"Brave NewÄ lgebraic Geometry
and global derived moduli spaces
of ring spectra
Bertrand To"en Gabriele Vezzosi
Laboratoire Emile Picard Dipartimento di Matematica
UMR CNRS 5580 Universit`a di Bologna
Universit'e Paul Sabatier, Toulouse Italy
France
September 8, 2003
Abstract
We develop homotopical algebraic geometry ([To-Ve 1, To-Ve 2]) in the s*
*pecial context where
the base symmetric monoidal model category is that of spectra S, i.e. what*
* might be called, after
Waldhausen, brave new algebraic geometry. We discuss various model topolo*
*gies on the model
category of commutative algebras in S, the associated theories of geometri*
*c S-stacks (a geometric
S-stack being an analog of Artin notion of algebraic stack in Algebraic Ge*
*ometry), and finally show
how to define global moduli spaces of associative ring spectra structures *
*as geometric S-stacks.
Key words: Sheaves, stacks, ring spectra, model categories.
MSC-class: 55P43; 14A20; 18G55; 55U40; 18F10.
Contents
1 Introduction *
* 2
2 Brave new sites *
* 4
2.1 The brave new Zariski topology . . . . . . . . . . . . . . . . . . . .*
* . . . . . . . . . . . 4
2.2 The brave new 'etale topology . . . . . . . . . . . . . . . . . . . . *
*. . . . . . . . . . . . 10
2.3 Standard topologies . . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . . . . . . . . . 13
3 S-stacks and geometric S-stacks *
* 13
3.1 Some descent theory . . . . . . . . . . . . . . . . . . . . . . . . . *
*. . . . . . . . . . . . 15
3.2 The S-stack of perfect modules . . . . . . . . . . . . . . . . . . . .*
* . . . . . . . . . . . 17
3.3 Geometric S-stacks . . . . . . . . . . . . . . . . . . . . . . . . . .*
* . . . . . . . . . . . . 17
1
4 Derived moduli spaces in algebraic topology as S-stacks *
* 19
4.1 The brave new group scheme RAut_(M) . . . . . . . . . . . . . . . . .*
* . . . . . . . . . 20
4.2 Moduli of algebra structures . . . . . . . . . . . . . . . . . . . . . *
*. . . . . . . . . . . . 21
1 Introduction
Homotopical Algebraic Geometry is a kind of algebraic geometry where the affine*
* objects are given
by commutative ring-like objects in some homotopy theory (technically speaking,*
* in a symmetric
monoidal model category); these affine objects are then glued together accordin*
*g to an appropri-
ate homotopical modification of a Grothendieck topology, and, more generally, w*
*e allow ourselves to
consider more flexible objects like stacks, in order to deal with appropriate m*
*oduli problems. This
theory is developed in full generality in [To-Ve 1, To-Ve 2](see also [To-Ve 3]*
*). Our motivations for
such a theory came from a variety of sources: first of all, on the algebro-geom*
*etric side, we wanted
to produce a sufficiently functorial language in which the so called Derived Mo*
*duli Spaces foreseen
by Deligne, Drinfel'd and Kontsevich could really be constructed; secondly, on *
*the topological side,
we thought that maybe the many recent results in Brave New Algebra, i.e. in (co*
*mmutative) algebra
over structured ring spectra (in any one of their brave new symmetric monoidal *
*model categories, see
e.g. [Ho-Sh-Sm, EKMM ]), could be pushed to a kind of Brave New Algebraic Geo*
*metry in which
one could take advantage of the possibility of gluing these brave new rings tog*
*ether into an actual
geometric object, much in the same way as commutative algebra is helped (and ge*
*neralized) by the
existence of algebraic geometry. Thirdly, on the motivic side, following a sug*
*gestion of Y. Manin,
we wished to have a sufficiently general theory in order to study algebraic geo*
*metry over the recent
model categories of motives for smooth schemes over a field ([Hu , Ja, Sp]).
The purpose of this paper is to present the first steps in the second type *
*of applications mentioned
above, i.e. a specialization of the general framework of homotopical algebraic *
*geometry to the context
of stable homotopy theory. Our category S - Affof brave new affine objects will*
* therefore be defined
as the the opposite model category of the category of commutatve rings in the c*
*ategory S of symmetric
spectra ([Ho-Sh-Sm]).
We first define and study various model topologies defined on S - Aff. They*
* are all extensions,
to different extents, of the usual Grothendieck topologies defined on the categ*
*ory of (affine) schemes,
like the Zariski and 'etale ones.
With any of these model topologies ø at hand, we define and give the basic *
*properties of the cor-
responding model category of S-stacks, understood in the broadest sense as non *
*necessarily truncated
presheaves of simplicial sets on S - Affsatisfying a homotopical descent (i.e. *
* sheaf-like) condition
with respect to ø-(hyper)covers. As in algebraic geometry one finds useful to s*
*tudy those stacks which
arise as quotients by smooth groupoids (these are called Artin algebraic stacks*
*), we also define a brave
new analog of these and call them geometric S-stacks, to emphasize that such S-*
*stacks host a rich
geometry very close to the geometric intuition learned in algebraic geometry. I*
*n particular, given a
geometric S-stack F , it makes sense to speak about quasi-coherent and perfect *
*modules over F , about
the K-theory of F , etc.; various properties of morphisms (e.g. smooth, 'etale,*
* proper, etc.) between
geometric S-stacks can likewise be defined.
