial presheaves of spectra,
and we define the hypercohomology of X with coefficients in E by
H. (X; E) = holim T .E(X)
In fact H. (; E) is itself a presheaf on the site. Note that ss*T .E= T .ss*
**E .
Hence the BousfieldKan machinery discussed above yields a conditionally
convergent spectral sequence
Ep;q2= Hp(X; "ssqE) ) ssqpH .(X; E)
Many basic results about cohomology can now be extended rather easily
to hypercohomology spectra. We illustrate this with some examples that will
44
be important later. Given a site , a presheaf of spectra E and an object
X of the site, we say that cd (X; E) is bounded, or cd < 1, if for some d
Hp(X; "ssqE) = 0 for all p > d and all q. If the presheaf and/or the object are
varying over a family, the phrase "uniformly bounded" means that a fixed
d works for all presheaves and/or objects in the family. A condition is said
to hold stalkwise if it holds for the stalks of the presheaf in question. Often
this makes sense even without reference to points of the site  for example,
a presheaf of spectra is stalkwise bounded above if for some m the sheaves "ssq
vanish for all q > m.
Theorem 3.7 Suppose that is a noetherian site, Eia pseudofiltered system
of presheaves of spectra on . Fix an object X of and suppose that either
(a) cd (X; Ei) is uniformly bounded, or (b) the E iare stalkwise uniformly
bounded above. Then the natural map
colimiH. (X; Ei) ! H. (X; colimiEi)
is a weak equivalence.
Proof: One simply checks the conditions of the colimit lemma 3.3. Con
dition (i) of that lemma holds by assumption, and condition (ii) is immediate
from the algebraic version 2.10 .
This result is important because it will allow us to commute smash prod
ucts and Bousfield localizations with hypercohomology (see the end of this
section).
Similarly, we can promote the continuity theorem 2.11 to the spectrum
level. Thus suppose Xi is an admissible system of schemes (i.e. a cofiltered
system of quasicompact and quasiseparated schemes, with affine transition
maps), with X = limiXi. Recall that the main case to keep in mind is that
of a filtered system of rings, such as the system used to define henselizations.
Let E be a presheaf of `torsion spectra on a sufficiently largeetale site, and
suppose that E is continuous in the sense that for any admissible system Ui
in the site, the natural map colimiE (Ui) ! E(U) is a weak equivalence.
Theorem 3.8 Suppose that Xi is an admissible system of schemes, E a con
tinuous presheaf of `torsion spectra as above. Suppose also that either (a)
cd (Xi; E) is uniformly bounded, or (b) E is stalkwise bounded above. Then
the natural map
45
colimiH .et(Xi; E) ! H.et(X; E)
is a weak equivalence.
Again, the proof is immediate from the colimit lemma together with the
continuity theorem 2.11. The one point to note is that the presheaves ssqE
are continuous by assumption, and hence so are the associated sheaves "ssqE.
The main example of a continuous presheaf of spectra is the Ktheory
presheaf. It is continuous by a theorem of Quillen ([35], 7,2.2). The point is
that the category of locally free sheaves on the limit scheme X is the "colimit"
of the corresponding categories on the Xi.
Our main application of continuity is to the determination of the stalks
of hypercohomology presheaves.
Theorem 3.9 Let X be a separated noetherian scheme of finite Krull di
mension. Suppose X is uniformly `bounded. Let E be a presheaf of `torsion
spectra on Xet and regard F = H .et(; E) as a presheaf on XZar, XNis, or
Xet. Then
(i) The Zariski stalk of F at x is H.et(O x; E)
(ii) The Nisnevich stalk of F at x0! x is H.et(O hx0; E)
(iii) Theetale stalk of F at x! x is H.et(O shx; E)
Proof: Consider for example case (iii), where we can work on the re
strictedetale site. If U = Spec Oshx, then U = limiUi, where Ui ranges
over the cofiltered system ofetale neighborhoods of x. The Ui can be taken
affine and hence form an admissible system. Moreover the assumptions on
X are inherited by the Ui, so that cdet`(Ui) is bounded uniformly in i. Hence
Theorem 3.8 applies, completing the proof.
Now suppose W is a fixed spectrum. Then for any presheaf of spectra E
there is a natural map
W ^ H. (X; E) ! H. (X; W ^ E)
which in general need not be a weak equivalence. However smashing with
a finite spectrum commutes with arbitary homotopy limits and products,
hence:
46
Proposition 3.10 If W is a finite spectrum, then for any presheaf of spectra
E and object X the map
W ^ H. (X; E) ! H. (X; W ^ E)
is a weak equivalence.
If W is not finite then some condition has to be imposed on the site, the
object, the presheaf or all three. The next result is not the most general one
can imagine, but takes the form that we will typically need.
Proposition 3.11 Suppose cd`(X) < 1,and that either W is an `torsion
spectrum or E is a presheaf of `torsion spectra. Then the map
W ^ H. (X; E) ! H. (X; W ^ E)
is a weak equivalence.
Proof: W is a directed colimit of finite spectra Wi, which can be taken
`torsion if W is. Then
W ^ H. (X; E) = colimiWi^ H. (X; E) = colimiH .(X; Wi^ E)
by the preceeding proposition. Now each presheaf Wi^ E is a presheaf of
`torsion spectra, so Theorem 3.7 applies.
Finally, recall that a spectrum E is smashing if LE Z = Z ^ LE S0 for
all Z, or equivalently LE commutes with directed colimits. For example the
complex Ktheory spectrum is smashing, but the mod ` Moore spectrum is
not (completion doesn't commute with direct limits).
Corollary 3.12 Suppose cd`(X) < 1, E is smashing, and that either LE S0
is an `torsion spectrum or E is a presheaf of `torsion spectra. Then the
natural map
LE H. (X; E) ! H. (X; LE E)
is a weak equivalence.
47
3.3 Jardine's closed model category
Let be a site and consider the category P of presheaves of spectra on .
We would like to define a closed model category structure on P in a way that
depends on the Grothendieck topology, not just the underlying category. In
particular the weak equivalences should depend on the topology, and the
fibrant objects should be the objects satisfying descent.
The most natural definition of weak equivalence is stalkwise weak equiva
lence. We say that a map of presheaves E ! F is a stalkwise weak equivalence
if the induced map of associated abelian sheaves "ss*E!"ss*Fis an isomor
phism. Note this definition makes sense even when doesn't have enough
points. If does have enough points, as is the case in all the examples we
care about, then a map is a stalkwise weak equivalence if and only if the
induced maps on stalks are weak equivalences of spectra. However even in
this case it will be convenient to have available the definition as stated. Note
that every pointwise weak equivalence is a stalkwise weak equivalence, since
pointwise weak equivalence just means that the induced map on homotopy
presheaves is already an isomorphism. The converse is obviously false.
Remark. Any stalkwise weak equivalence is also a stalkwise homology iso
morphism. This is clear for a site with enough points, but not at all obvious
in the general case. See [22], 2.9.
A map of presheaves is a cofibration if it is a pointwise cofibration, and
is a fibration if it has the right lifting property with respect to cofibrations
that are also stalkwise weak equivalences. The following theorem is due to
Jardine [21]. Earlier results of this type, but for sheaves of simplicial sets,
were proved BrownGersten [7] in the Zariski case and Joyal in the general
case.
Theorem 3.13 Let be an arbitrary site. Then with the above definitions,
the category P of presheaves of spectra on is a closed model category.
In fact Jardine shows that P is a proper simplicial model category. The
proof of Theorem 3.13 is discussed at the end of this section.
In order to avoid possible confusion with either pointwise fibrations or the
local fibrations of [20], the fibrations in this closed model category structure
