ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF
SCHEMES
DANIEL C. ISAKSEN
Abstract.We compare Friedlander's definition of 'etale homotopy for simp*
*li-
cial schemes to another definition involving homotopy colimits of pro-si*
*mplicial
sets. This can be expressed as a notion of hypercover descent for 'etale*
* homo-
topy. We use this result to construct a homotopy invariant functor from *
*the
category of simplicial presheaves on the 'etale site of schemes over S t*
*o the
category of pro-spaces. After completing away from the characteristics o*
*f the
residue fields of S, we get a functor from the Morel-Voevodsky A1-homoto*
*py
category of schemes to the homotopy category of pro-spaces.
Contents
1. Introduction 2
2. Etale Realizations 3
2.1. Simplicial Presheaves 3
2.2. Etale Realization on sPre(Sch=S) 5
2.3. Etale Realization on the A1-Homotopy Category of Schemes 7
2.4. Excision for the Etale Topological Type 9
3. Hypercovers of Simplicial Schemes 10
3.1. Finite Limits of Schemes 10
3.2. Simplicial Schemes 11
3.3. Skeleta and Coskeleta 11
3.4. Hypercovers 13
3.5. Rigid Pullbacks 15
3.6. Rigid Limits 16
4. Pro-Spaces 20
4.1. Preliminaries on Pro-Categories 20
4.2. Homotopy Theory of Pro-Spaces 22
4.3. n-Truncated Realizations 22
4.4. Realizations of pro-spaces 24
5. Hypercover Descent for the Etale Topological Type 25
References 27
____________
Date: June 19, 2001.
1991 Mathematics Subject Classification. 14F42 (Primary), 14F35 (Secondary).
Key words and phrases. A1-homotopy theory of schemes, 'etale topological typ*
*e, simplicial
presheaves, hypercovers, pro-spaces.
The author was partially supported by an NSF Postdoctoral Research Fellowshi*
*p and partially
supported by Universität Bielefeld, Germany. The author thanks Vladimir Voevods*
*ky for suggest-
ing the problem. The author also thanks Ben Blander and Dan Dugger for useful c*
*onversations.
1
2 DANIEL C. ISAKSEN
1.Introduction
In the recent proof of the Milnor conjecture [31], a certain realization func*
*tor
from the A1-homotopy category of schemes over C [25] to the ordinary homotopy
category of spaces plays a useful role. The basic idea is to detect that a cer*
*tain
map in the stable A1-homotopy category is not homotopy trivial by checking that
its image in the ordinary stable homotopy category is not homotopy trivial.
This realization functor is defined by extending the notion of the underlying
analytic space of a complex variety, so we call it analytic realization. The an*
*alytic
realization functor as defined in [25, x 3.3] has two shortcomings. First, it i*
*s defined
directly on the homotopy categories. It would be much preferable to have a func*
*tor
on the point-set level that is homotopy invariant and therefore induces a funct*
*or
on the homotopy categories.
The second shortcoming is that the analytic realization does not work over fi*
*elds
with positive characteristic. Varieties over abstract fields have no underlying*
* an-
alytic topology. However, the 'etale topological type [2] [9] is a substitute.*
* In
characteristic zero, the 'etale topological type EtX of a variety X is the pro-*
*finite
completion of the underlying analytic space of X. In any characteristic, EtX ca*
*rries
information about the 'etale cohomology of X and the algebraic fundamental group
of X.
The main goal of this paper to fix the second of these two shortcomings. Beca*
*use
the category of pro-spaces is complicated, it is essential to use model structu*
*res to
establish the functor on the homotopy categories. Using a model structure for
A1-homotopy theory slightly different than the one in [25], the 'etale topologi*
*cal
type provides a functor from the category of simplicial presheaves on the Nisne*
*vich
site of smooth schemes over S to the category of pro-spaces. This functor is a *
*left
Quillen functor, which means that it automatically gives a functor on the homot*
*opy
categories.
The 'etale realization functor provides a calculational tool for A1-homotopy *
*the-
ory of schemes over fields of positive characteristic. In future work, we hope*
* to
take Galois group actions into account to obtain a realization functor into a h*
*omo-
topy category of equivariant pro-spaces. However, the foundations for a suitab*
*le
equivariant homotopy theory of pro-spaces have not yet been established. We also
hope to stabilize our techniques to obtain a functor on stable A1-homotopy theo*
*ry.
Although some progress on the foundations of the homotopy theory of pro-spectra
has been made [5] [18], it is not yet clear whether these theories are suitable*
* for the
current application.
In this paper, the first shortcoming of the analytic realization described ab*
*ove
remains unfixed. We plan to fill this gap in future work with Dan Dugger. The
general program of this paper will work provided that for any open hypercover U
of a topological space X, the geometric realization of U is weakly equivalent t*
*o X.
This generalizes a result of [28, x 4] about Cech complexes of open covers.
A summary of the contents of the paper follows. Section 2 begins with a revie*
*w of
simplicial presheaves and their homotopy theory. We assume familiarity with clo*
*sed
model structures. General references on this topic include [13], [14], or [26].*
* We
conform to the conventions of [13] as closely as possible. See also [6] for mor*
*e details
on model structures as applied to simplicial presheaves. Next comes the definit*
*ion
of the 'etale realization functor, and the first major result is that it is hom*
*otopy
invariant on the local projective model structure for simplicial presheaves on *
*the
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 3
'etale site. Specializing to the Nisnevich site of smooth schemes, 'etale reali*
*zation
is also homotopy invariant with respect to A1-weak equivalences but only after
completing away from the characteristics of the residue fields of the base sche*
*me S.
The reason for this completion is that EtA1 is non-trivial in positive characte*
*ristic.
Completion takes care of this problem.
Section 2 closes with a corollary concerning the behavior of the 'etale topol*
*ogical
type on elementary distinguished squares. This result can be interpreted as exc*
*ision
for 'etale topological types.
This finishes the main thrust of the paper. The remaining sections are dedica*
*ted
to developing language and machinery suitable for proving Theorem 5.4, which is*
* a
key ingredient in the earlier sections. This theorem states that if U is a hype*
*rcover
of a scheme X, then the natural map
hocolimnEtUn ! EtX
of pro-spaces is a weak equivalence.
Section 3 is dedicated to the study of rigid hypercovers. First come some tec*
*h-
nical results about finite limits of schemes and an introduction to the languag*
*e of
simplicial schemes. Next is the definition hypercovers and rigid hypercovers a*
*nd
then some redefinitions and clarifications of the constructions concerning the *
*'etale
topological type that first appeared in [9].
Section 4 concerns pro-spaces. We review only the bare essentials of pro-spac*
*es
and their homotopy theory. See [15] for details. Some results from [16] on calc*
*u-
lating colimits of pro-spaces are also necessary. A k-truncated realization fun*
*ctor
is a necessary tool because the infinite colimits that are used to construct or*
*dinary
realizations are hard to handle in the category of pro-spaces.
Finally, Section 5 gives the hypercover descent theorem for 'etale topological
types.
We make a few final remarks on terminology. We always mean simplicial sets
[21] whenever we refer to spaces. We write sSetfor the category of simplicial s*
*ets.
An 'etale map U ! X is any map such that U is a (possibly infinite) disjoint
union of schemes Ui and each map Ui ! X is 'etale. This allows us to discuss 'e*
*tale
covers in terms of single maps rather than collections.
Throughout, we assume that the base scheme S is locally Noetherian. Since
all of our schemes are locally of finite type over S, every scheme that we cons*
*ider
is locally Noetherian. This is a technical requirement for the machinery of 'e*
*tale
topological types [9, Ch. 4].
2.Etale Realizations
2.1. Simplicial Presheaves. Let S be a locally Noetherian scheme. Consider the
big 'etale site Sch=S of S [23, x II.1]. The objects of this category are sche*
*mes
locally of finite type over S. Morphisms in Sch=S are just morphisms of schemes
over S. Covers in this category are collections of 'etale maps that have surjec*
*tive
images. The site Sch=S is suitable for studying 'etale cohomology in the sense *
*that
for every X in Sch=S, the 'etale cohomology functors H*et(X; .) are the derived
functors of the functor taking a presheaf F to its group F (X) of sections over*
* X.
Let sPre(Sch =S) be the category of simplicial presheaves on Sch=S. Objects of
sPre(Sch=S) are contravariant functors from Sch=S to simplicial sets; equivalen*
*tly,
4 DANIEL C. ISAKSEN
they are simplicial objects in the category of set-valued presheaves on Sch=S. *
*Mor-
phisms of sPre(Sch=S) are natural transformations of functors.
2.1.1. Objectwise Model Structures on sPre(Sch=S). Let us recall several model
structures for sPre(Sch=S). In the first two, the weak equivalences are maps of
presheaves F ! G such that F (X) ! G(X) is a weak equivalence for every X
in Sch=S; we call such maps objectwise weak equivalences. An injective
cofibration is an objectwise cofibration. An injective fibration is a map of
presheaves having the right lifting property with respect to all objectwise acy*
*clic
injective cofibrations.
On the other hand, a projective fibration is an objectwise fibration, and a
projective cofibration is a map of presheaves having the left lifting property
with respect to all objectwise acyclic projective fibrations.