Stacks were introduced in algebraic geometry mainly to study moduli problem*
*s of various sorts;
they provide actual geometric objects (rather than sets of isomorphisms classes*
* or coarse moduli
schemes) which store all the fine details of the classification problem and on *
*which a geometry very
similar to that of algebraic varieties or schemes can be developed, the two asp*
*ects having a fruitful
2
interplay. In a similar vein, in our brave new context, we give one example of *
*a moduli problem arising
in algebraic topology (the classification of A1 -ring spectra structures on a g*
*iven spectrum M) that
can be studied geometrically through the geometric S-stack RAss_M it represents*
*. We wish to empha-
size that instead of a discrete homotopy type (like the ones studied, for diffe*
*rent moduli problems,
in [Re, B-D-G, G-H]), we get a full geometric object on which a lot of interest*
*ing geometry can be
performed. The geometricity of the S-stack RAss_M is actually the main theorem *
*of this paper (see
theorem 4.2.1). We think that this approach can be extended to other, more inte*
*rested and involved,
moduli problems algebraic topologists are interested in, and perhaps this riche*
*r geometry could be of
some help in answering, or at least in formulating in a clearer way, some of th*
*e deep questions raised
by the recent progress in stable homotopy theory (see [G ]).
Acknowledgments. We wish to thank Haynes Miller for his invitation to spea*
*k at the INI
Workshop on Elliptic Cohomology and Higher Chromatic Phenomena (Cambridge UK, D*
*ecember
2003) and Bill Dwyer for his encouraging comments. We would also like to thank *
*Michael Mandell,
Haynes Miller, Peter May, John Rognes, Stefan Schwede and Neil Strickland for e*
*xtremely helpful
discussions and suggestions.
Notations. To fix ideas, we will work in the category S := Sp of symmetr*
*ic spectra (see
[Ho-Sh-Sm]), but all the constructions of this paper will also work, possibly w*
*ith minor variations
(see [Sch]), for other equivalent theories (e.g for the category of S-modules o*
*f [EKMM ]). We will
consider S as a symmetric monoidal simplicial model category (for the smash pro*
*duct - ^ -) with
the Shipley-Smith S-model structure (see [Shi, Thm. 2.4]).
We define S - Alg as the category of (associative and unital) commutative m*
*onoids objects in
S, endowed with the S-alg model structure ([Shi, Thm. 3.2]); we will simply cal*
*l them commutative
S-algebras instead of the more correct but longer, commutative symmetric ring s*
*pectra. For any
commutative S-algebra A, we will denote by A - Alg the under-category A=S - Alg*
*, whose objects
will be called commutative A-algebras. Finally, if A is a commutative S-algebra*
*, A - Mod will be the
category of A-modules with the A-model structure ([Shi, Thm. 2.6 (2)]). This mo*
*del category is also
a symmetric monoidal model category for the smash product - ^A - over A.
For a morphism of commutative S-algebras, f : A -! B one has a Quillen adju*
*nction
f* : A - Mod -! B - Mod A - Mod - B - Mod : f*,
where f*(-) := - ^A B is the base change functor. We will denote by
Lf* : Ho (A - Mod ) -! Ho (B - Mod ) Ho (A - Mod ) - Ho (B - Mod ) :*
* Rf*
the induced derived adjunction on the homotopy categories.
Our references for model category theory are [Hi, Ho]. For a model category*
* M with equivalences
W , the set of morphisms in the homotopy category Ho (M) := W -1M will be denot*
*ed by [-, -]M ,
or simply by [-, -] if the context is clear. The (homotopy) mapping spaces in M*
* will be denotedby
Map M(-, -). When M is a simplicial model category, the simplicial Hom's (resp*
*. derived simplicial
Hom's) will be denoted by Hom__M(resp. RHom__M), or simply by Hom__(resp. RHom_*
*_) if the context
is clear. Recall that in this case one can compute Map M(-, -) as RHom__M(-, -).
Finally, for a model category M and an object x 2 M we will often use the c*
*oma model categories
x=M and M=x. When the model category M is not left proper (resp. is not right p*
*roper) we will
always assume that x has been replaced by a cofibrant (resp. fibrant) model bef*
*ore considering x=M
(resp. M=x). More generally, we will not always mention fibrant and cofibrant*
* replacements and
suppose implicitly that all our objects are fibrant and/or cofibrant when requi*
*red.
3
Since we wish to concentrate on applications to stable homotopy theory, som*
*e general construc-
tions and details about homotopical algebraic geometry will be omitted by refer*
*ring to [To-Ve 1]. For
a few of the results presented we will only give here sketchy proofs; full proo*
*fs will appear in [To-Ve 2].
2 Brave new sites
In this section we present two model topologies on the (opposite) category of c*
*ommutative S-algebras.
They are brave new analogs of the Zariski and 'etale topologies defined on the *
*category of usual com-
mutative rings and will allow us to define the brave new Zariski and 'etale sit*
*es.
We denote by S - Affthe opposite model category of S - Alg.
If M is a model category we say that an object x in M is finitely presented*
* if, for any filtered
direct system of objects {zi}i2Jin M, the natural map
colimiMap M(x, zi) -! MapM (x, colimizi)
is an isomorphism in the homotopy category of simplicial sets.