are called global fibrations. Similarly a presheaf is globally fibrant if it is
48
fibrant in the closed model category structure of Theorem 3.13. Note that
the cofibrant objects are just the presheaves which are pointwise cofibrant
in the sense of BousfieldFriedlander. In particular any presheaf is pointwise
weak equivalent to a cofibrant presheaf, so there is no harm in assuming that
our presheaves are cofibrant.
Now by CM5, given any presheaf E we can find a globally fibrant presheaf
GE and a map E ! GE which is both a cofibration and a stalkwise weak
equivalence. It follows from the proof of Theorem 3.13 that GE can be
chosen functorially. It is convenient, however, not to be tied down to any
particular choice; any such stalkwise weak equivalence (whether or not it
is a cofibration) will be called a globally fibrant model for E. This globally
fibrant model (which we will see shortly is welldefined up to pointwise weak
equivalence) will serve as an alternative to, and generalization of, Thomason's
hypercohomology spectrum. Before justifying this assertion, we summarize a
few properties of Jardine's closed model category structure, properties which
begin to illustrate the elegance of this approach. We first consider the be
haviour of the three classes of morphisms under direct image, inverse image,
and extension by zero.
Proposition 3.14 Let f :  ! 0 be a site morphism. Then the presheaf
inverse image functor f# preserves cofibrations and stalkwise weak equiv
alences, and the direct image functor f# preserves cofibrations and global
fibrations.
The proof of the first statement is easy, using the fact that f# is defined
in terms of pseudofiltered colimits of spectra. Obviously f# preserves cofibra
tions. That f# preserves global fibrations is a formal consequence of the fact
that it has a left adjoint, namely f# , that preserves cofibrations and weak
equivalences. It is clear that f# does not generally preserve global fibration*
*s,
and f# does not generally preserve stalkwise weak equivalences (consider for
example a scheme with the Zariski topology and a map to Spec of a field).
Here is a simple example that will be used later. Consider a scheme X
and a presheaf E which is globally fibrant on theetale site. Applying the
proposition to the restriction morphisms Xet! XNis! XZar shows that E
is globally fibrant on the Nisnevich and Zariski sites as well.
49
Corollary 3.15 Let j : =U ! denote inclusion of a local site as in sec
tion 2.5. Then the extension by zero functor j! preserves cofibrations and
stalkwise weak equivalences, and j# preserves global fibrations.
This follows immediately because j# = k# and j!= k# , where k is as in
section 2.5. We also deduce:
Corollary 3.16 Every global fibration is a pointwise fibration.
For suppose given a global fibration ss : E ! F . We wish to show that
for every object U of the site, E (U) ! F (U) is a fibration of spectra. But
the "sections on U" functor has a left adjoint that preserves cofibrations and
stalkwise weak equivalences  namely, the functor E 7! j!cE , where cE is
constant presheaf on =U associated to the spectrum E.
As noted above, a stalkwise weak equivalence need not be a pointwise
weak equivalence. There is one case, however, where we do get pointwise
weak equivalence. I thank Paul Goerss and the referee for providing the
proof of the following important proposition.
Proposition 3.17 Suppose f : E ! F is a stalkwise weak equivalence of
globally fibrant presheaves. Then f is a pointwise weak equivalence.
Using CM5 and CM2, we reduce immediately to two cases: either f is a
global fibration, or f is a cofibration. If f is a global fibration, the adjoi*
*nt
functor argument used above shows that f is a pointwise weak equivalence.
If f is a cofibration, then f is a homotopy equivalence  in fact, E is a de
formation retract of F . To explain this assertion, we need a tiny bit of
the "simplicial" model category1structure on presheaves of spectra. If A is
any presheaf of spectra, let A denote the path object of simplicial maps
1 ! A (defined pointwise and levelwise). Homotopies can then be defined
using this path object, in the usual way. Now since E is globally fibrant,
there is a lift r in the diagram
=
E ______E
 pp 
f  pp 
 pp r 
?p ?
F ______*
50
so that rf = 1E . On the other hand, since F is globally fibrant, the obvious
1
map F  ! F x F is a global fibration (this is a simple special case of
Quillen's axiom "SM7" for simplicial model categories). Hence there is a lift
in the diagram
E ______F1
 p 
 pp 
f  pp 
 pp 
?p ?
F _______F(1;fr)
where the top map is the constant homotopy of f. This lift is a homotopy
1F ~= fr. Thus f is a homotopy equivalence as claimed. It follows imme
diately that for any object U, E (U) ! F (U) is a homotopy equivalence of
fibrant spectra and hence a weak equivalence.
In particular this shows that globally fibrant models are welldefined up
to pointwise weak equivalence.
Elegant though this machinery may be, it is still in need of justification.
The crucial test is that there should be a descent spectral sequence for the
pointwise homotopy of a globally fibrant presheaf. The key step in setting
up such a spectral sequence is an analysis of globally fibrant models for
EilenbergMaclane spectra. So let A be a presheaf of abelian goups. An
EilenbergMaclane presheaf of type (A; n) is a presheaf of spectra A such
that ssnA is isomorphic to A and sskA is identically zero for k 6= n.
Proposition 3.18 Let A be an EilenbergMaclane presheaf of spectra of type
(A; n), with globally fibrant model GA . Then for all k 2 Z and all objects U
of the site,
ssk(GA )(U) ~=Hnk (U; aA)
We sketch a proof (see [22], Ch.4 for a proof that is more in the spirit of
Thomason's "Scholium of Great Enlightenment"). It is convenient to work
with spaces (=simplicial sets), so we first reduce to that case and use the
closed model category structure for presheaves of spaces (see the end of this
section). Since formal suspension and desuspension preserve globally fibrant
51
models, we can assume k 0. Since A can be assumed pointwise fibrant,
and the zeroth space presheaf of GA is a globally fibrant presheaf of spaces,
we reduce to the analogous statement for EilenbergMaclane presheaves of
spaces. We will continue to use the notation A for such a presheaf.
Now let P ss; Psag; Pch denote respectively the categories of presheaves of
simplicial sets, simplicial abelian groups, and nonnegatively graded chain
complexes. There is an evident pair of adjoint functors
Z : P ss! P sag; i : P sag! Pss
as well as a pair of mutually inverse equivalences of categories
N : P sag! Pch; T : P ch! P sag
(See [27]; N is the normalized chain complex functor.) Replacing A by the
nth Postnikov section of ZA , we can assume A is a simplicial abelian group.
In fact up to natural pointwise weak equivalence we can even assume A =
T (CA ), where CA is the chain complex consisting of the presheaf A in degree
n and zero elsewhere. Now let aA ! I.be an injective resolution of the sheaf
aA, and form the chain complex JA which is zero above degree n, is equal
to Ink for 0 < k < n and with I0 = Ker (In! In1). Then the natural map
CA ! JA is a stalkwise homology isomorphism, and the pointwise homology
of JA is given by HkJA = Hnk (U; aA).
We claim that iT JA is a globally fibrant model for A . Since homology in
Pch corresponds to homotopy in P sag, we have a stalkwise weak equivalence
A ! T (JA ). Finally, it is not hard to show that iT JA is globally fibrant 
for example, by using the argument for the analogous theorem in [15].
To construct the descent spectral sequence, we start from a Postnikov
tower {P nE} of the presheaf E. Let Fn denote the fibre of P nE! P n1E.
Easy formal arguments lead to a commutative ladder
:::_________Pon1Ee ________PonEe__________o:::e
 