Theorem 2.1. [4, Prop. XI.8.1] [12, x II.4] The definitions of objectwise weak
equivalences, injective cofibrations, and injective fibrations satisfy the axio*
*ms for a
simplicial proper model structure. The definitions of objectwise weak equivalen*
*ces,
projective cofibrations, and projective fibrations satisfy the axioms for a sim*
*plicial
proper model structure.
These are the objectwise injective and objectwise projective model struc-
tures respectively. The simplicial structure comes from objectwise tensoring a*
*nd
cotensoring.
Both model structures for represent the same homotopy category. This means
we can use whichever model structure is most convenient for a particular purpos*
*e.
For example, the injective cofibrations are simple to describe. In particular, *
*every
object is injective cofibrant. This is sometimes a convenient property. However*
*, the
price for this convenience is that there is no explicit description of the fibr*
*ations.
On the other hand, the projective fibrations are simple to describe, but the *
*pro-
jective cofibrations are more complicated. Not every object is projective cofib*
*rant.
However, there is still a partially explicit description of the projective cofi*
*brations.
For every X in Sch=S, let the representable presheaf rX be the presheaf given
by the formula
rX(Y ) = Hom S(Y, X).
Note that rX is discrete in the sense that each space rX(Y ) is discrete. More
generally, if X is a simplicial scheme over S, then rX is the (not necessarily *
*discrete)
presheaf given by the formula
rX(Y )n = Hom S(Y, Xn).
Proposition 2.2. Every map of the form rX @ [k] ! rX [k] is a projective
cofibration. The maps of this form are a set of generating projective cofibrati*
*ons.
Proof.By the Yoneda lemma, a map F ! G of simplicial presheaves has the right
lifting property with respect to the map rX @ [k] ! rX [k] if and only if *
*the
map F (X) ! G(X) of simplicial sets has the right lifting property with respect*
* to
the map @ [k] ! [k]. A map of simplicial sets has the right lifting property w*
*ith
respect to the maps @ [k] ! [k] if and only if it is an acyclic fibration.
2.1.2. Local Model Structures on sPre(Sch=S). The two objectwise structures of *
*the
previous section are intermediate stages to constructing the two model structur*
*es
of chief interest. Usually, the weak equivalences of these interesting structur*
*es are
defined using sheaves of homotopy groups [19, x 2] or weak equivalences of stal*
*ks.
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 5
Our approach is to define the local weak equivalences by a left Bousfield local*
*ization
[13, Defn. 3.2.1] because this serves our particular purposes best. We follow [*
*6] in
this viewpoint.
Start with the objectwise projective model structure and define a set T of ma*
*ps
that are to become weak equivalences. First, let T contain the map rU ! rX
for every hypercover U of X. See`Definition`3.12 for the definition of hypercov*
*ers.
Second, let T contain the map rVi ! r( Vi) for every collection of objects in
Sch=S.
Definition 2.3. The local projective model structure on sPre(Sch=S) is the T -
localization of the objectwise projective model structure in the sense of [6, D*
*efn. 5.2].
The local weak equivalences are the weak equivalences in this model structure.
The following proposition tells us that our definition of local weak equivale*
*nces
is the same as the usual one.
Proposition 2.4. [7] The local weak equivalences are the same as the topological
weak equivalences of [19, x 2].
Remark 2.5. Note that the same kind of T -localization can be applied to the ob-
jectwise injective model structure. We do not use this theory.
The local projective cofibrations are the same as the projective cofibrations.
A local projective fibration is a map of presheaves that has the right lifting
property with respect to all local acyclic projective cofibrations.
2.2. Etale Realization on sPre(Sch =S). The 'etale topological type functor from
Sch=S to pro-spaces gives a method for constructing a homotopy invariant functor
from sPre(Sch=S) to pro-spaces. See Section 5 or [9] for the definition and pro*
*perties
of this functor.
Recall the following constructions from [6, Prop. 2.1]. Suppose given any fun*
*ctor
F : Sch=S ! C, where C is a simplicial cocomplete category. Define the singular
complex functor SF : C ! sPre(Sch=S) by the formula
SF Z(X) = Map C(F X, Z).
The realization functor ReF is the left adjoint of SF . Moreover, the functors *
*ReF
and SF are simplicial adjoints in the sense that
Map (ReF G, Z) ~=Map (G, SF Z)
for every simplicial presheaf G and every object Z of C.
For any X in Sch=S, note that ReF (rX) = F X. In fact, ReF is the unique
colimit-preserving functor with this property. This follows from the fact that *
*every
presheaf is a colimit of representable presheaves.
The above paragraphs apply in particular to the functor Et : Sch=S ! pro-sSet
that takes a scheme X to the 'etale topological type [9, Defn. 4.4] of the cons*
*tant
simplicial scheme cX. Since pro-sSetis a simplicial cocomplete category [15], we
can apply the constructions of the previous paragraphs to obtain adjoint functo*
*rs
Reet: sPre(Sch=S) ! pro-sSet
and
Set: pro-sSet! sPre(Sch=S).
For every scheme X, the pro-space Reet(rX) is equal to EtX.
The next theorem is one of the main results of this paper.
6 DANIEL C. ISAKSEN
Theorem 2.6. With respect to the local projective model structure on sPre(Sch=S)
and the model structure on pro-sSetgiven in [15], the functors Reetand Setform
a Quillen pair.
Remark 2.7. The theorem is not true if we consider the local injective model st*
*ruc-
ture on sPre(Sch=S). There are too many injective cofibrations.
Proof.By the universal property of T -localizations [6, Defn. 5.2], we need only
show that Reet takes maps in T to weak equivalences. Cofibrant replacements
are no problem because the targets and sources of every map in T are already
projective cofibrant. To show that rU is projective cofibrant for every hyperco*
*ver
U, use Lemma 3.13 to conclude that U is a split simplicial scheme.
First consider a collection {Vi} of objects in Sch=S. Since Reetcommutes with
coproducts of simplicial presheaves and`Et commutes with`coproducts of schemes
[9, Prop. 5.2],`it follows that Reet( rVi) and Reetr( Vi) are both isomorphic*
* to
the pro-space EtVi.
Next consider a hypercover U of X. The simplicial presheaf rU is isomorphic
to the realization |n 7! rUn|. Since Reetis a simplicial left adjoint, it commu*
*tes
with realizations; therefore, ReetrU is equal to the realization |n 7! EtUn| in*
* the
category of pro-spaces. By Theorem 5.4, the map ReetrU ! ReetrX is a weak
equivalence of pro-spaces.
The point of the previous theorem is that Reetinduces a homotopy invariant
derived functor LReet. In order to evaluate LReeton a simplicial presheaf, one *
*first
takes a projective cofibrant replacement for the presheaf and then apply Reet.
Corollary 2.8. The functor LReetinduces a functor from the local homotopy cat-
egory of simplicial presheaves to the homotopy category of pro-spaces. On the l*
*evel
of homotopy categories, it has a right adjoint RSet. Moreover, LReet(rX) = EtX
for every scheme X in Sch=S.
Proof.The first two claims follow from the formal machinery of Quillen adjoint
functors [13, x 8.6]. The last claim follows from the construction of LReetand *
*the
fact that every representable presheaf is projective cofibrant.
The following corollary implies that the original definition of the 'etale to*
*pological
type is recovered by the functor LReet.
Corollary 2.9. Let X be a simplicial scheme in Sch=S. Then LReetrX is weakly
equivalent to EtX.
Proof.The simplicial presheaf rX is equal to the realization |n 7! rXn|. This r*
*eal-
ization is weakly equivalent to hocolimnrXn only if n ! rXn is a Reedy cofibrant
diagram [13, Defn. 16.3.3] of simplicial presheaves. This diagram is not Reedy
cofibrant with respect to the local projective model structure because X is not
necessarily a split simplicial scheme. However, the diagram is Reedy cofibrant
with respect to the local injective model structure. Since the local injective *
*model
structure and the local projective model structure have the same homotopy cate-
gories, we are entitled to use either model structure to construct homotopy col*
*imits.
Therefore, rX is weakly equivalent to the simplicial presheaf hocolimnrXn. Since
homotopy colimits commute with left derived functors, it follows that LReetrX is
weakly equivalent to hocolimnEtXn. This homotopy colimit is weakly equivalent
to EtX by Theorem 5.3.
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 7
2.3. Etale Realization on the A1-Homotopy Category of Schemes. The
'etale realization functor Reetalso behaves well with respect to A1-local model*
* struc-
tures [25]. Begin with the projective objectwise model structure on the site Sm*
*=S of
smooth schemes over S equipped with Nisnevich covers and define a set T 0of maps
that are to become weak equivalences. First, let T 0contain the map rU ! rX for
every Nisnevich hypercover U of X. Nisnevich hypercovers are defined similarly *
*to
hypercovers except that we use Nisnevich covers, not 'etale covers. Since Nisne*
*vich
covers are a special`kind of`'etale cover, Theorem 5.4 applies to them. Second,*
* let T 0
contain the map rVi ! r( Vi) for every collection of objects in Sm =S. Third,
let T 0contain the map r(X x A1) ! rX for every scheme X in Sm =S.