Definition 2.0.1A morphism A ! B of commutative S-algebras is finitely presente*
*d if it is a finitely
presented object in the model under-category A=(S - Alg) = A - Alg. An A-module*
* E is finitely pre-
sented or perfect if it is a finitely presented object in the model category A *
*- Mod .
2.1 The brave new Zariski topology
Definition 2.1.1 o A morphism f : A -! B in S - Algis called a formal Zariski*
* open immer-
sion if the induced functor Rf* : Ho (B - Mod ) -! Ho (A - Mod ) is fully *
*faithful.
oA morphism f : A -! B is a Zariski open immersion if S - Algis it is a for*
*mal Zariski open
immersion and of finite presentation (as a morphism of commutative S-algeb*
*ras).
oA family {fi: A -! Ai}i2Iof morphisms in S - Algis called a (formal) Zaris*
*ki open covering
if it satisfies the following two conditions.
- Each morphism A -! Aiis a (formal) Zariski open immersion.
- There exist a finite subset J I such that the family of inverse ima*
*ge functors
{Lf*j: Ho (A - Mod ) -! Ho (Aj- Mod )}j2J
is conservative (i.e. a morphism in Ho (A - Mod ) is an isomorphism *
*if and only if its
images by all the Lf*j's are isomorphisms).
Example 2.1.2 If A 2 S - Algand E is an A-module such that the associated Bous*
*field localization
LE is smashing (i.e. the natural transformation LE(-) ! LEA ^LA(-) is an isomo*
*rphism), then
A ! LEA (which is a morphism of commutative S-algebras by e.g. [EKMM , xVIII.2*
*]) is a formal
Zariski open immersion. This follows immdiately from the fact that Ho (LEA - Mo*
*d) is equivalent to
the subcategory of Ho (A - Mod) consisting of LE-local objects, by [Wo ].
4
It is easy to check that (formal) Zariski open covering families define a m*
*odel topology in the sense
of [To-Ve 1, x4.3] on the model category S - Aff. For the reader's convenience *
*we recall what this
means in the following lemma.
Lemma 2.1.3 oIf A -! B is an equivalence of commutative S-algebras then th*
*e one element
family {A -! B} is a (formal) Zariski open covering.
oLet {A -! Ai}i2Ibe a (formal) Zariski open covering of S-algebras and A -!*
* B a morphism.
Then, the family of homotopy push-outs {B -! B ^LAAi}i2Iis also a (formal)*
* Zariski open
covering.
oLet {A -! Ai}i2Ibe a (formal) Zariski open covering of S-algebras, and for*
* any i 2 I let
{Ai -! Aij}j2Jibe a (formal) Zariski open covering of S-algebras. Then, t*
*he total family
{A -! Aij}i2I,j2Jiis again a (formal) Zariski open covering.
Proof: Left as an exercise. *
* 2
Therefore, by definition, Lemma 2.1.3 shows that (formal) Zariski open cove*
*rings define a model
topology on the model category S - Aff and so, as proved in [To-Ve 1, Prop. 4.*
*3.5], induce a
Grothendieck topology on the homotopy category Ho (S - Alg). This model topolog*
*y is called the
brave new (formal) Zariski topology, and endows S - Affwith the structure of a *
*model site in the
sense of [To-Ve 1, x4]. This model site, denoted by (S - Aff, Zar) for the brav*
*e new Zariski topology,
and (S - Aff, fZar) for the brave new formal Zariski topology. They will be ca*
*lled the brave new
Zariski site and the brave new formal Zariski site.
Let Alg be the category of (associative and unital) commutative rings. Let *
*us recall the existence
of the Eilenberg-Mac Lane functor
H : Alg -! S - Alg,
sending a commutative ring R to the commutative S-algebra HR such that ß0(HR) =*
* R and ßi(HR) =
0 for any i 6= 0. This functor is homotopically fully faithful and the followin*
*g lemma shows that our
brave new Zariski topology does generalize the usual Zariski topology.
Lemma 2.1.4 1.Let R -! R0 be a morphism of commutative rings. The induced *
*morphism
HR -! HR0is a Zariski open immersion of commutative S-algebras (in the sen*
*se of Definition
2.1.1) if and only if the morphism Spec R0-! Spec R is an open immersion o*
*f schemes.
2.A family of morphisms of commutative rings, {R -! R0i}i2I, induces a Zaris*
*ki covering family
of commutative S-algebras {HR -! HR0i}i2I(in the sense of Definition 2.1.1*
*) if and only if
the family {Spec Ri-! Spec R}i2Iis a Zariski open covering of schemes.
Proof: Let us start with the general situation of a morphism f : A -! B of*
* commutative S-
algebras such that the induced functor Rf* : Ho (B - Mod ) -! Ho (A - Mod ) is *
*fully faithful. Let
L = Rf* O Lf*, which comes with a natural transformation Id -! L. Then, the ess*
*ential image of
Rf* consist of objects M in Ho (A - Mod ) such that the localization morphism M*
* -! LM is an
isomorphism.
The Quillen adjunction (f*, f*) extends to a Quillen adjunction on the cate*
*gory of commutative
algebras
f* : A - Alg -! B - Alg A - Alg - B - Alg : f*,
also with the property that Rf* : Ho (B - Alg) -! Ho (A - Alg) is fully faithfu*
*l. Furthermore, the
essential image of this last functor consist of all objects C 2 Ho (A-Alg ) suc*
*h that the underlying A-
module of C satisfies C ' LC (i.e. the underlying A-module of C lives in the im*
*age of Ho(B -Mod )).