 
 
? ?
:::________GPon1Ee ______oGPenE _________o:::e
in which (i) the bottom row consist of global fibrations and globally fibrant
objects; (ii) limnGP nEis globally fibrant; and (iii) the fibre of GP nE! GP *
*n1E
52
is a globally fibrant model GFn of the EilenbergMaclane presheaf Fn. Hence
we have
ssk(GFn)(U) = Hnk (U; "ssnE)
If we assume the cohomology groups on the right vanish for nk >> 0, then
both towers are uniformly stable (see 3.1.3) and hence
E = limnP nE! limnGP nE
is a stalkwise weak equivalence. In other words, limnGP nE is a globally
fibrant model for E . The tower of fibrations GP nE then yields a spectral
sequence with
Ep;q1= H2pq(X; "sspE)
The indexing may look bizarre to some readers, but we can formally
reindex the spectral sequence to get the usual form of the E2term (see
[42],p.542; note the same peculiar indexing shows up if one constructs the
AtiyahHirzebruch spectral sequence by a Postnikov resolution of the target
instead of a skeletal filtration of the source). The result is:
Theorem 3.19 Let E be a presheaf of spectra on a site . Let X be an object
of the site and suppose cd (X; E) = d < 1. Then there is a convergent right
halfplane cohomology spectral sequence
Ep;q2= Hp(X; "ssqE) ) ssqp(GE )(X)
From this spectral sequence it is morally clear that the constructions of
Jardine and Thomason must agree on a site with enough points, at least
under finite cohomological dimension hypotheses. To prove this, we need to
check two things.
Proposition 3.20 Let be a site with enough points, E a presheaf of spec
tra on . Then the hypercohomology presheaf H .(; E) is globally fibrant. If
cd E < 1, then the natural map E ! H .(; E) is a stalkwise weak equiva
lence.
53
Thus the hypercohomology presheaf of E is a globally fibrant model of E.
To get the stalkwise weak equivalence in the proposition, consider the map
on stalks at p
f : colim E(U) ! colim H.(U; E)
where the colimits are over the filtered category N opp. The finite cohomo
logical dimension hypothesis ensures that the colimit of the descent spectral
sequences for H .(U; E) converges to the homotopy of the colimit (Proposi
tion 3.3). This colimit spectral sequence collapses to its vertical edge, and
ss*f can be identified with the natural isomorphism (ss*E )p ~=("ss*E)p.
The adjoint functor trick used in (3.14) shows that each cosimplicial term
of the cosimplicial presheaf T .Eis a globally fibrant presheaf. One then has to
show that the homotopy limit remains globally fibrant, using the techniques
of [6]; see [20],3.3.
We conclude this section with a discussion of the proof of Theorem 3.13.
The theorem was proved first for presheaves of simplicial sets in [20], and
then extended to spectra in [21], using the techniques of [5]. The key point is
to get a closed model category structure on the category P ssetsof presheaves
of simplicial sets.
In P ssetsthe definition of stalkwise weak equivalence is not as simple as it
is for spectra, because one cannot even define presheaves of homotopy groups
without a compatible family of basepoints. Suppose that E is a presheaf of
simplicial sets and U is an object of the site. Each choice of basepoint
e0 2 E(U)0 defines a compatible family of basepoints on the local site =U,
because U is a terminal object of the latter. Hence for n > 0 we get a presheaf
of homotopy groups ssn(E U ; e0) on =U. If n = 0 we have a presheaf of sets
ss0E on . We then say that a map f : E ! F is a stalkwise weak equivalence
if (i) "ss0(f) is an isomorphism of sheaves; and (ii) for all objects U, basepo*
*ints
e0 2 E(U)0, and n > 0, the induced map "ssn(E U ; e0) ! "ssn(F U ; f(e0)) is
an isomorphism of sheaves on =U.
It is not hard to check that if the site has enough points, then f is a
stalkwise weak equivalence if and only if p*E ! p*F is a weak equivalence
for a sufficient set of points p. In this case several steps of the proof of
Theorem 3.13 become much easier.
As in the case of spectra, the cofibrations are the pointwise cofibrations
and the global fibrations are defined by the right lifting property with re
54
spect to acyclic cofibrations. Axioms CM1 and CM3 (as well as the mostly
redundant requirement that the three classes are closed under composition
and contain all identities) are then easily verified. If the site has enough
points, CM2 is obvious (CM2 is also obvious for spectra). In the general
case, however, one of the three implications of CM2 runs into a problem
with basepoints, as the reader will quickly discover on inspection (see [20],
1.11 and 2.1). Half of CM4 is true by definition. It turns out that the other
half follows formally from CM5 by a trick due to Joyal, using nothing more
than the fact that cofibrations are stable under cobase change ([20], p. 64).
The crux of the matter, then, is CM5. This axiom is usually proved using
Quillen's "small object argument". Consider first the problem of factoring a
map f : E ! F as an acylic cofibration followed by a global fibration. To
make the small object argument work, two ingredients are needed. First, we
need to know that acyclic cofibrations are stable under (i) cobase change,
and (ii) pseudofiltered colimits. If the site has enough points, (i) is obvious
((i) is also obvious for spectra, because of the MayerVietoris sequence in
homotopy associated to a pushout square). The general case of (i) is much
harder ([20], 2.2). Condition (ii) is clear in all cases.
Second, we need to know that there is a set S of acyclic cofibrations such
that (i) S detects global fibrations (in the obvious sense); and (ii) S is "sma*
*ll",
in a sense to be indicated. Here we fix an infinite cardinal number ff greater
than the cardinality of the set of morphisms of the underlying category of the
site. Then for S we take a set of representatives of the isomorphism classes
of acyclic cofibrations A ! B such that the cardinality of B(U)n is at most
ff for all objects U and all n 0. The key fact is that this S detects global
fibrations; see [20], Lemma 2.4, for the interesting proof.
Returning to an arbitrary morphism f : E ! F , the "small object ar
gument" (or "infinite glueing construction" in [11]) begins by forming the
pushout
a
Ai ______E
 
 
 j
 
a ? ?
Bi ______E1
55
in which Ai! Biis ranging over all maps of elements of S into f. This yields
a factorization E ! E1 ! F in which the first map is an acyclic cofibration.
The second map is not a global fibration, but at least all the lifts associated
to the given Ai! Bi have been "filled in". Iterating this construction some
transfinite number of times (taking colimits at the limit ordinals) yields a
factorization
p
E !i E1 ! F
in which E1 is a certain colimit over some large ordinal. The point now is
that if we choose this ordinal to be sufficiently large (bigger than the cardi
nality of the power set of the set of morphisms will do), then the cardinality
bound defining the elements of S forces every map of an element A ! B of
S into f to factor through some lower stage of the colimit. Then the desired
lift exists by construction.
The second part of CM5 can also be achieved by a small object argument,
or by the following cheap trick. In the closed model category structure on
simplicial sets, the factorizations of CM5 can be chosen in a functorial way.
Hence any map of presheaves f : E ! F can be factored as
j ______q
E ______D F