Definition 2.10. The A1-local projective model structure on sPre(Sm =S) is the *
*T 0-
localization of the objectwise projective model structure in the sense of [6, D*
*efn. 5.2].
The A1-local weak equivalences are the weak equivalences in this model struc-
ture.
These A1-local weak equivalences agree with the definition of [25, Defn. 3.2.*
*1].
This follows from Proposition 2.4 and the fact that the A1-local model structur*
*e of
[25] is defined as a localization.
Remark 2.11. The same process yields the T 0-localization of the objectwise inj*
*ec-
tive model structure; this leads to an A1-local injective model structure, whic*
*h is
exactly the model structure presented in [25]. In this paper we use the A1-loc*
*al
projective model structure on Sm =S; although the model structures are differen*
*t,
the homotopy categories are the same.
Remark 2.12. The Nisnevich-local projective model structure is formed by start-
ing with the objectwise projective model structure on sPre(Sm =S) and inverting*
* all
the maps in T 0except for the projections X x A1 ! X. This is the Nisnevich ver-
sion of the local projective model structure considered in Section 2.1.2. Of co*
*urse,
the A1-local projective model structure is the localization of the Nisnevich-lo*
*cal
projective structure at the projections X x A1 ! X.
The importance of Nisnevich hypercovers here suggests that a "Nisnevich topo-
logical type" may give an interesting realization functor on sPre(Sm =S). Howev*
*er,
it turns out that the Nisnevich topological type of any scheme is always homo-
topy discrete. The problem is that the Nisnevich topological type only captures
Nisnevich cohomology with locally constant coefficients, but these always vanis*
*h.
Therefore, we continue to use the 'etale topological type.
Now we discuss the A1-homotopy invariance of the 'etale realization functor. *
*In
order to have an A1-homotopy invariant functor, it is necessary to complete away
from the characteristics of the residues fields of S. We use here a functorial *
*model for
Z=p-completion of pro-spaces as described in [17], where p is a prime not occur*
*ring
as a characteristic of a residue field of S. Below is a summary of the details *
*of this
construction.
Let X be a pointed space. Then ^Xis a tower
. .!.(Z=p)2X ! (Z=p)1X ! (Z=p)0X ! X
of fibrations [4, I.4.3]. The inverse limit of this tower is one of the usual n*
*otions of
the Z=p-completion of X. However, we always consider X^as a pro-space, not as
an ordinary space. This definition extends to pro-spaces.
Definition 2.13. Let X be a pointed pro-space (i.e., a pro-object in the catego*
*ry
of pointed spaces). The Z=p-completion X^ of X is the pro-space given by the
8 DANIEL C. ISAKSEN
functor
(n, s) 7! (Z=p)nXs.
The following result from [17] reminds us of the most important properties of
Z=p-completion.
Theorem 2.14. Let f : X ! Y be a map of pointed pro-spaces. Then the map
^f: X^! ^Yis a weak equivalence of pro-spaces in the sense of Section 4.2 if and
only if for every q 0 and every Z=p-vector space V , the map
Hq(Y ; V ) ! Hq(X; V )
is an isomorphism.
The above theorem uses the traditional formula of cohomology of pro-objects
(see Section 4.2).
Now let p be a fixed prime that does not occur as the characteristic of any r*
*esidue
field of S, and let E^tbe the functor from schemes to pro-spaces that takes X to
et et
the Z=p-completion of EtX. As in Section 2.2, let R^e and ^S be the realizati*
*on
and singular complex functors corresponding to ^Et.
Theorem 2.15. With respect to the A1-local projective model structure on sPre(S*
*m =S)
et
and the model structure on pro-sSetdescribed in Section 4.2, the functors R^e a*
*nd
^Setform a Quillen pair.
Remark 2.16. The theorem is not true when considering the A1-local injective
model structure on sPre(Sm =S). There are too many injective cofibrations.
Proof.The argument is basically the same as in the proof of Theorem 2.6. The
only significantly different part is in showing that
^ReetrX ! ^Reetr(X x A1)
is a weak equivalence for every scheme X in Sm =S. In other terms, we must show
that ^Et(X x A1) ! ^EtX is a weak equivalence of pro-spaces. Since the functor *
*Et
commutes with coproducts and X x A1 ! X induces an isomorphism of connected
components, it suffices to assume that X is connected. Moreover, we may choose
an arbitrary basepoint for X so that we have a map of pointed connected pro-
spaces. Invoking Theorem 2.14, it is necessary only to show that this map induc*
*es
an isomorphism in cohomology with coefficients in Z=p-vector spaces. In order to
understand these cohomology maps, [9, Prop. 5.9] allows us to consider the map *
*on
'etale cohomology induced by the projection
X x A1 ! X.
The projection induces an isomorphism in 'etale cohomology by [23, Cor. VI.4.20*
*].
Remark 2.17. It is also possible to use the completion of [24] in order to defi*
*ne a
slightly different A1-homotopy invariant 'etale realization functor. See [17] f*
*or more
details.
The next corollary follows from Theorem 2.15 in the same way that Corollary
2.8 follows from Theorem 2.6.
et
Corollary 2.18. The functor LR^e induces a functor from the homotopy cate-
gory of schemes to the homotopy category of pro-spaces. On the level of homotopy
categories, it has a right adjoint RS^et.
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 9
2.4. Excision for the Etale Topological Type. This section gives an interesting
corollary about 'etale topological types and elementary distinguished squares. *
*Recall
that an elementary distinguished square [25, Defn. 3.1.3] is a diagram
U xX V ____//_V
| p|
| |
|fflffl fflffl|
U ___i___//_X
of smooth schemes over S in which i is an open inclusion and p : p-1(X - U) !
X - U is an isomorphism (where the schemes p-1(X - U) and X - U are given the
reduced structure). The relevance of such squares is that the maps i and p form
a Nisnevich cover of X. One way of interpreting the next theorem is that these
special Nisnevich covers generate all Nisnevich covers in a certain sense.
Theorem 2.19. [3, Lem. 4.1] A simplicial presheaf F in sPre(Sm =S) is Nisnevich-
local projective fibrant (see Remark 2.12) if and only if F is objectwise fibra*
*nt and
takes elementary distinguished squares to homotopy pullback squares.
This leads immediately to the following excision theorem for 'etale topologic*
*al
types.
Theorem 2.20. Let
U xX V _____//V
| |
| |
fflffl| fflffl|
U _______//_X
be an elementary distinguished square of smooth schemes over S. Then the square
Et(U xX V )____//_EtV
| |
| |
fflffl| fflffl|
EtU ________//_EtX
is a homotopy pushout square of pro-spaces.
Proof.The argument of the proof of Theorem 2.6 shows that Reetand Set are a
Quillen pair of adjoint functors between the category sPre(Sm =S) (equipped with
the Nisnevich-local projective model structure) and the category of pro-spaces.
Therefore, SetZ is Nisnevich-local projective fibrant for every fibrant pro-spa*
*ce Z.
By Theorem 2.19, the square
Map (Et(UOxXOV ), Z)oo__Map (EtV,OZ)O
| |
| |
| |
Map (EtU, Z)oo_______Map (EtX, Z)
is a homotopy pullback square for every fibrant Z, so
Et(U xX V )____//_EtV
| |
| |
fflffl| fflffl|
EtU ________//_EtX
10 DANIEL C. ISAKSEN
is a homotopy pushout square.
Remark 2.21. The previous theorem can also be viewed in terms of the cohomo-
logical excision theorem of [23, III.1.27], at least with locally constant coef*
*ficients,
because the 'etale cohomology of a scheme is isomorphic to the singular cohomol*
*ogy
of its 'etale topological type.
3.Hypercovers of Simplicial Schemes
The point of this section is to study and define rigid hypercovers of simplic*
*ial
schemes and to make some useful constructions concerning them.
3.1. Finite Limits of Schemes. We first study how finite limits interact with
'etale maps and separated maps. The results here are not particularly striking,*
* but
they do not appear in the standard literature [10] [11] [23] [30].
A technical result about fiber products comes first. The more general claim
about arbitrary finite limits follows relatively easily.
Lemma 3.1. Consider a diagram of schemes
U _____//Voo___W
| | |
| | |
fflffl|fflffl| fflffl|
X _____//Yoo___Z
such that the three vertical maps are 'etale (resp., separated). Then the induc*
*ed map
U xV W ! X xY Z
is also 'etale (resp., separated).
Proof.We prove the lemma for 'etale maps. The proof for separated maps is iden-
tical. See [10, Prop. I.5.3.1] for the necessary properties of separated maps.
Recall that base changes preserve 'etale maps [23, Prop I.3.3(c)]. Let f be t*
*he
map in question. Factor f as
U xV W _____//U xY W_____//X xY W_____//X xY Z.
The second map and third maps are 'etale because of the pullback squares
U xY W _____//U X xY W _____//W
| | | |
| | | |
fflffl| fflffl| fflffl| fflffl|
X xY W ____//_X X xY Z _____//_Z.