5
From these observations, we deduce that for any commutative A-algebra C, th*
*e mapping space
RHom______A-Alg(B, C) is either empty or contractible; it is non-empty if and o*
*nly if the underlying A-
module of C belongs to the essential image of Rf*.
To prove (1), let us first suppose that f : Spec R0- ! Spec R is an open im*
*mersion of schemes.
Then, clearly the induced functor on the derived categories f* : D(R0) -! D(R) *
*is fully faithful. As
there are natural equivalences ([EKMM , IV Thm. 2.4])
Ho(HR - Mod ) ' D(R) Ho (HR0- Mod ) ' D(R0)
this implies that the functor Rf* : Ho (HR0- Mod ) -! Ho (HR - Mod ) is also fu*
*lly faithful. It
only remains to show that HR -! HR0is finitely presented in the sense of Defini*
*tion 2.0.1.
We will first assume that R0 = Rf for some element f 2 R. The essential im*
*age of Rf* :
Ho (HR0- Mod ) -! Ho (HR - Mod ) then consists of all objects E 2 Ho (HR - Mod*
* ) ' D(R)
such that f acts by isomorphisms on the cohomology R-module H*(E). By what we *
*have seen at
the beginning of the proof, this implies that for any commutative HR-algebra C *
*the mapping space
RHom______HR-Alg(HR0, C) is contractible if f becomes invertible in ß0(C), and *
*empty otherwise. From
this one easily deduces that RHom__HR-Alg(HR0, -) commutes with filtered colimi*
*ts, or in other words
that HR0is a finitely presented HR-algebra in the sense of Definition 2.0.1.
In the general case, one can write Spec R0 as a finite union of schemes of *
*the form Spec Rf for
some elements f 2 R. A bit of descent theory (see x2.1) then allows us to reduc*
*e to the case where
R0= Rf and conclude.
Let us now assume that HR -! HR0is a Zariski open immersion of commutative *
*S-algebras. By
adjunction (between H and ß0 restricted on connective S-algebras) one sees easi*
*ly that R -! R0is a
finitely presented morphism of commutative rings.
The induced functor on (unbounded) derived categories
f* : D(R0) ' Ho (HR0- Mod ) -! D(R) ' Ho (HR - Mod )
is fully faithful. Through the Dold-Kan correspondence, this implies that the Q*
*uillen adjunction on
the model category of simplicial modules (see [G-J])
f* : sR - Mod -! sR0- Mod sR - Mod - sR0- Mod : f*
is such that Lf* O f* ' Id. Let sR - Alg and sR0- Alg be the categories of simp*
*licial commuta-
tive R-algebras and simplicial commutative R0-algebras, endowed with their natu*
*ral model structures
(equivalences are and fibration are detected in the category of simplicial modu*
*les). Then, the Quillen
adjunction
f* : sR0- Alg -! sR - Alg sR0- Alg - sR - Alg : f*
also satisfies Lf* O f* ' Id, as this is true on the level on simplicial module*
*s. In particular, for any
simplicial R0-module M, the space of derived derivations
LDerR(R0, M) := RHom__sR-Alg=R0(R0, R0 M) ' RHom__sR0-Alg=R0(R0, R0 M*
*) ' *
is acyclic (here R0 M is the simplicial R0-algebra which is the trivial extens*
*ion of R0by M). As
a consequence one sees that Illusie's contangent complex LR0=R is acyclic, whic*
*h implies that the
morphism R -! R0is an 'etale morphism of rings.
Finally, using that the functor on the category of modules R0-Mod -! R-Mod*
* is fully faithful,
one sees that Spec R0-! Spec R is a monomorphism of schemes. Therefore, the mor*
*phism of schemes
6
Spec R0-! Spec R is an 'etale monomorphism, and so is an open immersion by [EGA*
*-IV , Thm. 17.9.1].
Finally, point (2) is clear if one knows (1) and that Ho (HR - Mod ) ' D(R)*
*. 2
Remark 2.1.5 The argument at the beginning of the proof of Lemma 2.1.4 shows t*
*hat if f : A ! B
is a Zariski open immersion, the functor L(f) := Rf*Lf* is a localization funct*
*or on the homotopy
category of A-modules in the sense of [HPS , Def. 3.1.1]. And it is also clear *
*by definition that L(f)
is also smashing ([HPS , Def. 3.3.2]). Let us call a localization functor L on *
*Ho (A - Mod) a formal
Zariski localization functor over A if L ' L(f) for some formal Zariski open im*
*mersion f. Let us also
say that a localization functor L on Ho (A - Mod) is a smashing algebra Bousfie*
*ld localization over A
if L ' LB for some A-algebra B such that LB is smashing (over A). Then it is ea*
*sy to verify that in
the set of equivalence classes of localization functors on Ho (A - Mod), the su*
*bset consisting of formal
Zariski localization functors over A coincides with the subset consisting of sm*
*ashing algebra Bousfield
localizations over A. In fact, if f : A ! B is a Zariski open immersion, LB de*
*notes the Bousfield
localization with respect to the A-module B, and `B=A : A ! LBA the correspondi*
*ng morphism
of commutative A-algebras, we have L(f) ' LB ' L(`B=A) because all three locali*
*zations have the
same category of acyclics. Viceversa, if LC is a smashing algebra Bousfield loc*
*alization over A, and
`C=A : A ! LCA is the corresponding morphism of commutative A-algebras, one has*
* LB ' L(`C=A).