k
 r
?
C
where j is a cofibration, q is a pointwise weak equivalence, and q has been
factored as an acylic cofibration k followed by a global fibration r, using the
part of CM5 already proved. Then kj is a cofibration and r is an acyclic
global fibration (using CM2). See also the more interesting argument in [20],
which shows that the acyclic fibration can be taken to be a pointwise weak
equivalence.
3.4 C ech hypercohomology spectra
Our discussion of C ech hypercohomology will be brief and somewhat impre
cise. See [42],1, for details. Fix a site and an object X of the site. Let
56
U = {Ui! X} be a covering family. As a cosimplicial object, the com
plex defining ordinary C ech cohomology (2.9) makes perfectly good sense
for presheaves with values in any category with products. In particular,
given a presheaf of spectra E, we obtain in this way a cosimplicial spectrum
C.(U =X; E). Taking an appropriate filtered colimit over a cofinal family of
covers leads to the absolute C ech complex C.(X; E). The corresponding C ech
hypercohomology spectra are then defined by
H.(U =X; E) = holim C.(U =X; E)
and
H.(X; E) = holim colimU C.(U =X; E)
As with the Godement complex, each of these formulae lead to a descent
spectral sequence with E2term given by the appropriate C ech cohomology
groups. Of course one cannot, in general, reverse the order of the holim and
the colim in the second formula above. Using (3.3), however, we obtain:
Proposition 3.21 If E is pointwise bounded above, then the natural map
colimU H.(U =X; E) ! H.(X; E)
is a weak equivalence.
(Note that if ssqE = 0 for q > m, then for all U we have ssqH .(U =X; E) = 0
for q > m.)
Example 3.22 Consider theetale site of a field k. We can take the finite
Galois extensions of k as cofinal system of covers. Then for E pointwise
bounded above, we have (reverting to the notation of section 1)
colimffH.(Gff; E) ~=H .(k; E)
where Gffranges over the Galois groups of such extensions. More generally
this works with any Galois extension M of k in place of the separable closure.
"C ech descent for sheaf cohomology" takes the following form:
57
Theorem 3.23 Suppose that either E is stalkwise bounded above, or that
there is a uniform bound on cd(U; E) (uniform in U, as U ranges over the
individual objects of all covering families in sight). Then there is a natural
weak equivalence
H.(X; E) ~=H .(U =X; H.(; E))
Example 3.24 In the notation of the preceeding example, suppose that ei
ther E is stalkwise bounded above or that E is an `torsion spectrum and
cd`k = d < 1. Let k0=k be a Galois extension with group G. Then
H.(k; E) ~=H .(G; H.(k0; E))
Note cd`k0 d.
As a final and very pretty example we show how Artin's Theorem 2.9
extends to hypercohomology spectra ([42], Proposition 1.54).
Theorem 3.25 Suppose X is quasiprojective over a noetherian ring of finite
Krull dimension, and E is an additive presheaf of spectra on Xet. Then there
is a natural chain of weak equivalences between H.et(X; E) and H.et(X; E).
To prove this one first reduces to the case where E is pointwise bounded
above, by using a Postnikov resolution. Then consider the natural maps
H.et(X; E) ff!H .et(X; H.et(; E)) fi H .et(X; E)
Note that fi is a weak equivalence by Theorem 3.23. As for ff, note
first that the C ech hypercohomology spectral sequences converge, so it is
enough to show that ff is an isomorphism on E2terms  i.e., the map of
coefficient presheaves ssqE ! ssqH .et(; E) induces an isomorphism on C ech
cohomology. But this map is an isomorphism on stalks by the second part
of Proposition 3.20, and hence induces an isomorphism on sheaf cohomology
of the associated sheaves. Then Artin's Theorem 2.9 completes the proof.
In conclusion, we note that it might be interesting to study Verdier hy
percohomology spectra (see section 2.9).
58
4 Descent theorems in Ktheory
Let E be a presheaf of spectra on a site , X an object of the site. If
the site has enough points, so that Thomason's hypercohomology spectrum
H. (X; E) is defined, then the problem of descent is to determine whether or
not the natural map j : E(X) ! H. (X; E) is a weak equivalence. If it is a
weak equivalence, then we get a descent spectral sequence as in section 3,
but now converging to ss*E (X). Typically X is a terminal object of the site,
and in the cases of interest any proof of descent for X will prove descent for
an arbitrary object of the site as well. So we may drop the assumption of
enough points, and (bearing in mind Proposition 3.20) rephrase the descent
problem in terms of Jardine's closed model category structure.
Descent Problem: Let j : E ! GE be a globally fibrant model for the
presheaf E. Is j a pointwise weak equivalence?
Recall that the map j is always a stalkwise weak equivalence, by definition.
Now let X be a separated noetherian regular scheme of finite Krull di
mension,and let E = K denote the Ktheory presheaf on one of the sites
XZar; XNis; Xet. Then K satisfies descent on XZar by the BrownGersten
theorem, and satisfies descent on XNis by a theorem of Nisnevich; these re
sults are discussed in the first two sections below. Descent on theetale site
is precisely the subject of the LichtenbaumQuillen conjectures. In this case,
however, there are two complicating factors. The first is that, for technical
reasons, we need to work at a fixed prime `with the `adic completions. The
second and more serious complication is that it simply isn't true that K
theory satisfies `adic descent. In fact, in the form stated above descent fails
even for finite fields F, because the `adic hypercohomology spectrum has a
nonvanishing ss1, isomorphic to H1et(Spec F ; Z`(0)) ~=Z`. A natural reaction
to this annoyance is to replace GK (or equivalently, H.) by its (1)connected
cover. But then for example when F is a number field, we find that ss0(GK)^
contains a contribution from the Brauer group that is simply not present in
K0F . It seems that the best one can hope for is an equivalence above some
dimension depending on the `adicetale cohomological dimension d of the
scheme X.
`adic Etale Descent Problem for Ktheory: Is the globally fibrant
model j : K ! GetK a pointwise weak equivalence after `adic completion
59
and passing to some connected cover?
One guess is that sskj^ is an isomorphism for k > 2d; for another see [22],
Conjectures 7.35 and 7.36. The BeilensonLichtenbaum conjectures suggest
yet another guess; see for example [40].
Thomason's descent theorem [42], which forms the main subject of this
paper, says that j induces an isomorphism on "Bottperiodic" homotopy
mod powers of `. Equivalently, ^L(j) is a weak equivalence, where ^Ldenotes
Bousfield localization with respect to mod ` topological complex Ktheory.
Spectra of the form ^L() are always nonconnective (or trivial!), so connec
tivity obstructions of the sort discussed above do not arise. The proof we
will sketch has two main steps  (i) the proof for fields, all discussion of
which is postponed to the final section 5; and (ii) the deduction of the gen
eral case from (i). With the ThomasonJardine machinery of section 3 and
the Nisnevich descent theorem in hand, the proof of step (ii) is remarkably
simple.
For practical applications, the descent problem as formulated above is
still missing two ingredients. First, the main point of establishing a pointwise
weak equivalence E ! GE is not merely to get a descent spectral sequence,
but to get one whose E2term is actually computable. This means that we
need to at least identify the sheaves associated to the presheaves ssqE. We
state this as:
Descent Problem, Addendum: "Determine" the associated sheaves "ss*E.
In practice this amounts to identifying the stalks, and by the continuity
theorem 2.11 this in turn reduces to to computing ss*E on the appropriate
local rings. Thus in the Ktheory setting, we need to know the Ktheory of the
Zariski, hensel, or strict hensel local rings, as the case may be. In the strict
hensel case the answer is completely known, thanks to the GilletThomason
SuslinGabber rigidity theorem and Suslin's theorem on algebraically closed
fields. This is discussed further in section 4.11 below; in particular we show
how Thomason's descent theorem yields a spectral sequence with the desired
E2term.
Second, one needs to know that the descent spectral sequence converges.
The cheapest way to ensure convergence is to assume that the presheaf (or
even the site) in question has finite cohomological dimension. We generally do
60
so in order to simplify the exposition. It is worth noting, however, that there
are interesting cases where the cohomological dimension is infinite but one has
convergence nevertheless. For example, suppose `= 2 and X = Spec R with
R either a number field admitting a real embedding or a ring of Sintegers
in such a field. Then cdet2X = 1. Nevertheless, the 2adic descent spectral
sequence for H.et(X; K) converges  in fact, the filtration has length at most
3 and the spectral sequence collapses at E4. This follows from the analogous
statement for R , together with classical calculations of Tate. As a result,
many of the general results of the last section still apply (see Remark 3.6).
Finally, some comments may be in order concerning our schemetheoretic
hypotheses. The three Ktheory descent theorems stated here all assume that
X is separated and regular. The sole reason for this assumption is to ensure
that KX ~= K0X, where K0X is the Ktheory spectrum associated to the
exact category of coherent sheaves on X. This is an instance of "reduction by
resolution" [35], which in turn depends on Theorem 2.1. Thus the "separated
and regular" hypothesis can be dropped if we are only interested in K0X. It
can also be dropped if we replace Ktheory with Bass Ktheory, using the
deep theorem of Thomason [44].
For Thomason'setale descent theorem, we also assume that X is uni
formly `bounded (i.e. there is a uniform bound on cd`k as k ranges over the
residue fields of X). One reason for this assumption (which is satisfied in most
cases of interest) is to ensure that cd`X < 1, so that the descent spectral
sequence converges. Moreover it is a property inherited by every object of the
(restricted)etale site of X. Hence we are free to pass back and forth between
hypercohomology presheaves a la Thomason and globally fibrant models a
la Jardine, using Proposition 3.20.
Some further (but mild) restriction on the residue fields is necessary.
Our assumption will be that the residue fields are "`good" as defined in
section 4.4. This hypothesis, which is taken from [22], could undoubtedly be
weakened somewhat; it has the advantage that it is easily remembered and
applied.
4.1 BrownGersten descent
Let X be a scheme, E a presheaf of spectra on the Zariski site of X. We say
that E is a MayerVietoris presheaf if for every pair of open sets U; V X,
the diagram
61
E(U [ V ) ______E(U)
 
 
 