It remains to show that the first map is also 'etale. The diagram
U xV W ________//V
| |
| |
fflffl| fflffl|
U xY W _____//V xY V
is a pullback square, where is the diagonal map. It suffices to observe that *
* is
'etale [23, Prop. I.3.5].
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 11
Proposition 3.2. Let f : U ! X be a map of finite diagrams of schemes such
that the map fa : Ua ! Xa is 'etale (resp., separated) for every a. Then the map
limf : limU ! limX is 'etale (resp., separated).
Proof.This is a formal consequence of Lemma 3.1 because every finite limit can *
*be
expressed in terms of finite products and fiber products.
3.2. Simplicial Schemes. This section makes more precise many of the construc-
tions and notations of [9]. We work in the category of schemes or more generally
in the category of schemes over a fixed base scheme S; these two cases are actu*
*ally
the same since the category of schemes has a terminal object SpecZ.
Let be the category whose objects are the non-empty ordered sets [n] = {0 <
1 < 2 < . .<.n} and whose morphisms are the weakly monotonic maps. This is
the usual indexing category for simplicial objects.
Let [n] be the simplicial set represented by [n]. Thus, [n]m = Hom ([m], [*
*n]).
Note that [ ] is a cosimplicial space.
Let + be the category with an initial object [-1] adjoined. The opposite of
+ is the usual indexing category for augmented simplicial objects.
Definition 3.3. A simplicial scheme is a functor from op to schemes. An
augmented simplicial scheme is a functor from op+to schemes. A bisimplicial
scheme is a functor from ( x )op to schemes. An augmented bisimplicial
scheme is a functor from ( x + )op to schemes.
Note that augmented bisimplicial schemes are augmented in only one direction.
Augmented bisimplicial schemes are perhaps more correctly but awkwardly called
simplicial augmented simplicial schemes.
For every scheme X, let cX be the constant simplicial scheme with value X.
Recall the nth latching object LnX of a simplicial object X [13, Defn. 16.2.5*
*].
It is a finite colimit indexed by the subcategory of consisting of the degene*
*racy
maps [m] ! [m-1] for 1 m n-1. Beware that LnX does not necessarily exist
for every simplicial scheme X because the category of schemes is not cocomplete.
Definition 3.4. A simplicial scheme X is split if LnX exists for every n 0 and
the canonical map LnX ! Xn is the inclusion of a direct summand. If X is split,
let NXn be the subscheme of Xn such that Xn = LnX q NXn.
A simplicial scheme is split up to dimension n if the map Lm X ! Xm is the
inclusion of a direct summand for m n.
The idea is that NXn is the non-degenerate part of Xn and that Xn splits into
a direct sum of its degenerate part and its non-degenerate part. Note that NXn *
*is
well-defined because the category of schemes is locally connected [2, x 9].
3.3. Skeleta and Coskeleta. Let (n)be the full subcategory of on the objects
[m] for m n. Note that (n)is a finite category.
op
Definition 3.5. An n-truncated simplicial scheme is a functor from (n)
to schemes.
Definition 3.6. If X is a simplicial scheme, then the truncated n-skeleton
sknX is the n-truncatedosimplicialpscheme given by restriction of X along the
inclusion (n) ! op. The n-skeleton Skn X is the simplicial scheme given
in dimension m by
(SknX)m = colim Xk,
OE:[m]![k]
k n
12 DANIEL C. ISAKSEN
provided that it exists.
In general, SknX does not exist because the necessary colimits may not exist *
*in
the category of schemes. However, SknX does exist when X is split up to dimensi*
*on
n. In this case, (SknX)m is a disjoint union of one copy of NXk for each surjec*
*tive
map [m] ! [k] with k n.
Remark 3.7. It is not usually necessary to distinguish between the functors skn*
*and
Sknin the category of simplicial sets. This is because Sknexists for every simp*
*licial
set since every simplicial set is split. For technical precision, the distinct*
*ion is
important in the category of simplicial schemes.
Definition 3.8. The nth coskeleton functor coskn from n-truncated simplicial
schemes to simplicial schemes is right adjoint to the functor skn.
We abuse notation and write cosknX instead of coskn(sknX) for a simplicial
scheme X. To avoid confusion, write coskSnfor the nth coskeleton functor in the
category of schemes over S. By convention, define cosk-1X to be the constant si*
*m-
plicial scheme cSpecZ. More generally, coskS-1X is the constant simplicial sche*
*me
cS. In particular, (coskS-1X)0 is equal to S. This convention makes our definit*
*ion
of hypercovers in Section 3.4.2 more concise.
In order to make the notation consistent, we should define another functor Co*
*skn
from simplicial schemes to simplicial schemes that is right adjoint to Skn. Bec*
*ause
Sknis not defined on the full category of simplicial schemes, it is somewhat aw*
*kward
to precisely state this adjointness. If X is a simplicial scheme such that SknX
exists, then for every simplicial scheme Y , maps SknX ! Y are in one-to-one
correspondence with maps X ! cosknY . Hence our abuse of notation in the
previous paragraph makes it unnecessary to introduce another functor Coskn.
The functor cosknplays a critical role, so we recall some of its properties. *
*Very
importantly for us, each object (cosknX)m is a finite limit of the objects Xk f*
*or
k n. Also important is that (cosknX)m is isomorphic to Xm when m n. In
other words, cosknX and X agree up to dimension n.
For every simplicial scheme X, the unit map X ! coskn(sknX) induces a natural
map
Xm ! (coskkX)m .
These maps will appear again and again.
Note that (cosknX)n+1 is the nth matching object MnX of X [13, Defn. 16.2.5].
Proposition 3.9. Let f : U ! X be a map of n-truncated simplicial schemes such
that for every m n, the map fm is 'etale. Then
(cosknf)m : (cosknU)m ! (cosknX)m
is 'etale for every m.
Proof.This is just a special case of Proposition 3.2.
Remark 3.10. For any finite simplicial set K and`any simplicial scheme X, define
X K to be the simplicial scheme isomorphic to KnXn in dimension n. Then
define the cotensor XK such that the functors (.) K and (.)K are adjoints. In
these terms, the scheme (cosknX)m is isomorphic to XSkn [m] 0.
The simplicial structure on simplicial schemes can be a very useful language.
However, we will not need it here.
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 13
3.4. Hypercovers. Much of the material in this section can be found in [9]. We
review the basic notions of hypercovers and rigid hypercovers. In the next few
sections we formalize some useful constructions concerning them.
For any_point x0 of a scheme X, a_geometric point of X over x0 is a map
x : Speck! X with image x0, where_k is the separable closure of the residue fie*
*ld
k(x0). For any scheme X, let_X(k)_be the set of geometric points. If f : X ! Y
is a map of schemes and_y : k ! Y is a geometric point of Y , then a lift of y *
*is a
geometric point_x : k_!_X such_that y = f O x. Equivalently, x goes to y under
the set map f(k) : X(k) ! Y (k). In this situation, we abuse notation and write
f(x) = y.
3.4.1. Rigid Covers.
Definition 3.11. A rigid cover of a scheme X is a map of schemes f : U ! X such
that f is a separated 'etale surjective map satisfying the following properties*
*. The
connected components of U are in one-to-one correspondence with the geometric
points of X. Write Ux for the component of U corresponding to the geometric poi*
*nt
x, and it has a basepoint ux, which is a geometric point of Ux such that f(ux) *
*= x.
If f : U ! X and f0 : U0 ! X0 are rigid covers of X and X0, then a rigid cover
map g : f ! f0 over a scheme map h : X ! X0 is a commuting square
g
U _____//U0
f|| |f0|
fflffl| fflffl|
X __h_//_X0
such that g(ux) = u0h(x)for every geometric point x of X. The idea is that g
preserves the basepoints of each component.
The importance of rigid covers is that there exists at most one rigid cover m*
*ap
between any two rigid covers of a scheme [9, Prop. 4.1].
3.4.2. Hypercovers and Rigid Hypercovers.
Definition 3.12. A hypercover of a scheme X is an augmented simplicial scheme
U such that U-1 = X and the map
Un ! (coskXn-1U)n
is 'etale surjective for all n 0. A hypercover of a simplicial scheme X is an
augmented bisimplicial scheme U such that U ,-1= X and Un, is a hypercover of
Xn for each n.
By convention, the map
Un ! (coskXn-1U)n
is equal to the map U0 ! X when n = 0. It is important to remember that U0 ! X
must be 'etale surjective.
If U and U0 are hypercovers of schemes (resp., simplicial schemes) X and X0,
then a hypercover map g : U ! U0 over a map h : X ! X0 is an augmented
simplicial scheme map (resp., augmented bisimplicial scheme map) such that g-1
(resp., g ,-1) is equal to h.
The following lemma is a key property of hypercovers. It provides a technical
ingredient in the construction of rigid pullbacks and rigid limits of rigid hyp*
*ercovers
in Sections 3.5 and 3.6.3.
14 DANIEL C. ISAKSEN
Lemma 3.13. Every hypercover of a scheme is split.
Proof.Let U be a hypercover of X. By induction and Proposition 3.9, U is a
simplicial object in the category of 'etale schemes over X. The remark after [*
*2,
Defn. 8.1] finishes the argument.