Let Affbe the opposite category of commutative rings, and (Aff, Zar) the bi*
*g Zariski site. The site
(Aff, Zar) can also be considered as a model site (for the trivial model struct*
*ure on Aff). Lemma 2.1.4
implies in particular that the functor H : Aff- ! S - Affinduces a continuous m*
*orphism of model
sites ([To-Ve 1, Def. 4.8.4]). In this way, the site (Aff, Zar) becomes a sub-m*
*odel site of (S - Aff, Zar).
To finish with the Zariski topology we will now describe a general procedur*
*e in order to construct
interesting open Zariski immersions of commutative S-algebras using the thechni*
*ques of Bousfield
localization for model categories.
Let A be a commutative S-algebra and M be a A-module. We will assume that M*
* is a perfect
A-module (in the sense of Definition 2.0.1), or equivalently that it is a stron*
*gly dualizable object in the
monoidal category Ho (A - Mod ). Perfect A-modules are exactly the retract of f*
*inite cell A-modules,
see [EKMM , Thm. III-7.9]). Let M[n] = Sn L M the n-th suspension A-module of*
* M (here n 2 Z).
We denote by D(M[n]) the derived dual of M[n], defined as the derived inter*
*nal Hom's of A-
modules
D(M[n]) := RHOM A-Mod(M[n], A).
Consider now the (derived) free commutative A-algebra over D(M[n]), LFA(D(M[n])*
*), characterized
by the usual adjunction
[LFA(D(M[n])), -]A-Alg ' [D(M[n]), -]A-Mod .
The model category A - Alg is a combinatorial and cellular model category, and *
*therefore one can
apply the localization techniques (see e.g. [Hi, Sm ]) in order to invert the *
*natural augmentations
FA(D(M[n])) -! A for all n 2 Z. One checks easily, using that M is strongly du*
*alizable, that
the local objects for this localization are the commutative A-algebras B such t*
*hat M ^LAB ' 0 in
Ho (B - Mod ). The local model of A for this localization will be denoted by A*
*M . By definition, it
is characterized by the following universal property: for any commutative A-alg*
*ebra B, the mapping
space RHom__A-Alg(AM , B) is contractible if B ^LAM ' 0 and empty otherwise. In*
* other words, for
any commutative S-algebra B the natural morphism
RHom__S-Alg(AM , B) -! RHom__S-Alg(A, B)
7
is equivalent to an inclusion of connected components and its image consists of*
* morphisms A -! B
in Ho (S - Alg) such that B ^LAM ' 0.
Lemma 2.1.6 With the above notations, the morphism A -! AM is a Zariski open*
* immersion.
Proof: Let us start by showing that AM is a finitely presented commutative *
*A-algebra.
Let {Bi}i2Ibe a filtered system of commutative A-algebras and B = colimiBi.*
* We assume that
B ^LAM ' 0, and we need to prove that there exists an i 2 I such that Bi^LAM ' *
*0.
By assumption, the two points Id and 0 are the same in ß0(REnd_B-Mod (M ^LA*
*B)). But, as M
is a perfect A-module one has
ß0(REnd_B-Mod (M ^LAB)) ' colimi2Iß0(REnd_Bi-Mod(M ^LABi)).
This implies that there is some index i 2 I such that Id and 0 are homotopic in*
* REnd_Bi-Mod(M^LABi),
and therefore that M ^LABiis contractible.
It remains to prove that the induced functor Ho (AM - Mod ) -! Ho (A - Mod *
*) is fully faithful.
To see this, one first notice that by definition the functor Ho (AM - Alg) -! H*
*o (A - Alg) is fully
faithful, and therefore so is Ho (AM - Alg=AM ) -! Ho (A - Alg=AM ). For any tw*
*o AM -modules
N and P , one consider the trivial extensions AM _ N and AM _ P of AM by N and*
* P (these are
AM -augmented commutative AM -algebras). Then, one has a natural equivalence
RHom__AM -Alg=AM(AM _ N, AM _ P ) ' RHom__AM -Mod(N, P ).
Furthermore one has a natural fiber sequence
RHom____A-Mod(N, P_)__//RHom__A-Alg=AM(AM _ N, AM _ P_)_//_RHom__A-Alg=AM(AM ,*
* AM _.P )
But, as RHom__A-Alg=AM(AM , AM _ P ) ' RHom__AM -Alg=AM(AM , AM _ P ) ' * this *
*shows that the
natural morphism
RHom__AM -Mod(N, P ) -! RHom__A-Mod(N, P ),
is an equivalence and therefore that Ho (AM - Alg) -! Ho (A - Alg) is fully fai*
*thful. 2
An important property of the localization A -! AM is the following fact.
Lemma 2.1.7 Let A be a commutative S-algebra, and M be a perfect A-module. Th*
*en the essential
image of the fully faithful functor
Ho (AM - Mod ) -! Ho (A - Mod )
consists of all A-modules N such that M ^LAN ' D(M) ^LAN ' 0.
Note that since M is perfect, then for any A-module N, M ^LAN ' 0 iff D(M) *
*^LAN ' 0, so the
two conditions in the lemma are actually one. Moreover, AM ' AD(M) in Ho (A - A*
*lg).