? ?
E(V ) ______E(U \ V )
is a homotopy fibre square. Alternatively, we can express this as an excision
condition for closed pairs. Given closed subschemes Y,Z with Z Y , let
Y;ZE denote the presheaf assigning to each open U the homotopy fibre of
E(U  Z) ! E(U  Y ). We write Y for Y;OE. Then E satisfies excision if
Y;ZE depends only on Y  Z  that is, if (Y; Z) (Y 0; Z0) then the natural
map Y;ZE ! Y 0;Z0Eis a pointwise weak equivalence. It is easy to check
that E is MayerVietoris if and only if E satisfies excision.
Remark: Quillen's localization sequence for K0theory [35] shows that K0
satisfies excision and hence defines a MayerVietoris presheaf. If X is sep
arated and regular, then the Ktheory presheaf is also a MayerVietoris
presheaf, since then K ~=K0.
The next theorem is due to Brown and Gersten [7].
Theorem 4.1 Let X be a noetherian scheme of finite Krull dimension, and
suppose that E is a MayerVietoris presheaf of spectra on the Zariski site of
X. Then E satisfies Zariski descent: any globally fibrant model
E! GZarE = H.Zar(; E)
is a pointwise weak equivalence.
Corollary 4.2 If X is a separated noetherian regular scheme the Ktheory
presheaf satisfies Zariski descent. In particular there is a convergent spectral
sequence
Ep;q2= HpZar(X; "ssqK) ) ssqpK(X)
Here is a quick proof of the theorem, paraphrasing Thomason's exercise
2.5 [42]. Call a presheaf of spectra quasifibrant if it is pointwise weak equi*
*va
lent to a globally fibrant presheaf. We will use the following easy facts: (a) *
*if
62
any two terms of a pointwise fibre sequence are quasifibrant, so is the third;
and (b) a pseudofiltered colimit of quasifibrant presheaves is quasifibrant
(this follows from Proposition 3.7, since XZar is a noetherian site of finite
cohomological dimension).
Let Y be a closed subscheme of X. For each closed Z Y of codimension
at least one, the sequence
Z E! Y E! Y;ZE
is a pointwise fibre sequence by the "Quetzlcoatl lemma" (Thomason's pic
turesque term for the homotopytheoretic analogue of the snake lemma).
Passing to the colimit over the directed set of all such Z yields a pointwise
fibre sequence
colimZ Z E! Y E! CY E
where CY E = colimZ Y;ZE. If Y is irreducible with generic point x, we also
write xE = CY E (the reason for this apparently superfluous notation will
appear when we consider Nisnevich descent). Using excision, Quetzlcoatl,
and induction, one obtains:
Lemma 4.3 Suppose Y has irreducible components Y1; :::; Ym . Then there is
a pointwise weak equivalence
_ ~=
CYiE ! CY E
Now let
SpE = colimcodim Y pY E
We will show by descending induction on p that SpE is quasifibrant. This
will prove the theorem, since S0E = E. To start the induction, note that if
X has Krull dimension n, Sn+1E = *. At the inductive step, define CpE by
the pointwise cofibre sequence
Sp+1E ! SpE ! CpE
It is enough to show that CpE is quasifibrant.
63
Lemma 4.4 There is a pointwise weak equivalence
_
xE ! CpE
codim x=p
The proof is easy from Lemma 4.3. Thus we have reduced to showing
that xE is quasifibrant for all x 2 X.
Lemma 4.5 Let i : x! X denote the inclusion. Then the natural map
xE ! i# i# xE is a pointwise weak equivalence.
To prove this lemma, it is enough to check (i) (colimZ Y;ZE)(U) is weakly
contractible if x is not in U; and (ii) (colimZ Y;ZE)(U) ! (colimZ Y;ZE)(W )
is a weak equivalence whenever x 2 W U. Condition (i) is obvious. For
condition (ii), consider the diagram
Y;ZE(U) _______E(U  Z) _______E(U  Y )
  
  
j   
  
? ? ?
Y;ZE(W ) ______E(W  Z) ______E(W  Y )
Note that (W  Z) \ (U  Y ) = W  Y . If we have Y  Z W ,
then U  Z = (U  Y ) [ (W  Z) and hence j is a weak equivalence by a
direct application of the MayerVietoris condition. But the set of Z satisfying
Y Z W is easily seen to be cofinal, so j is a weak equivalence after passage
to the colimit, proving (ii).
A presheaf on the Zariski site of a field is quasifibrant if and only if its
value on the empty set is weakly contractible. It follows that i# xE is quasi
fibrant, and since direct image functors preserve quasifibrant presheaves,
this completes the proof of Theorem 4.1.
4.2 Nisnevich descent
Let X be a noetherian scheme of finite Krull dimension. By excision data we
mean a morphism f : U ! V between objects of the Nisnevich site (recall
that the underlying category is the same as that of theetale site), together
64
with a closed subscheme Z V such that U xV Z ! Z is an isomorphism.
A presheaf of spectra E on XNis satisfies Nisnevich excision if for all such
excision data, the square
E(V ) ________E(V  Z)
 
 
 
 
? ?
E(U) ______E(f1 (V  Z)
is a homotopy fibre square,`and E(OE) ~=*. It is easy to check that then E is
additive  that is, E(U V ) ~= E(U) x E(V )  and that the restriction of E
to the Zariski site of any object U satisfies Zariski excision (i.e. the Mayer
Vietoriscondition of the preceeding section). It is also clear that K0theory
satisfies Nisnevich excision, and hence so does Ktheory if X is separated and
regular.
The following theorem is the Nisnevich descent theorem [34].
Theorem 4.6 Let X be a noetherian scheme of finite Krull dimension, E
a presheaf of spectra on the Nisnevich site of X. If E satisfies Nisnevich
excision, then E satisfies Nisnevich descent  that is, the globally fibrant mo*
*del
E! GNisE = H.Nis(; E)
is a pointwise weak equivalence.
Corollary 4.7 If X is a separated noetherian regular scheme of finite Krull
dimension, the Ktheory presheaf satisfies Nisnevich descent.
The proof that we will sketch here follows [22], and very closely parallels
the proof of the BrownGersten theorem. For each object U ! X of XNis
and each closed subscheme Y U, we can form the spectra Y E(U) and
CY E(U) as before. Note that these are not presheaves on XNis. We can,
however, define for each x 2 X a presheaf xE by
xE (ss : U ! X) = C____ss1xE(U)
65
The spectrum SpE (U) we defined in the Zariski case is functorial in U, so
we also get a presheaf SpE on XNis. Let us again define CpE by the pointwise
cofibre sequence
Sp+1E ! SpE ! CpE
Lemma 4.8 There is a pointwise weak equivalence
_
xE ! CpE
codim x=p
The proof is easy from Lemma 4.3. Thus we have again reduced to
showing that xE is quasifibrant.
Lemma 4.9 Let i : x! X denote the inclusion. Then the natural map
xE ! i# i# xE is a pointwise weak equivalence.
Assuming this for the moment, the proof of Theorem 4.6 is completed as
follows. A presheaf of spectra on the Nisnevich site of a field is quasifibrant
if and only if it is additive (the proof is straightforward  this is the only *