Definition 3.14. A rigid hypercover of a scheme X is a hypercover of X such
that the map
Un ! (coskXn-1U)n
is a rigid cover for all n 0.
If U and U0 are rigid hypercovers of schemes X and X0, then a rigid hypercover
map U ! U0 over a map X ! X0 is a hypercover map such that the square
Un _____________//_U0n
| |
| |
fflffl| fflffl|0
(coskXn-1U)n____//(coskXn-1U0)n
is a rigid cover map for every n 0.
Definition 3.15. A rigid hypercover of a simplicial scheme X is a hypercover
of X such that Un, is a rigid hypercover of Xn for each n and Un, ! Um, is a
rigid hypercover map over Xn ! Xm for every [m] ! [n].
If U and U0 are rigid hypercovers of simplicial schemes X and X0, then a rigid
hypercover map g : U ! U0 over a map h : X ! X0 is a hypercover map such that
gn is a rigid hypercover map over hn for each n.
Similarly to rigid covers, there exists at most one map between two rigid hyp*
*er-
covers of a scheme (or simplicial scheme) [9, Prop. 4.3]. On the other hand, ma*
*ps
between hypercovers are unique only in a certain homotopical sense [2, Cor. 8.1*
*3].
Definition 3.16. For a scheme (or simplicial scheme) X, let HRR (X) be the
category of rigid hypercovers of X.
The notation comes from [9]. The critical property of this category is that i*
*t is
cofiltered [9, Prop. 4.3]. Since there is at most one map between any two objec*
*ts
of HRR (X), this category is actually a directed set.
Lemma 3.17. Let X be a scheme. The categories HRR (X) and HRR (cX) are
equivalent.
Proof.Consider the functor HRR (X) ! HRR (cX) that takes a rigid hypercover
U of X to the hypercover V of cX given by the formula Vm,n = Un. This functor
is full and faithful, so it suffices to show that every rigid hypercover of cX *
*belongs
to the image of this functor.
Let V be an arbitrary rigid hypercover of cX. Then V is a simplicial diagram
in the category HRR (X). There is at most one rigid hypercover map between any
two rigid hypercovers of X. It follows that the map Vn, ! Vm, is the identity m*
*ap
for all [m] ! [n].
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 15
3.5. Rigid Pullbacks. Suppose that f : X ! Y is a map of schemes and U ! Y
is 'etale surjective. Then the base change f*U ! X is the projection X xY U ! X,
which is again 'etale surjective [23, Prop. I.3.3(c)]. This idea generalizes t*
*o rigid
covers.
Definition 3.18. Let f : X ! Y be any map of schemes and let U ! Y be a
rigid cover. Then the rigid pullback f*U ! X is the rigid cover defined by
the following construction. For each geometric point x of X, let (f*U)x be the
component of X xY U containing x x uf(x), and let x x uf(x)be the basepoint of
(f*U)x.
Note that (f*U)x is a component of X xY Ux, but f*U is not a restriction of
X xY U to certain components since some components of X xY U may occur more
than once as components of f*U. Also note that there is a canonical rigid cover
map
f*U _____//U
| |
| |
fflffl| fflffl|
X ______//Y.
Lemma 3.19. Let f : X ! Y be any map of schemes and let U ! Y be a rigid
cover. Then the rigid cover f*U ! X has the following universal property. Let
V ! Z be an arbitrary rigid cover. Rigid cover maps
V _____//f*U
| |
| |
fflffl| fflffl|
Z ______//X
correspond to rigid cover maps
V _____//U
| |
| |
fflffl|fflffl|
Z _____//Y
together with a factorization of the map Z ! Y through X.
Proof.The category of connected pointed schemes has finite limits. To construct
such limits, just take the basepoint component of the usual limit of schemes. T*
*he
lemma now follows from this observation and the universal property of pullbacks
of schemes.
Rigid pullbacks of rigid covers generalizes to rigid hypercovers.
Definition 3.20. Suppose f : X ! Y is a map of schemes and U is a rigid
hypercover of Y . Then the rigid pullback f*U is the hypercover of X constructed
as follows. Let (f*U)0 be the rigid pullback along f of the rigid cover U0 !
Y . Inductively define (f*U)n to be the rigid pullback along (coskXn-1f*U)n !
(coskYn-1U)n of the rigid cover Un ! (coskYn-1U)n.
In order to describe the face and degeneracy maps of f*U, it suffices to give
inductively a factorization of g : Lnf*U ! (coskXn-1f*U)n through (f*U)n for
each n [13, Thm. 16.2.1]. Recall that Lnf*U is the nth latching object of f*U
and (coskXn-1f*U)n is the nth matching object. Note that Lnf*U exists because
16 DANIEL C. ISAKSEN
(by induction) the (n - 1)-truncated scheme f*U is a truncated hypercover and is
therefore split by Lemma 3.13. Also, LnU exists since U is a hypercover and is
therefore split.
The diagram
(f*U)n ____________//_Un""9
| | 999
| | 99
fflffl| fflffl|999
(coskXn-1f*U)n____//(coskYn-1U)n77999
pppp eeLLL99
pppp LLLL99
ppp LLL99
Lnf*U _______________________________________//_LnU
induces a map
h0: Lnf*U ! Un x(coskYn-1U)n(coskXn-1f*U)n.
We can compose with the projection map to (coskXn-1f*U)n to get a factorization
of g, but this is not quite the desired factorization because (f*U)n is not exa*
*ctly
equal to the fiber product
Un x(coskYn-1U)n(coskXn-1f*U)n.
In order to produce the desired factorization h : Lnf*U ! (f*U)n, we must speci*
*fy
which component of (f*U)n is the target of each component of Lnf*U.
Since Lnf*U is a disjoint union of copies of (f*U)m for m < n, each component
has a basepoint. Let C be a component of Lnf*U with basepoint c. Then h is
defined to take C into the component ((f*U)n)g(c)of (f*U)n.
Remark 3.21. Defining the face and degeneracy maps of f*U is technically complex
and unimportant in the big picture, but we see no way of avoiding it. This issu*
*e is
not dealt with in [9, p. 37].
A careful inspection of the definitions yields the critically important prope*
*rty
that rigid pullbacks of rigid hypercovers are functorial. This means that the d*
*efi-
nition of rigid pullbacks extends to rigid hypercovers of simplicial schemes.
Also note that there is a canonical rigid hypercover map f*U ! U over the map
f : X ! Y .
Lemma 3.22. Let U be a rigid hypercover and let f : X ! U-1 be any map of
schemes. The rigid hypercover f*U of X has the following universal property. Let
V be an arbitrary rigid hypercover. Rigid hypercover maps V ! f*U correspond to
rigid hypercover maps V ! U together with a factorization of the map V-1 ! U-1
through X.
Proof.This follows from Lemma 3.19 and induction. Because V , U, and f*U are
all split by Lemma 3.13, the degeneracy maps take care of themselves.
3.6. Rigid Limits. In this section, Proposition 3.2 is generalized to rigid cov*
*ers
and rigid hypercovers.
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 17
3.6.1. Limits of Covers and Hypercovers.
Proposition 3.23. Let f : U ! X be a finite diagram of rigid cover maps. Then
the map
limafa : limaUa ! limaXa
is 'etale surjective.
Proof.The map is 'etale by Proposition 3.2, so we need only show that every geo-
metric point x of limX lifts to limU. Let xa be the composition of x with the
obvious projection map limX ! Xa. Since each fa is a rigid cover, there exist
canonical lifts ua of each xa to Ua. They assemble to give a geometric point u *
*of
limU because f is a diagram of rigid cover maps.
The above proposition is not true if each fa is only 'etale surjective. The p*
*roblem
is that a limit of surjective maps is not necessarily surjective. Note that lim*
*f is
only 'etale surjective; it is not a rigid cover. As the proof above indicates, *
*there are
canonical lifts for each geometric point of limX, but the components of limU may
not correspond one-to-one to the geometric points of limX.
Proposition 3.24. Suppose that U is a finite diagram of rigid hypercovers. Then
limaUa is a hypercover.
Proof.For convenience, write X for U-1. We want to show that limaUa is a
hypercover of limaXa, so we must prove that
(limU)n-1 ! cosklimXn(limU)n-1
is 'etale surjective for all n 0. This follows from Proposition 3.2 and the f*
*act that
a a
cosklimXn(limU)n-1 ~=lima(coskXnU )n-1.
Again, the above proposition is not true if each Ua is only a hypercover. Als*
*o,
limU is only a hypercover, not a rigid hypercover.
3.6.2. Rigid Limits of Rigid Covers. As seen in the previous section, ordinary *
*finite
limits do not preserve rigid covers and rigid hypercovers. Thus, the notion of *
*limit
must be refined in order to get a rigid cover-preserving construction.
Definition 3.25. Let f : U ! X be a finite diagram of rigid cover maps. Then
the rigid limit
Rlimafa : RlimaUa ! limaXa
is the rigid cover defined as follows. For each geometric point x = limaxa of
limaXa, let (Rlima Ua)x be the connected component of limaUa containing ux =
limauaxa, and let ux be the basepoint of (Rlima Ua)x.
It is important to begin with a diagram of rigid cover maps, not just cover m*
*aps.