Proof: As every AM -module N can be constructed by homotopy colimits of fre*
*e AM -modules and
- ^LAM commutes with homotopy colimits, it is clear that AM ^LAM ' 0 implies N *
*^LAM ' 0. Since
AM ^LAD(M) ' D(AM ^LAM) ' 0 (here the second derived dual is in the category of*
* AM -modules),
the same argument shows that N ^LAD(M) ' 0.
Conversely, let N be an A-module such that N ^LAM ' N ^LAD(M) ' 0. By defi*
*nition, the
commutative A-algebra A -! AM is obtained as a local model of A ! A when one in*
*verts the set of
8
morphisms LFA(D(M[n])) -! A, for any n 2 Z. It is well known (see e.g. [Hi, x4]*
*) that such a local
model can be obtained by a transfinite composition of homotopy push-outs of the*
* form
Aff___________________//_Aff+1OOOO
| |
| |
| |
@ p L LFA(D(M[n])) ____//_ p L LFA(D(M[n]))
in the category of A-algebras. From this description, and the fact that - ^LAM *
*commutes with ho-
motopy colimits, one sees that the adjunction morphism N -! N ^LAAM is an equi*
*valence because
by assumption on N, the natural morphism N ' N ^LAA -! N ^LALFA(D(M[n])) is an *
*equivalence. 2
Lemma 2.1.7 allows us to interpret geometrically AM as the open complement*
* of the support of
the A-module M. Lemma 2.1.7 also has a converse whose proof is left as an exerc*
*ise.
Lemma 2.1.8 Let f : A -! B be a morphism of commutative S-algebras and M be a*
* perfect A-
module. We suppose that the functor Rf* : Ho (B -Mod ) -! Ho (A-Mod ) is full*
*y faithful and that
its essential image consists of all A-modules N such that N ^LAM ' 0. Then, the*
* two commutative
A-algebras B and AM are equivalent (i.e. isomorphic in Ho (A - Alg)).
Remark 2.1.9 One should be careful that even if the Eilenberg-Mac Lane functor*
* H embeds (Aff, Zar)
in (S - Aff, Zar) as model sites, there exist commutative rings R and Zariski o*
*pen coverings HR -! B
in S - Affsuch that B is not of the form HR0for some commutative R-algebra R0. *
*One example
is given by taking R to be C[X, Y ], and considering the localized commutative *
*HR-algebra (HR)M
(in the sense above), where M is the perfect R-module R=(X, Y ) ' C. If (HR)M *
*were of the form
HR0for a Zariski open immersion Spec R0-! Spec R, then for any other commutativ*
*e ring R00, the
set of scheme morphisms Hom(Spec R00, Spec R0) would be the subset of Hom(Spec *
*R00, A2) consisting
of morphisms factoring through A2 - {0}. This would mean that Spec R0' A2 - {0}*
*, which is not
possible as A2-{0} is a not an affine scheme. This example is of course the sam*
*e as the example given
in [To, x2.2] of a 0-truncated affine stack which is not an affine scheme. Thes*
*e kind of example shows
that there are much more affine objects in homotopical algebraic geometry than *
*in usual algebraic
geometry.
Remark 2.1.10
1.Note that Lemma 2.1.7 shows that the localization process (A, M) /o////ooA*
*Mis in some sense
ö rthogonal" to the usual Bousfield localization process (A, M) /////oooL*
*MiAn that the local
objects for the former are exactly the acyclic objects for the latter. To*
* state everything in
terms of Bousfield localizations, this says that LAM -local objects are ex*
*actly LM -acyclic objects
(compare with Remark 2.1.5). Note that however, while the Bousfield local*
*ization is always
defined for any A-module M, the commutative A-algebra AM probably does not*
* exist unless M
is perfect.
2.Let Sp be the p-local sphere. If f : Sp ! B is any formal Zariski open imm*
*ersion then L :=
Rf*Lf* is clearly a smashing localization functor in the sense of [HPS , x*
*3]. Its category C
of perfect1 acyclics (i.e. perfect objects X in Ho (Sp - Mod ) such that *
*LX is null) is then
a localizing thick subcategory of the homotopy category Ho (Sp - Mod perf)*
* of the category of
perfect Sp-modules, and therefore by [H-S] it is equivalent to the categor*
*y Cn of perfect E(n)-
acyclics, for some 0 n < 1, where E(n) is the n-th Johnson-Wilson Sp-mod*
*ule (see e.g. [Rav]);
1____________________________________
The word finite instead of perfect would be more customary in this setting.
9
in other words L and Ln := LE(n)are both smashing localization functors on*
* Ho (Sp - Mod )
having the same subcategory of finite acyclics. Therefore, if we assume (*
*one of the form of)
the Telescope conjecture (see [Mil]), we get that Ln and L have equivalent*
* categories of acyclics
and so have equivalent categories of local objects. But the category of l*
*ocal objects for L is
equivalent to the category Ho (B - Mod ) (since Rf* is fully faithful by h*
*ypothesis) and the
category of local objects for Ln is equivalent to the category Ho ((LnSp) *
*- Mod ), by [Wo ] since
Ln is smashing. This easily implies that the two commutative Sp-algebras *
*B and LnSp are
equivalent (i.e. isomorphic in Ho (Sp- Alg)).
In conclusion, one sees that if the Telescope conjecture is true, then, up*
* to equivalence of Sp-
algebras, the only (non-trivial) formal Zariski open immersions for Sp are*
* given by the family
U := {Sp ! LnSp}0 n<1.