*place
where the Nisnevich topology is used!). One easily checks that xE is additive
and hence also i# xE is additive. Since direct images preserve quasifibrant
presheaves, this completes the proof.
We briefly indicate the proof of Lemma 4.9, so the reader can at least
see where the Nisnevich excision condition comes in. By unravelling the
definitions of i# and i# , one reduces to checking the following:
Lemma 4.10 Suppose f : U ! V is a morphism in XNis and z 2 V . Sup
pose that f1 z consists of a single point y, and that y! z is an isomorphism
of schemes (i.e., an isomorphism on residue fields). Then the induced map
CzE (V ) ! CyE (U) is a weak equivalence.
If y! zis an isomorphism of schemes, this follows by a direct application
of Nisnevich excision. But we can always reduce to this case because y ! z
is an isomorphism on generic points by assumption, and hence restricts to
an isomorphism between suitable open subschemes of y; z. See [22],Lemma
7.21, for further details.
66
4.3 Gabber rigidity and the Ktheory sheaf
On theetale site, the `adic Ktheory sheaf is explicitly known. Let
n
Rn = Z[t]=t`  1
and let `n denote the sheaf on the bigetale site (all schemes) represented by
Spec Rn. In other words, `n(U) is the group of units of order dividing `n in
the ring of regular functions on U. Next, let fin 2 ss2(BZ =`n; Z=`n) correspond
to the standard generator under the Bockstein isomorphism
ss2(BZ =`n; Z=`n) ~=ss1BZ =`n = Z=`n
The natural map
1 BZ =`n+! KRn
then yields a "Bott element" in ss2(KRn; Z=`n), also denoted fin.
Now define a map of presheaves e : `n ! ss2(K; Z=`n) by
e(f : U ! Spec Rn) = f*fin
If U is affine  say U = Spec S  then e(f) is just the image of fin under the
composite
BZ =`n ! BGL1Rn ! BGL1S ! BGLS+
From this description it is clear that e is a group homomorphism. Using the
ring spectrum structure on K(; Z=`n), we obtain more generally maps of
presheaves
ei : Z=`n(i) i`n ! ss2iK(; Z=`n)
Proposition 4.11 Let X be a scheme over Z[1=`]. Then the maps ei induce
isomorphisms of sheaves
q ~= n
Z =`n(__) ! "ssqK(; Z=` )
2
The group on the left is to be interpreted as zero if q is odd. The proof
of the proposition depends on the Gabber Rigidity Theorem ([13]; earlier
versions are due to GilletThomason [14] and Suslin ([39])):
67
Theorem 4.12 Let R be a hensel local ring with residue field k of char
acteristic different from `. Then the reduction map KR ! Kk is a weak
equivalence on `adic completions.
Combining Gabber rigidity and Suslin's theorem 1.1 shows that for any
strictly henselian ring S over Z[1=`], Z=`n(q=2) ~=ssq(KS; Z=`n). Hence by the
continuity theorem 2.11, ei induces an isomorphism on stalks. This proves
the proposition.
In other words, we have settled the question raised in the "Addendum"
to the descent problem mentioned in the introduction. As a consequence
the `adic descent spectral sequence for hypercohomology with coefficients in
Ktheory takes the form
q . ^
Ep;q2= Hpet(X; Z`(__)) ) ssqp(H et(X; K) )
2
where as alwaysetale cohomology means continuous cohomology in the sense
of Jannsen [19] and the groups on the right are the homotopy groups of the
completion, not the completion of the homotopy groups. To see why we get
continous cohomology, note first that
(H .et(X; K))^ = holimnH .et(X; K) ^ MZ =`n = holimnH .et(X; K ^ MZ =`n)
by (3.11), and this in turn is equivalent to
holimnholim T .(K(; Z=`n))(X) = holim holimnT .(K(; Z=`n))(X)
Therefore we get a spectral sequence whose E2term is the cohomotopy of
the cosimplicial abelian group ss*holimnT .(K(; Z=`n))(X). Now
q
ssqT .(K(; Z=`n)) = T .ssq(K(; Z=`n)) = T .`n(__)
2
Since the reduction maps `n+1! `n are epimorphisms of sheaves, and the
composite functor (X; ) O T .is exact, it follows that the Milnor lim1 van
ishes and
ss*holimnT .(K(; Z=`n))(X) = limnssqT .(K(; Z=`n))
Since T .takes values in acyclic sheaves, it also follows that T .ssq(K(; Z=`n*
*))
is a resolution of `n(q_2) by lim0 O (X; )acyclic objects. Hence the coho
motopy yields continous cohomology as claimed.
68
4.4 Thomason'setale descent theorem
We recall the following definition from Section 1: a field F is `good if
char F 6= ` and F has finite transcendence degree over a subfield E satis
fying cd`E1 1, where E1p is_obtained from E by adjoining all the `power
roots of unity; and with 1 2 E if ` = 2. A scheme X is then said to be
`good if all of its residue fields are `good.
Recall also (see Theorem 2.8) that a scheme X is uniformly `bounded if
there is a uniform bound d < 1 on theetale cohomological dimension of
the residue fields of X. For example, if X is "uniformly `good" in the sense
that the transcendence degrees occurring in the definition of `good scheme
are uniformly bounded (e.g. any variety over an `good field), then X is
uniformly ` bounded.
We can now state the descent theorem of Thomason [42]:
Theorem 4.13 Let X be a separated noetherian regular scheme of finite
Krull dimension. Assume that X is `good and uniformly `bounded. Then
the natural map KX ! H.et(X; K) induces a weak equivalence
^LK(X) ~=!^LH.et(X; K)
Equivalently, in the terminology of Jardine, the localization of globally fibra*
*nt
models
^LK ! ^L(GetK)
is a pointwise weak equivalence on Xet.
Corollary 4.14 There is a convergent descent spectral sequence
q
Ep;q2= Hpet(X; Z`(__)) ) ssqp^LKX
2
For the corollary, note that ^LH.et(X; K) ~=(H .(X; LK))^ by Corollary 3.12.
The indicated form of the E2term then follows from Gabber rigidity as in
section (4.11).
The two forms of Theorem 4.13 are equivalent by Proposition 3.20, and
because all of the hypotheses on X are inherited by any object U! X of the
(restricted)etale site of X.
69
Now, assuming Theorem 4.13 has been proved for `good fields,
the general case is a rather easy consequence of the machinery thus far as
sembled. It will be convenient to use the abbreviation
L = LK ^ MZ `= K ^ LMZ =` = K ^ LS0 ^ MZ =`
Then it is enough to show that the globally fibrant model j : L ! GetL,
which is by definition a stalkwise weak equivalence for theetale topology, is
a pointwise weak equivalence (that is, a stalkwise weak equivalence for the
chaotic topology!). Let us first show:
Lemma 4.15 j : L ! GetL is a stalkwise weak equivalence for the Nisnevich
topology.
Proof: By continuity, this reduces to showing that when X = Spec R, R
a henselian ring, the map L X ! (GetL)(X) = H.et(X; L) is a weak equiva
lence. Let k as usual denote the residue field of the maximal ideal and let
x = Spec k. Consider the commutative diagram
L X ____________Lx
 