Otherwise, the geometric points uaxaare not necessarily compatible and do not
induce a geometric point ux of limaUa.
RQ R
The symbols and x denote rigid limits in the case of products or fiber prod*
*ucts.
Note that there is a natural map RlimU ! limU over limX.
Lemma 3.26. The rigid limit of a finite diagram of rigid covers is a rigid cove*
*r.
18 DANIEL C. ISAKSEN
Proof.The map RlimaUa ! limaXa factors as a local isomorphism RlimaUa !
limaUa followed by the map limaUa ! limaXa. The latter is 'etale and separated
by Proposition 3.2, so the composition is also 'etale and separated. The other *
*parts
of the definition of a rigid cover are satisfied by construction.
Lemma 3.27. Let f : U ! X be a finite diagram of rigid covers. Then Rlimafa
is universal in the following sense. Let g : V ! Y be any rigid cover of a sche*
*me
Y . Rigid cover maps g ! Rlimf are in one-to-one correspondence with collections
of rigid cover maps g ! fa such that for every map fa ! fb, the diagram
g ____//_>fa
>> |
>>> |
>OEOE>fflffl|
fb
of rigid cover maps commutes.
Proof.As in the proof of Lemma 3.19, it is important that the category of conne*
*cted
pointed schemes has finite limits. The lemma now follows from this observation *
*and
the universal property of limits.
Remark 3.28. Rigid limits have the same kind of functoriality as ordinary limit*
*s.
We make this more precise. Let f : U ! X and g : V ! Y be diagrams of rigid
cover maps indexed by finite categories A and B respectively. Suppose given a
functor F : B ! A, and let F *f be the diagram of rigid cover maps indexed by
B given by the formula (F *f)b = fF(b). Suppose given a natural transformation
j : F *f ! g. Then j induces a natural map RlimA f ! RlimB g. This is precisely
what happens for ordinary limits.
Remark 3.29. Given a diagram U of rigid cover maps over a fixed scheme X,
arbitrary finite rigid limits are not really necessary. Since the category of *
*rigid
covers of X is actually a directed set, the rigid limit of U is just theQleast *
*upper
bound of the rigid covers Ua. This is isomorphic to the rigid product RUa.
More generally, suppose now that X is not a constant diagram. For every a,
consider the rigid pullback (ßa)*Ua ! limbXb of Ua ! Xa along the projectionQ
ßa : limbXb ! Xa. Then RlimaUa is isomorphic to the rigid product R(ßa)*Ua.
Hence arbitrary rigid limits can be constructed with rigid products and rigid
pullbacks. This observation relies heavily on the fact that there is at most on*
*e rigid
cover map between any two rigid covers.
Remark 3.30. Suppose that U and X are n-truncated schemes and f : U ! X is a
diagram of rigid cover maps. Write
(Rcosknf)k : (RcosknU)k ! (cosknX)k
for the rigid limit of the finite diagram whose ordinary limit is (cosknf)k. Be*
*cause
of the functoriality expressed in Remark 3.28, these constructions assemble int*
*o a
map
Rcosknf : RcosknU ! cosknX
of simplicial schemes that is a simplicial object in the category of rigid cove*
*rs.
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 19
3.6.3. Rigid Limits of Rigid Hypercovers. Let U be a finite diagram of rigid hy*
*per-
cover maps, and let X equal U-1. Each Ua is a rigid hypercover of Xa, and each
Ua ! Ub is a rigid hypercover map over Xa ! Xb.
Proposition 3.24 implies that V = limaUa is a hypercover of Y = limaXa. We
use rigid limits to define a canonical rigid hypercover W = RlimaUa of Y and a
natural hypercover map W ! V over Y .
Begin by defining W0 to be RlimaUa0. There is a canonical map from W0 to
V0 = limaUa0.
Suppose for sake of induction that Wm and the map Wm ! Vm have been defined
for m < n. Thus there is a map (coskYn-1W )n ! (coskYn-1V )n. Let x be a geomet*
*ric
point of (coskYn-1W )n, andalet y be its image in (coskYn-1V )n. Since (coskYn-*
*1V )n
is isomorphic to lima(coskXn-1Ua)n,ay gives compatible geometric points ya in e*
*ach
of the schemes (coskXn-1Ua)n. Each ya has a canonical lift za in Uansince each *
*Ua
is a rigid hypercover. Moreover, these lifts are compatible since U is a diagra*
*m of
rigid hypercover maps. This means that they assemble to give a geometric point z
of Vn = limaUan, and z is a lift of y.
Now define (Wn)x to be the connected component of
Vn x(coskYn-1V()ncoskYn-1W )n
containing zxx, and let zxx be the basepoint of (Wn)x. This extends the definit*
*ion
of W to dimension n.
Remark 3.31. To describe the face and degeneracy maps of W , one must use an
argument similar to that given in Section 3.5 for describing rigid pullbacks of*
* rigid
hypercovers. Although it is technically complex and unimportant in the big pict*
*ure,
we know of no way of avoiding it.
Lemma 3.32. Rigid limits of rigid hypercovers have the following universal prop-
erty. Suppose that U is a diagram of rigid hypercover maps, and let V be an
arbitrary rigid hypercover of a scheme Y . Rigid hypercover maps from V to RlimU
are in one-to-one correspondence with collections of rigid hypercover maps V ! *
*Ua
such that for every map Ua ! Ub, the diagram
V ____//_AUa
AA |
AAA |
AA__fflffl|
Ub
of rigid hypercover maps commutes.
Proof.This follows from Lemma 3.27 and induction. Because V , limU, and each
Ua are all split by Lemma 3.13, the degeneracy maps take care of themselves.
Remark 3.33. As for rigid limits of rigid covers, rigid limits of rigid hyperco*
*vers
have the same kind of functoriality as ordinary limits. See Remark 3.28 for more
details.
Arbitrary finite rigid limits of rigid hypercovers are not really necessary. *
*In fact,
rigid products and rigid pullbacks suffice to construct all rigid limits. See R*
*emark
3.29 for more details.
We use the notation Rcoskn for rigid hypercovers analogously to our use of th*
*is
notation for rigid covers as in Remark 3.30.
20 DANIEL C. ISAKSEN
3.6.4. Cofinal Functors of Rigid Hypercovers. The necessary constructions for r*
*igid
hypercovers have now been established. Our investment in the previous sections
clarifies some of the technical complexities in the proofs of [9, Ch. 4].
For every simplicial scheme X and every n 0, there is a forgetful functor
HRR (X) ! HRR (Xn) taking a rigid hypercover U of X to the rigid hypercover
Un, of Xn. These functors assemble to give a functor
HRR (X) ! HRR (X0) x HRR (X1) x . .x.HRR (Xn).
The idea is that this functor forgets the face and degeneracy maps and only re-
members the objects Um, for m n.
Proposition 3.34. Let X be a simplicial scheme. The functor
HRR (X) ! HRR (X0) x HRR (X1) x . .x.HRR (Xn).
is cofinal.
This proposition is closely related to [9, Cor. 4.6], which show that the fun*
*ctor
HRR (X) ! HRR (Xn) is cofinal for every simplicial scheme X and every n 0.
Proof.For convenience, let I be the category
HRR (X0) x HRR (X1) x . .x.HRR (Xn).
Since each HRR (Xm ) is actually a directed set, so is I. The category HRR (X) *
*is
also a directed set, so it suffices to show that for every object U = (U0,, U1,*
*, . .,.Un,)
of I, there is an object V of HRR (X) and a rigid hypercover map Vm, ! Um, over
Xm for every m n.
For each m, define Vm, to be
Rlim Uk,.
OE:[k]![m]
k n
The idea is that Vm, is a "rigid right Kan extension". The rigid limit is fini*
*te
because k is at most n.
The functoriality of rigid limits as expressed in Remark 3.33 assures us that*
* V is
in fact a rigid hypercover of X. The identity map id: [m] ! [m] gives a project*
*ion
Vm, ! Um, .
These maps are the desired ones.
4.Pro-Spaces
Having established the necessary background on hypercovers, we switch topics
and proceed to develop some ideas about pro-spaces and realizations. Hypercovers
reappear in Section 5.
4.1. Preliminaries on Pro-Categories. We begin with a review of the necessary
background on pro-categories. This section contains only standard material on p*
*ro-
categories [1] [2] [8] [16].
Definition 4.1. For a category C, the category pro-C has objects all cofiltering
diagrams in C, and
Hom pro-C(X, Y ) = limscolimtHomC(Xt, Ys).
Composition is defined in the natural way.
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 21
A category I is cofiltering if the following conditions hold: it is non-empty*
* and
small; for every pair of objects s and t in I, there exists an object u togethe*
*r with
maps u ! s and u ! t; and for every pair of morphisms f and g with the same
source and target, there exists a morphism h such that fh equals gh. Recall tha*
*t a
category is small if it has only a set of objects and a set of morphisms. A dia*
*gram
is said to be cofiltering if its indexing category is so.
Let Y : I ! C and X : J ! C be arbitrary pro-objects. Then X is cofinal in Y
if there is a cofinal functor F : J ! I such that X is equal to the composite Y*
* F .