This example shows that the formal Zariski topology might be better suited*
* in certain contexts
than the Zariski topology itself (e.g. it is not clear that there exists a*
*ny non-trivial Zariski open
immersion of Sp, i.e. that the morphisms of commutative S-algebras Sp -! L*
*nSp are of finite
presentation). Note however that the family U is not a formal Zariski cov*
*ering according to
Definition 2.1.1 because the family of base-change functors
n o
(-) ^LSpLnSp : Ho (Sp- Mod) -! Ho (LnSp- Mod)
0 n<1
is not conservative; in fact, as Neil Strickland pointed out to us, the Br*
*own-Comenetz dual I of
Sp is a non-perfect non-trivial Sp-module which is nonetheless Ln-acyclic *
*for any n. However, it
is true that the family of base-changes above is conservative when restric*
*ted to the (homotopy)
categories of perfect modules. Therefore, one could modify the second cov*
*ering condition in
Definition 2.1.1, by only requiring the property of being conservative on *
*the subcategories of
perfect modules and relaxing the finiteness of J; let us call this modifie*
*d covering condition
formal Zariski covering-on-finites condition. Then, U is a formal Zariski*
* covering-on-finites
family and indeed the unique one, up to equivalences of Sp-algebras, if th*
*e Telescope conjecture
holds.
3.The previous example also shows that the commutative S-algebras LnSp are l*
*ocal for the formal
Zariski topology (again assuming the Telescope conjecture). Indeed, for an*
*y formal Zariski open
covering {LnSp -! Bi}i2Ithere is an i such that LnSp -! Biis an equivalenc*
*e of commutative
S-algebras.
2.2 The brave new 'etale topology
Notions of 'etale morphisms of commutative S-algebras has been studied by sever*
*al authors ([Ro1,
MC-Min ]). In this paragraph we present the definition that appeared in [To-Ve*
* 1] and was used there
in order to define the 'etale K-theory of commutative S-algebras.
We refer to [Ba ] for the notions of topological cotangent spectrum and of *
*topological Andr'e-Quillen
cohomology relative to a morphism A ! B of commutative S-algebras, except for s*
*lightly different
notations. We denote by L B=A 2 Ho (B - Mod ) the topological cotangent spectru*
*m (denoted as
B=A in [Ba ]) and, for any B-module M, by
LDerA(B, M) := RHom__A-Alg=B(B, B _ M)
the derived space of topological derivations from B to M (B _ M being the trivi*
*al extension of B by
M). Note that there is an isomorphism LDerA(B, M) ' RHom__B-Mod(L B=A, M), natu*
*ral in M.
10
Definition 2.2.1 o Let f : A -! B be a morphism of commutative S-algebras.
- The morphism f is called formally 'etale if L B=A ' 0.
- The morphism f is called 'etale if it is formally 'etale and of finit*
*e presentation (as a mor-
phism of commutative S-algebras).
oA family of morphisms {fi : A -! Ai}i2Iin S - Alg is called a (formal) 'et*
*ale covering if it
satisfies the following two conditions.
- Each morphism A -! Aiis (formally) 'etale.
- There exists a finite subset J I such that the family of inverse im*
*age functors
{Lf*j: Ho (A - Mod ) -! Ho (Aj- Mod )}j2J
is conservative (i.e. a morphism in Ho (A - Mod ) is an isomorphism *
*if and only if its
images by all the Lf*j's are isomorphisms).
As shown in [To-Ve 1, x5.2], (formal) 'etale covering families are stable b*
*y equivalences, composi-
tions and homotopy push-outs, and therefore define a model topology on the mode*
*l category S - Aff.
Therefore one gets two model topologies called the brave new 'etale topology an*
*d the brave new formal
'etale topology. The corresponding model sites will be denoted by (S - Aff, 'et*
*) and (S - Aff, f'et), and
will be called the brave new 'etale site and the brave new formal 'etale site.
As for the brave new Zariski topology one proves that the brave new 'etale *
*topology is a general-
ization of the usual 'etale topology.
Lemma 2.2.2 1.Let R -! R0 be a morphism of commutative rings. The induced *
*morphism
HR -! HR0is an 'etale morphism of commutative S-algebras (in the sense of *
*Definition 2.2.1)
if and only if the morphism Spec R0-! Spec R is an 'etale morphism of sche*
*mes.
2.A family of morphisms of commutative rings, {R -! R0i}i2I, induces an 'eta*
*le covering family
of commutative S-algebras {HR -! HR0i}i2I(in the sense of Definition 2.2.1*
*) if and only if
the family {Spec Ri-! Spec R}i2Iis an 'etale covering of schemes.
Proof: This is proved in [To-Ve 1, x5.2]. *
* 2
Let Aff be the opposite category of commutative rings, and (Aff, 'et) the b*
*ig 'etale site. The site
(Aff, 'et) can also be considered as a model site (for the trivial model struct*
*ure on Aff). Lemma 2.2.2
shows in particular that the Eilenberg-Mac Lane functor H : Aff -! S - Affinduc*
*es a continuous
morphism of model sites ([To-Ve 1, x4.8]). In this way, the site (Aff, 'et) bec*
*omes a sub-model site of
(S - Aff, 'et).
Another important fact is that the brave new 'etale topology is finer than *
*the brave new Zariski
topology.
Lemma 2.2.3 1.Any formal Zariski open immersion of commutative S-algebras i*
*s a formally
'etale morphism.