j  
 
 
? ?
H.et(X; L) ______H.et(x;hL)
The top map is a weak equivalence by Gabber rigidity, and the righthand
map is a weak equivalence by our assumption that the theorem has been
proved for fields. The bottom map is also a weak equivalence. To see this,
note that L is a presheaf on the bigetale site, and h factors as
H .et(X; LX ) a! H .et(x; j# (L X)) b! H .et(x; Lx)
Here the notations LX ; Lx indicate restriction from the bigetale site to the
appropriate small site, and j : x! X is the inclusion. The first map is a weak
equivalence by a spectral sequence comparison argument using Theorem 2.5.
The second map is induced by the map of presheaves j# (L X) ! Lx (which
exists because L is a presheaf on the big site). This map is a stalkwise weak
equivalence by Gabber rigidity, so b is a weak equivalence as claimed. Hence
j is a weak equivalence, completing the proof of the lemma.
70
Now consider the commutative diagram of presheaves
j
L ___________GetL
 
 
 
 
? ?
GNisL ______GNis(GetL)G
Nisj
We wish to show that j is a pointwise weak equivalence. The lefthand
map is a pointwise weak equivalence by the Nisnevich descent theorem. The
righthand map is a pointwise weak equivalence since GetL is globally fibrant
on the Nisnevich site (because theetale topology is finer than the Nisnevich
topology; see Proposition 3.14). Since j is a Nisnevich stalkwise weak equiv
alence by the lemma, the induced map GNisj on globally fibrant models is a
pointwise weak equivalence (Proposition 3.17). This completes the proof of
Theorem 4.13  assuming that it has been verified for fields, as will be done
in the next section.
In conclusion, we remark that Thomason's original proof used the Brown
Gersten theorem, as Nisnevich descent was not available at the time. This
approach does not work quite so neatly, because of the lack of an analogue of
Gabber rigidity. See [42] or [22] for the Zariski descent proof. See also [44].
5 The proof of Thomason's theorem for fields
In this final section we sketch the proof of Theorem 4.13 for fields, following
Chapter 7 of [22]. Since we no longer need to consider general schemes, we
write in terms of rings rather than schemes and use the following notation:
F is a field, and L is a finite separable extension of F  usually Galois with
group G. X is a spectrum, a Gspectrum or a presheaf of spectra on theetale
site of F. Let k() denote the mod ` Ktheory presheaf K ^ MZ =`. In order
to avoid distracting technicalities, we assume ` > 3.
Thus our goal is to show that the natural map kF ! H .(F ; k) induces a
weak equivalence LkF ! LH .(F ; k), or equivalently
~= 1 . . 1
A1kF  ! A H (F ; k) ~=H (F ; A k)
71
Here A is the Adams selfmap of the Moore spectrum, and since cd`F < 1
we are free to commute A1 and H.by Theorem 3.7. A theorem of Snaith and
Dwyer says that inverting the Adams map is equivalent to inverting the Bott
element fi`1 2 k2`2F . By a standard (and trivial) transfer argument we
can assume F contains the `th roots of unity. Thus we have a Bott element
fi 2 ss2kF , and Theorem 4.13 for fields is equivalent to:
Theorem 5.1 Let F be an `good field containing the `th roots of unity.
Then kF ! H .(F ; k) induces a weak equivalence on Bottperiodic spectra
fi1kF ! fi1H .(F ; k) ~=H .(F ; fi1k)
The mapping telescopes in this theorem are to be interpreted in terms
of the evident kF module spectrum structures. In particular H .(F ; k) is a
kF module spectrum with multiplication
kF ^ H.(F ; k) ! H.(F ; kF ^ k) ! H.(F ; k)
Before we can even outline the proof, we need some preliminaries on the
hypernorm and hypertransfer (subsections 14). The fifth subsection outlines
the proof modulo two key lemmas, whose proofs are sketched in the sixth
and last subsection.
5.1 Hyperhomology spectra
Suppose G is a discrete group and X is a Gspectrum. The hyperhomology
spectrum H .(G; X) is the homotopy colimit of the Gaction, which we can
take to mean the spectrum EG+ ^G X. Filtering EG by skeleta, one easily
obtains a first quadrant homology spectral sequence
E2p;q= Hp(G; ssqX) ) ssp+qH .(G; X)
Let F (X; Y ) denote the function spectrum for spectra X,Y. Then if Y has
trivial Gaction, there is a more or less obvious universal coefficient formula
H.(G; F(X; Y )) ~=F (H .(G; X); Y )
72
5.2 The hypernorm
Let G be a finite group, X a Gspectrum. There is a "norm" map N : X! X,
in the stable category that is just the sum of the actions of the various
elements of G. An explicit, functorial strict representative of this map is
given by the composite
OE r
X G! Map (G+ ; X)  G+ ^ X ! X
where G (x)(g) = gx, r is the ordinary fold map, and OE(g ^ x)(h) = x if
h = g and OE(g ^ x)(h) = * otherwise. Note that Map (G+ ; X) and G+ ^ X
are just the product and wedge respectively of  G  copies of X. Hence OE is a
weak equivalence and the diagram represents a map in the stable category.
Now observe that there are G x Gactions on all four terms of the above
diagram, such that the three maps are G x Gequivariant  although the
actions on the two instances of X are not the same! Explicitly, for (g; h) 2
G x G we have
Map (G; X) : [(g; h) ](a) = h (h1ag)
G+ ^ X : (g; h)(a ^ x) = hag1 ^ hx
X (in source): (g; h)x = gx
X (in target): (g; h)x = hx
Thus the norm N is a G x Gequivariant map.
There is a functor from GxGspectra to spectra that takes the homotopy
limit with respect to the second copy of G and then the homotopy colimit
with respect to the first copy of G. Equivalently, we can think of this functor
as
EG1+^G1 (MapG2(EG2+; Y ))
for a G x Gspectrum Y . Here the superscripts serve to indicate which
copy of G is acting. Applying this to the norm map and making use of the
augmentation BG+ ! S0 yields
EG+ ^G X ! EG+ ^G (Map (BG+ ; X) ! BG+ ^MapG (EG+ ; X) ! MapG (EG+ ; X)
This composite is the hypernorm which we write as
73
Nh : H.(G; X) ! H.(G; X)
It has the property that the composite map
X ! H.(G; X) ! H.(G; X) ! X
is the ordinary norm N. As a consequence we have:
Lemma 5.2 Suppose the Tate cohomology groups ^Hp(G; ssqX) vanish for all
p; q. Then the hypernorm is a weak equivalence:
~= .
Nh : H.(G; X) ! H (G; X)
For both spectral sequences collapse to an edge, and the induced map on
homotopy can be identified with the algebraic norm map
n : (ssqX)G ! (ssqX)G
But Ker n = ^H1 and Coker n = ^H0, whence the lemma.
5.3 An analogue for spectra of a theorem of Tate
Let G be a finite group. Recall that Tate cohomology is defined in the
following way. Given a Gmodule M, let P. and I. denote the standard
projective and injective resolutions (respectively) of M. The Tate complex
o.M is obtained by splicing together (P.)G and (I.)G via the norm map
oe : (P0)G ! MG  n!MG ! (I0)G
In other words, o.M is the mapping cone of oe. Tate cohomology H^*(G; M)
is then defined as the cohomology of this complex.
The same construction works for hypercohomology of cochain complexes
 given a cochain complex C: (which we picture running up the qaxis, in
the upper half of the (p; q)plane) we get a second quadrant double com
plex (P ::)G (C:) and a first quadrant double complex (I::)G (C:) which can be
spliced to produce an upper halfplane double complex o::(G; C:) of cohomo
logical type.
Now suppose G is a profinite group and M is a discrete Gmodule. We
say that cd M d if for all open subgroups V of G, Hn(V ; M) = 0 for n > d.
Then there is the following lemma of Tate ([37], Annexe 1 to Chapitre I):
74
Lemma 5.3 The discrete Gmodule M satisfies cd M = 0 if and only if for
all open normal subgroups U of G, H^*(G=U; MU ) 0.
Now suppose cd M d. Then one easily constructs a finite resolution
0 ! M ! N0 ! N1 ! ::: ! Nd ! 0
with cd Ni = 0 for all i (Tate, loc. cit.). Fix an open normal subgroup U and
form the associated Tate double complex T ::(G=U; (N:)U ) as above. Since
the horizontal cohomology of this complex vanishes by Tate's lemma, the
cohomology of the total complex is also trivial. We conclude:
Corollary 5.4 The norm (P ::)G=U (N:U) ! (I::)G=U (N:U) induces an isomor
phism on total cohomology.
Since the total cohomology of the target is H*(G; M), it follows that
there is a spectral sequence Ep;q2= Hp(G=U; Hq(U; M)) ) H*(G; M). Re
indexing by setting s = p and t = d  q yields a first quadrant homology
spectral sequence  the Tate spectral sequence:
E2s;t= Hs(G=U; Hdt(U; M)) ) Hdst(G; M)
We will not make any use of this spectral sequence, however. We have
stated Corollary 5.4 only to motivate its analogue for spectra. This analogue
could of course be stated purely in terms of profinite groups, but for future
reference we record it in terms of Galois extensions.
Theorem 5.5 Suppose E is a Galois extension of F and X a presheaf of
spectra on (E=F ) with cd (E=F ; X) < 1. Then for any finite Galois subex
tension L/F with group G, the hypernorm
Nh : H.(G; H.(E=L; X)) ! H.(G; H.(E=L; X))
is a weak equivalence.
Here cd (E=F ; X) d if cd (G(E=F ); "ssqX) d for all q, in the sense
described above.
Thomason's original proof is an induction on cd (E=F ; X) that proceeds
as follows. If this dimension is zero, then ssq(H .(E=L; X) = H0(E=L; ssqXE ).
Hence Nh is a weak equivalence by Tate's lemma combined with Lemma 5.2.
75
At the inductive step, form the cofibre sequence X! T X! CX, where
TX is the ThomasonGodement construction as usual and CX is the cofibre.
Then cd T X = 0 and cd CX < cd X; the result follows easily from this. See
[22],7.1, for a proof using Postnikov towers.
5.4 Transfer and hypertransfer
Let L be a finite field extension of F. In addition to the map i : KF ! KL
arising from extension of scalars, there is a transfer map t : KL ! KF arising
from restriction of scalars. We recall that the transfer has the following basic
properties:
o Index formula. The composite KF  i! KL !t KF is multiplica
tion by the index [L : F ]
o Invariance under base change. The transfer commutes with base
change  that is, given an arbitrary field extension F 0=F there is a
commutative diagram in the stable category
KL ______KL0
 
 
 
 
 
? ?
KF ______KF 0
where L0 = L F F 0and the vertical maps are transfers (of course L0
may not be a field, but the transfer is defined more generally).
o Norm formula. If L is Galois over F with group G, then the composite
KL t! KF  i!KL is the norm.
o Linearity. The transfer is a map of KFmodule spectra.
We note that if M/F is an arbitrary Galois extension with trivial pro`
~=
Sylow subgroup, it follows from the index and norm formulas that (KF )(`)!
((KM)(`))G(M=F) and therefore the Ktheory presheaf, localized at `, satisfies
76
descent on M/F. In particular this suffices to prove the LichtenbaumQuillen
conjectures for fields F with cd`F = 0, since a profinite group G satisfies
cd`G = 0 if and only if the pro`Sylow subgroup of G is trivial.
Now consider the transfer t : KL ! KF in the case of a finite Galois
extension. Then t is not Gequivariant with respect to the trivial action on
KF, because the forgetful functor L  mod ! F  mod is only Gequivariant
up to natural isomorphism. Here we are using the standard Gaction in
g
which g 2 G acts by tensoring up along L  ! L. Thus the transfer is
only equivariant up to homotopy. In fact, however, t is equivariant up to an
infinity of higher homotopies, in the sense that t extends canonically to a
map
th : H.(G; KL) ! KF
called the hypertransfer. The easiest way to think of this is to replace the
Gaction described above by the obvious equivalent action in which G acts by
leaving the underlying Fvector space strictly alone and twisting the scalar
multiplication. Then the transfer is strictly equivariant and we can define th
as the composite
H.(G; KL) = EG+ ^G KL ! KL=G ! KF
At the level of plus constructions, the hypertransfer boils down to inclusion
of the semidirect product of GLnL and G into GLdnF , where d = [L : F ].
For a rigorous account of all this, see [42],Example 2.30, and [22]. We note
that a dual discussion applies to extension of scalars i : KF ! KL, so that
the augmentation KF ! H .(G; KL) could just as well be called "hyperex
tension" ih.
The hypertransfer has the following basic properties, each of which will
be used at a crucial point in the proof of the main theorem below.
o Invariance under base change. If F 0is an arbitrary field extension
of F and L0 = L F F 0, then the following diagram commutes in the
stable category:
77
H .(G; KL) ______H.(G; KL0)
 