This means that for every s in I, the overcategory F # s is cofiltered. In the *
*case
when F is an inclusion of directed sets, F is cofinal if and only if for every *
*s in I
there exists t in J such that t s. The importance of this definition is that *
*X is
isomorphic to Y in pro-C.
A level representation of a map f : X ! Y is: a cofiltered index category I;
pro-objects ~Xand ~Yindexed by I and isomorphisms X ! ~Xand Y ! ~Y; and a
collection of maps fs : ~Xs! ~Ysfor all s in I such that for all t ! s in I, th*
*ere is a
commutative diagram
~Xt____//~Yt
| |
| |
fflffl| fflffl|
X~s_____//~Ys
and such that the maps fs represent a pro-map ~f: ~X! ~Ybelonging to a commu-
tative square
f
X _____//Y
| |
| |
fflffl|fflffl|
X~__~__//~Y
f
in pro-C. That is, a level representation is just a natural transformation such*
* that
the maps fs represent the element f of
limscolimtHomC(Xt, Ys) ~=limscolimtHomC(X~t, ~Ys).
Every map has a level representation [2, App. 3.2] [22].
More generally, suppose given any diagram A ! pro-C : a 7! Xa. A level
representation of X is: a cofiltered index category I; a functor X~ : A x I !
C : (a, s) 7! ~Xas; and isomorphisms Xa ! ~Xasuch that for every map OE : a ! b
in A, X~OEis a level representation for XOE. In other words, X~ is a uniform l*
*evel
representation for all the maps in the diagram X. Not every diagram of pro-
objects has a level representation, but every finite loopless diagram does have*
* a
level representation.
Level representations are important tools for calculating finite limits and c*
*olimits
in pro-categories. In order to calculate the limit or colimit of a finite diagr*
*am of
pro-objects, take a level representation for the diagram and then take the leve*
*lwise
limit or colimit.
Suppose that X : I ! C and Y : J ! C are two pro-objects. A strict repre-
sentation [9, p. 36] of a map f : X ! Y is a functor F : J ! I and a natural
transformation j : X O F ! Y such that the maps js : XF(s)! Ys represent the
22 DANIEL C. ISAKSEN
element f of
limscolimtHomC(Xt, Ys).
More generally, a strict representation of a diagram X in pro-C consists of
strict representations (F OE, jOE) for every map OE in X such that for every pa*
*ir of
composable maps OE and _ in X, the functor F _OEequals F OEF _and the natural
transformation j_OEequals j_ O jOEF _.
4.2. Homotopy Theory of Pro-Spaces. We now review from [15] the homotopy
theory of pro-spaces suitable for studying the 'etale topological type.
Given a cofiltered diagram of pointed connected spaces X, application of the
ith homotopy group functor yields a pro-group ßiX. A map f : X ! Y of such
pointed connected pro-spaces is a weak equivalence if ßif is an isomorphism of
pro-groups for every i 0.
In fact, one can define weak equivalences for pro-spaces that are not pointed*
* and
connected, but choices of basepoints become a complicated and messy issue. See
[15] for more details.
These weak equivalences belong to a proper simplicial model structure for pro-
spaces. The cofibrations are the maps that up to isomorphism have level represe*
*n-
tations that are levelwise cofibrations. It follows that every pro-space is cof*
*ibrant.
The fibrations are defined by a lifting property, but an explicit description i*
*s pos-
sible [15, Prop. 6.6].
Given a pro-space X, the formula
H*(X; M) = colimsH*(Xs; M)
defines cohomology [2, 2.2] [29, x 2.2]. In fact, there is an isomorphism betw*
*een
Hq(X; M) and the set [X, cK(M, q)]proof maps in the homotopy category of pro-
spaces [15, Lem. 8.1]. Here cK(M, q) is the constant pro-space with value an
Eilenberg-Mac Lane space.
4.3. n-Truncated Realizations. Let C be a simplicial category; this means that
objects of C can be tensored and cotensored with simplicial sets, and these ope*
*r-
ations satisfy appropriate adjointness conditions. We assume that C is complete
and cocomplete. Our application involves pro-spaces, which is a complete and
cocomplete category [15, Prop. 11.1].
Recall the definition of the realization of a simplicial object in C.
Definition 4.2. Given a simplicial object X in a simplicial category C, its rea*
*l-
ization ReX is the coequalizer of the diagram
` `
Xn [m] ____//_//_Xn [n],
OE:[m]![n] n
where the upper arrow is induced by maps id OE* : Xn [m] ! Xn [n] and
the lower arrow is induced by maps OE* id: Xn [m] ! Xm [m].
The realization of X can be expressed as the coend over of the simplicial o*
*bject
X with the cosimplicial object [ ]. The most important property of realization
is that it is left adjoint to the functor sending an object Y of C to the simpl*
*icial
object Y [.]
Remark 4.3. Beware of the difference between the realizations of Definition 4.2
and of Section 2.2. Although we treat them as distinct concepts, they really are
versions of the same idea. Since simplicial sets are contravariant functors on *
* , it is
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 23
possible to think of simplicial sets as presheaves. Then the realization of Def*
*inition
4.2 coincides with the realization of Section 2.2 where F is the functor [k] 7!*
* [k].
Remark 4.4. Rather than think of ReX as a coequalizer, we prefer to think of it*
* as
the colimit of the following diagram. The diagram has one object Xn [n] for ea*
*ch
n 0 and one object Xn [m] for each OE : [m] ! [n]. The maps of the diagram
are of two types. The first type is of the form id OE* : Xn [m] ! Xn [n], and
the second type is of the form OE* id : Xn [m] ! Xm [m]. The colimit of this
diagram is the realization ReX of X. Note that the diagram has no non-identity
endomorphisms. This fact makes the analysis of realizations of pro-spaces simpl*
*er.
Realizations present some problems because they are colimits of infinite dia-
grams. Sometimes it will be necessary to use techniques applicable only to coli*
*mits
of finite diagrams. Hence the following definitions are useful.
Definition 4.5. If X is a simplicial object in a simplicial category C, then the
n-truncated realization Ren X of X is the coequalizer of the diagram
`
Xk [m]_____//` X [m].
OE:[m]![k] _____//m nm
m,k n
This is essentially the same construction as ordinary realization except that*
* only
the objects Xm for m n are considered. It can be described as a coend over (*
*n)
of sknX with the n-truncated standard cosimplicial complex (n)[ ].
Remark 4.6. As for realizations, we prefer to think of n-truncated realizations*
* not
as coequalizers but as colimits of diagrams with no non-identity endomorphisms.
See Remark 4.4 for more details.
Like ordinary realization, n-truncated realization is also a left adjoint. Na*
*mely,
it is left adjoint to the functor sending an object Y of C to the simplicial ob*
*ject
that is the nth coskeleton of the simplicial object Y [.]
There is a canonical map RenX ! ReX for every simplicial object X. Of
course this map is not an isomorphism in general, but it is an isomorphism on
low-dimensional simplices as stated in the next proposition.
Proposition 4.7. Let X be a simplicial space. Then the natural map SknRenX !
SknReX is an isomorphism.
Proof.We show that both functors SknRen and SknRe have the same right adjoint.
The right adjoint of SknRe is the functor taking a space Y to the simplicial sp*
*ace
(cosknY ) [ ]. On the other hand, the right adjoint of SknRe nY is the functor
taking a space Y to the nth coskeleton of the simplicial space (cosknY ) [ ]. *
*By
direct computation, this right adjoint is isomorphic to the functor taking a sp*
*ace
Y to the simplicial space (cosknY )Skn [ ], which is is isomorphic to the simpl*
*icial
space (cosknY ) [ ].
Corollary 4.8. Let X be a simplicial space. Then for every i < n, the map
ßiRenX ! ßiReX is an isomorphism.
Proof.When i < n, the ith homotopy group of X only depends on SknX. Hence
Proposition 4.7 gives the result.
24 DANIEL C. ISAKSEN
4.4. Realizations of pro-spaces. We are interested in studying realizations and
k-truncated realizations of pro-spaces. One can define homotopy colimits in any
simplicial model category [13, Ch. 19], so homotopy colimits of pro-spaces can *
*be
formed.
Proposition 4.9. For any simplicial pro-space X, the realization ReX is weakly
equivalent to hocolimX.
Proof.Realization agrees with homotopy colimit up to weak equivalence for Reedy
cofibrant simplicial objects X [13, Thm. 19.6.4]. We show that every simplicial
pro-space is Reedy cofibrant. It is necessary to prove that the map LnX ! Xn is*
* a
cofibration for every n. In order to calculate LnX ! Xn, only the degeneracy ma*
*ps
Xm-1 ! Xm for 1 m n. are relevant. Since these maps form a finite loopless
diagram, we may take a level representation of the pro-spaces X0, X1, . .,.Xn a*
*nd
the degeneracy maps. Since finite colimits of pro-objects can be computed level*
*wise,
it follows that LnX ! Xn can be computed levelwise. For a simplicial space S,
LnS ! Sn is always a cofibration [13, Cor. 16.7.8]. Hence LnX ! Xn is a levelwi*
*se
cofibration, so it is a cofibration of pro-spaces.