 
 
 
 
? ?
KF ____________KF 0
Here the vertical maps are the hypertransfers; in the application L0will
in fact be a field, but the hypertransfer could have been defined more
generally.
o Norm formula. The composite
th ih .
H.(G; KL) ! KF  ! H (G; KL)
is the hypernorm.
o Linearity. The hypertransfer is a map of KFmodule spectra (with
respect to the obvious module structure on H.(G; KL)).
5.5 Outline of the proof
The proof is based on the two key lemmas 5.5 and 5.6, whose proofs are
discussed in the final subsection. A Bott inductor for a finite Galois extension
L/E is an element "fi2 ss2H .(G; kL) such that th(f"i) = fi. An arbitrary Galois
extension M/E admits Bott inductors if for every finite Galois subextension
a Bott inductor exists.
Lemma 5.6 Let M/E be a Galois extension such that either (a) M = Esep
and cd`E 1; or (b) M = E1 and cd`E1 1. Then M/E admits Bott
inductors.
Lemma 5.7 Let M/E be a Galois extension such that cd`(M=E) < 1 and
M/E admits Bott inductors. Then fi1k satisfies descent on M/E:
~= 1 .
fi1kE ! fi H (M=E; k)
78
We also need one more general fact about hypercohomology. Consider a
sequence of Galois extensions F M N, and a presheaf of spectra X on
N=F . Then there is a weak equivalence
~= . .
H .(N=F ; X) ! H (M=F ; H (N=; X))
This can be seen in several ways. The simplest is to observe that the coeffi
cient presheaf on the right is the direct image of H.(N=; X) under the site
morphism G(N=F )fsets! G(M=F )fsets; then one can use the fact that direct
image functors preserve globally fibrant presheaves (Proposition 3.14). It is
also easily proved directly in terms of homotopy fixed point sets (Section 1).
Using this we obtain:
Lemma 5.8 Suppose M/F has a filtration F = M0 M1 ::: Mn
with each layer Mi+1=Mi Galois and cd`Mi+1=Mi < 1, and suppose X is a
presheaf of spectra on Mn=F satisfying
~= .
XMk  ! H (Mk+1=Mk; X)
~= .
for all k. Then X ! H (F ; X).
Proof: We proceed by descending induction over the filtration. At the
inductive step, there is a commutative diagram
XMk ____________________H.(Mk; X)
 
 
 
 
 
 
? ?
H.(Mk+1=Mk; X) ______H.(Mk+1=Mk; H.(M=; X))
The lefthand map is a weak equivalence by assumption, and the righthand
map is a weak equivalence by the remarks above. The bottom map is a weak
equivalence because by inductive hypothesis the map on stalks is a weak
equivalence.
Now suppose F is an `good field containing `th roots of unity. We sup
pose that F has finite transcendence degree over a field F0 with cd`F0 = 1;
79
the proof in the other case is quite similar. Replacing F0 by its algebraic
closure in F if necessary, we can assume F is purely transcendental over
F0. Furthermore we can assume F is is a separable algebraic extension of
E = F0(t1; :::; tn) for some n (since both mod ` Ktheory andetale hyper
cohomology are invariant under purely inseparable extension), and moreover
that F is finite over E (by a direct limit argument). In other words, we can
assume F is a "function field" of dimension n over F0.
Now any such function field admits a "TateTsen filtration"  that is, a
filtration of Fsep=F of the form F = M0 M1 ::: Mn in which each layer
Mi+1=Mi is induced from an extension of the form Nsep=N with cd`N 1.
For example, if F = F0(t) we can take the filtration F0(t) (F0)sep(t) Fsep,
in which the first layer is induced from F0 (F0)sep and the second layer
has cd` 1 by Tsen's theorem. A similar but somewhat more complicated
argument constructs the filtration in the general case.
The extensions Nsep=N admit Bott inductors by Lemma 5.6. Since the
hypertransfer commutes with base change, each layer Mi+1=Mi therefore ad
~= .
mits Bott inductors. Hence XMk  ! H (Mk+1=Mk; X) by Lemma 5.7. Then
Lemma 5.8 completes the proof.
5.6 The key lemmas
Suppose cd`F 2, and consider the map j : kF ! H .(F ; k) in low degrees.
One can show:
o There is a short exact sequence
0 ! H2(F ; Z=`) ! ss0H .(F ; k) ! Z=` ! 0
and Z=` = ss0kF ! ss0H .(F ; k) maps onto the quotient term Z=`. Since
F contains the `th roots of unity, the norm residue symbol gives an
isomorphism from H2(F ; Z=`) to the `torsion in the Brauer group of
F.
o ss1H .(F ; k) = H1(F ; Z=`) = F x=` by Hilbert's Theorem 90, and ss1(j)
is an isomorphism.
~= .
o Multiplication by fi is an isomorphism ss0H .(F ; k) ! ss2H (F ; k). By
the MercurjevSuslin theorem [28], ss2(j) is an isomorphism.
80
o ssnkF = 0 for n < 0, but ss1H .(F ; k) ~=ss1H .(F ; k) and ss2H .(F ; k)*
* ~=
ss0H .(F ; k).
Now consider the commutative diagram in the stable category
H.(G; kL) ______H.(G;HH.(L;.k))(G;j)
 
 
oh Nh
 
 
? ?
kF _________H.(G; H.(L; k))
The bottom map is the composite
~= . .
kF ! H .(G; kL) ! H (G; H (L; k))
One might hope to prove Lemma 5.6 as follows. If we pretend for a
moment that ss*(j) is an isomorphism through degree 2, then the same is
true for H.(G; j), using the hyperhomology spectral sequence. Since Nh is a
weak equivalence by the spectrumlevel Tate theorem, it follows that ss2(oh)
is an isomorphism. Hence we have a (unique) Bott inductor.
Unfortunately, as noted above, we do not have an isomorphism through
degree 2. Even when cd`F 1, the group ss1H .(F ; k) gets in the way. In this
case, however, the problem is easily circumvented by replacing the presheaf
k by its 0connected cover. The reader can then easily complete the proof of
Lemma 5.6 when cd`F 1.
If cd`F = 2, then even the modified argument fails. The problem is with
the Brauer group term in ss0H .(F ; k). This term can be eliminated by passing
to more highly connected covers of k, but then we only succeed in eliminating
the very group we are attempting to analyze! Nevertheless, there is a similar 
but much more delicate  argument that carries the day ([22],Theorem 7.16).
One has to pick out just the right connected covers and Postnikov sections
to isolate the Bott element. Of course, the MercurjevSuslin theorem again
plays a crucial role.
We now turn to the proof of Lemma 5.7. We wish to show that j :
kF ! H .(M=F ; k) induces an isomorphism on fi1ss*. Let us first show that
it induces an injection. Let L/F be a finite Galois subextension with group
81
G. Fix a Postnikov section P nk of the presheaf k. Note that P nk is a presheaf
of kF module spectra.
Lemma 5.9 There is a map OE : H.(G; P nk) ! 2P nkF such that the com
posite
OE 2 n
P nkF ! H .(G; P nk) ! P kF
is homotopic to multiplication by fi.
In fact the lemma holds with P nk replaced by k itself. The point of
passing to a Postnikov section is that then
colimLH .(L=F ; P nk) ~=H .(M=F ); P nk)
.
by Example 3.22. To show that fi1j is injective, we will show that if a 2 ss*k
and j*a = 0, then fia = 0. For this purpose we may replace k by a Postnikov
section P nk, n >> 0, and then it follows immediately from the lemma that
fia = 0.
To construct OE, consider the composition of multiplication and the
transfer:
kL ^ P nkL ! P nkL ! P nkF
Taking adjoints yields a Gequivariant map P nkL ! F (kL; P nkF ), where
F(; ) denotes the function spectrum. We then define OE to be the compos
ite
"fi*2 n
H.(G; P nkL) ! H.(G; F(kL; P nkF )) ~=F (H .(G; kL); P nkF ) ! P kF
where the first map is induced by the coefficient map , the second is the
universal coefficient equivalence, and the last map is precomposition with a
Bott inductor "fi. Using linearity of the hypertransfer, it is not hard to show
that the composite map of the lemma is multiplication by fi.
The proof of surjectivity is similar. Essentially, one shows that the com
posite H.(G; P nk) ! 2P nkF ! 2H .(G; P nk) is also multiplication by fi; it
then follows that for all b 2 ss*H .(M=F ; k), fib is in the image of j.
82
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