Given any pro-space X, apply Sknto each Xs to obtain another pro-space SknX.
Define cosknX similarly. A straightforward computation shows that Sknand coskn
are adjoint functors from pro-spaces to pro-spaces.
Remark 4.10. Given a pro-space X, there are two ways to interpret the symbol
SknX. First, we may think of the pro-space formed by taking the n-skeleton of
each space Xs. From this viewpoint, Skn is left adjoint to coskn. This is what *
*we
always mean by the notation SknX for a pro-space X.
On the other hand, we may think of X as a simplicial pro-set and then apply
SknX in the sense of Definition 3.6 to obtain another simplicial pro-set. Subtl*
*eties
arise because the category of simplicial pro-sets is not equivalent to the cate*
*gory
of pro-spaces [16, Rem. 3.5]. The problem is that the simplicial indexing categ*
*ory
op is not cofinite. However, we have no need for this second construction, so *
*this
is no issue for us.
The following proposition is a direct analogue for pro-spaces of Proposition *
*4.7.
Proposition 4.11. Let X be a simplicial object in the category of pro-spaces. T*
*hen
the natural map SknRenX ! SknReX is an isomorphism.
Proof.The proof is similar to the proof of Proposition 4.7. The right adjoint *
*of
SknRe is the functor taking the pro-space Y to the simplicial pro-space (cosknY*
* ) [ ].
The right adjoint of SknRen is the functor taking the pro-space Y to the simpli*
*cial
pro-space that is the nth coskeleton of the simplicial pro-space (cosknY ) [ ].*
* Each
term of the nth coskeleton is a finite limit of pro-spaces, so they can be calc*
*ulated
levelwise. It follows by direct computation that the right adjoint takes Y to *
*the
simplicial pro-space (cosknY )Skn [ ]. Both functors coskn and (.)Skn [n]on pr*
*o-
spaces are defined levelwise, so the same calculation as in the proof of Propos*
*ition
4.7 tells us that the right adjoint takes Y to the simplicial pro-space (cosknY*
* ) [ ].
Corollary 4.12. Let X be a simplicial object in the category of pointed pro-spa*
*ces.
Then for every i < n, the map ßiRenX ! ßiReX is an isomorphism of pro-groups.
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 25
Proof.When i < n, the ith homotopy pro-group of X only depends on SknX.
Hence Proposition 4.11 gives the result.
5. Hypercover Descent for the Etale Topological Type
We now review the etale topological type functor. We record some of its basic
properties from [9].
For a simplicial scheme X, let EtX be the pro-space defined by the functor
Re O ß from HRR (X) to spaces [9, Defn. 4.4]. Here ß is the functor that takes a
scheme to its set of connected components, and the category of bisimplicial set*
*s is
identified with the category of simplicial spaces in order to interpret the rea*
*lization.
Recall the diagonal functor that takes a bisimplicial set T to its diagonal s*
*im-
plicial set n 7! Tn,n. This functor was used instead of realization in [9, Defn*
*. 4.4].
However, the diagonal of a simplicial space is the same as its realization [27,*
* p. 94],
so our definition is the same.
Given a simplicial scheme map f : X ! Y , rigid pullback as described in Defi-
nition 3.20 gives a functor HRR (Y ) ! HRR (X). If U is a rigid hypercover of Y*
* ,
then there is a canonical rigid hypercover map f*U ! U. These maps induce a
strict map EtX ! EtY . The fact that this map of pro-spaces is strict is critic*
*al for
the proof of Proposition 5.2.
When X is a scheme, define EtX to be Et(cX). In this case, there is a slightly
simpler formula for EtX. It is just the functor ß from HRR (X) to spaces. This
follows from Lemma 3.17.
If X is a pointed and connected scheme, then EtX is a pointed and connected
pro-space [9, Prop. 5.2]. By [15, Cor. 7.5], the pro-groups ßiEtX determine the
homotopy type of the pro-space EtX. The 'etale topological type commutes with
coproducts [9, Prop. 5.2], so the study of arbitrary schemes reduces easily to *
*the
study of pointed and connected schemes by considering one component at a time
and choosing an arbitrary basepoint for each component.
The realization in the definition of the 'etale topological type is an infini*
*te colimit.
This creates problems when trying to analyze the associated pro-spaces. Therefo*
*re,
n-truncated realizations enter into the picture.
Let EtnX be the pro-space given by the functor RenOß from HRR (X) to spaces.
In general, EtnX is not equivalent to EtX, but the next proposition tells us th*
*at
the pro-spaces EtnX are close enough to EtX to determine its homotopy type.
Proposition 5.1. Suppose that X is a pointed and connected scheme. The pro-map
ßiEtnX ! ßiEtX is an isomorphism of pro-groups whenever i < n.
Proof.This follows immediately from Corollary 4.8.
If X is a simplicial scheme, then we can calculate EtXn for each n separately.
These constructions assemble into a simplicial pro-space. This diagram is actua*
*lly
a strict representation.
We would like to compare EtX with hocolimnEtXn, where the homotopy col-
imit is calculated in the category of pro-spaces. In general they are not isomo*
*rphic.
The problem is that the realization in the definition of EtX is constructed lev-
elwise. Since this is an infinite colimit, it is not equal to the realization *
*of the
simplicial pro-space n 7! EtXn. Nevertheless, we shall prove that the natural m*
*ap
hocolimnEtXn ! EtX is a weak equivalence of pro-spaces.
Proposition 5.2. The pro-space EtnX is isomorphic to the pro-space Ren(m 7! EtX*
*m ).
26 DANIEL C. ISAKSEN
Proof.For simplicity of notation, let Y be the pro-space Ren(m 7! EtXm ).
As described in Remarks 4.4 and 4.6, the diagram for calculating Y is a cofin*
*ite
diagram of strict maps of pro-spaces. Moreover, each of the categories HRR (Xn)
has finite limits because of the existence of rigid limits. According to [16, x*
* 3.1],
the index set K for Y is the product category
HRR (X0) x HRR (X1) x . .x.HRR (Xn).
For each V = (V0,, V1,, . .,.Vn,) in K, the space YV is the coequalizer of the
diagram
` R `
ß(Vk, x OE*Vm, ) [m]___////_ß(Vm, ) [m].
OE:[m]![k] m n
m,k n
In this diagram, the upper map is induced by the maps OE* : [m] ! [k] and the
R
projections Vk, x OE*Vm, ! Vk,, while the lower map is induced by the maps
R * *
Vk, x OE Vm, ! OE Vm, ! Vm, .
The functor HRR (X) ! K is cofinal by Proposition 3.34. Therefore, take
HRR (X) to be the indexing category for Y . If V is a rigid hypercover of X,
then YV is the coequalizer of the diagram
` R `
ß(Vk, x OE*Vm, ) [m]___////_ß(Vm, ) [m].
OE:[m]![k] m n
m,k n
For every OE : [m] ! [k], the rigid hypercover map Vk, ! Vm, gives us a map
R
Vk, ! OE*Vm, . Since HRR (Xk) is actually a directed set, this means that Vk, x
OE*Vm, is isomorphic to Vk,. It follows that YV is isomorphic to the coequalize*
*r of
the diagram
`
ß(Vk,) [m]_____//` ß(V ) [m].
OE:[m]![k] _____//m n m,
m,k n
In other words, YV is Ren(m 7! ßVm, ). This is precisely the definition of EtnX.
The next theorem describes the 'etale topological type of a simplicial scheme*
* X
in terms of the 'etale topological types of each scheme Xn and homotopy colimits
of pro-spaces.
Theorem 5.3. For any simplicial scheme X, the natural map
hocolimnEtXn ! EtX
is a weak equivalence in the category of pro-spaces.
Proof.By Proposition 4.9, it suffices to consider the realization of the simpli*
*cial
pro-space n 7! EtXn.
First suppose that X is pointed and connected. By [15, Cor. 7.5], it suffices*
* to
show that the natural map Re(n 7! EtXn) ! EtX induces an isomorphism of pro-
homotopy groups in all dimensions. By Corollary 4.12 and Proposition 5.1, we may
as well consider the map Rem (n 7! EtXn)! EtmX to study the homotopy groups
in dimension less than m. This map induces an isomorphism on pro-homotopy
ETALE REALIZATION ON THE A1-HOMOTOPY THEORY OF SCHEMES 27
groups by Proposition 5.2. Since m was arbitrary, the map ßiRe(n 7! EtXn) !
ßiEtX is a pro-isomorphism for all i.
Now suppose that X is not necessarily connected and pointed. Since Et com-
mutes with coproducts, we reduce to the case when X is connected but not neces-
sarily pointed. However, choosing an arbitrary basepoint for X makes EtX into a
based and connected pro-space.
Now we come to the key ingredient for the proof of Theorem 2.6. The following
result is a kind of hypercover descent theorem for the 'etale topological type.
Theorem 5.4. Let U be a hypercover of a scheme X. Then the natural map
hocolimnEtUn ! EtX
is a weak equivalence of pro-spaces.
Proof.This follows immediately from Theorem 5.3 and [9, Prop. 8.1].
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Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA
E-mail address: isaksen.1@nd.edu