2.Any Zariksi open immersion of commutative S-algebras is an 'etale morphism.
3.Any (formal) Zariski open covering of a commutative S-algebra is a (formal*
*) 'etale covering.
11
Proof: Only (1) requires a proof, and the proof will be similar to the one*
* of Lemma 2.1.4 (2).
Let f : A -! B be a formal Zariski open immersion of commutative S-algebras. A*
*s the functor
Rf* : Ho (B - Mod ) -! Ho (A - Mod ) is a full embedding so is the induced func*
*tor Rf* : Ho (B -
Alg ) -! Ho (A - Alg). By definition of topological derivations one has for an*
*y B-module M,
LDerA(B, M) = RHom__A-Alg=B(B, B _ M). This and the fact that Rf* is fully fait*
*hful imply that
LDerA(B, M) = RHom__A-Alg=B(B, B _ M) ' RHom__B-Alg=B(B, B _ M) ' *,
and therefore that L B=A ' 0. *
* 2
Lemma 2.2.3 implies that the identity functor of S - Affdefines a continuou*
*s morphism between
model sites
(S - Aff, Zar) -! (S - Aff, 'et),
which is a base change functor from the brave new Zariski site to the brave new*
* 'etale site. The same
is true for the formal versions of these sites.
To finish this part, we would like to mention a stronger version of the bra*
*ve new 'etale topology,
called the thh-'etale topology, which is sometimes more convenient to deal with.
Definition 2.2.4 o Let f : A -! B be a morphism of commutative S-algebras.
- The morphism f is called formally thh-'etale if for any commutative A*
*-algebra C the map-
ping space RHom__A-Alg(B, C) is 0-truncated (i.e. equivalent to a dis*
*crete space).
- The morphism f is called thh-'etale if it is formally thh-'etale and *
*of finite presentation (as
a morphism of commutative S-algebras).
oA family of morphisms {fi: A -! Ai}i2Iin S - Algis called a (formal) thh-'*
*etale covering if
it satisfies the following two conditions.
- Each morphism A -! Aiis (formally) thh-'etale.
- There exists a finite subset J I such that the family of inverse im*
*age functors
{Lf*j: Ho (A - Mod ) -! Ho (Aj- Mod )}j2J
is conservative (i.e. a morphism in Ho (A - Mod ) is an isomorphism *
*if and only if its
images by all Lf*jare isomorphisms).
It is easy to check that (formal) thh-'etale coverings define a model topol*
*ogy on the model cate-
gory S - Aff, call the (formal) thh-'etale topology. The model category S - Aff*
*together with these
topologies will be called the brave new thh-'etale site and the brave new forma*
*l thh-'etale site, denoted
by (S - Aff, thh-'et) and (S - Aff, fthh-'et), respectively. An equivalent way *
*of stating the formal thh-
'etaleness condition for A ! B is to say that the natural map B ! S1 LB in Ho(A*
*-Alg ) is an isomor-
phism. By [MSV ], this is therefore equivalent to require that the canonical ma*
*p B ! THH (B=A, B) is
an isomorphism in Ho(A-Alg ), where THH denotes the topological Hochschild coho*
*mology spectrum
(see e.g. [EKMM , xIX]). This explains the name of this topology and, since as*
* observed in [MC-Min ]
the Goodwillie derivative of THH is the suspension of the topological Andr'e-Q*
*uillen spectrum TAQ
(where, for any B-module M, TAQ (B=A; M) is defined as the derived internal Hom*
* from L B=A to
M in the model category of B-modules) also shows that (formal) thh-'etale morph*
*isms are (formal)
'etale morphisms. Therefore the identity functor induces continuous morphisms o*
*f model sites
(S - Aff, thh-'et) -! (S - Aff, 'et) (S - Aff, fthh-'et) -! (S - *
*Aff, f'et).
We refer to [MC-Min ] for more details on the notion of thh-'etale morphisms.
12
2.3 Standard topologies
Standard model topologies on S - Affare obvious extensions of usual Grothendiec*
*k topologies on
affine schemes. They are defined in the following way.
Let ø be one of the usual Grothendieck topologies on affine schemes (i.e. Z*
*ariski, Nisnevich, 'etale
or faithfully flat).
Definition 2.3.1A family of morphisms of commutative S-algebras {A -! Bi}i2Iis *
*a standard
ø-covering (also called strong ø-covering) if it satisfies the following two co*
*nditions.
oThe induced family of morphisms of schemes {Spec ß0(Bi) -! Spec ß0(A)}i2Ii*
*s a ø-covering
of affine schemes.
oFor any i 2 I the natural morphism of ß0(Bi)-modules
ß*(A) i0(A)ß0(Bi) -! ß*(Bi)
is an isomorphism.
Its easy to check that this defines a model topology øs on S - Aff, called *
*the standard ø-topology.
The model site (S - Aff, øs) may be called the brave new standard-ø site. The i*
*mportance of stan-
dard topologies is that all øs-coverings of commutative S-algebras of the form *
*HR comes from usual
ø-coverings of the scheme Spec R. Its behavior is therefore very close to the g*
*eometric intuition one
gets in Algebraic Geometry.
Finally, let us also mention the semi-standard (or semi-strong) model topol*
*ogies. A family of mor-
phisms of commutative S-algebras {A -! Bi}i2Iis a semi-standard ø-covering (als*
*o called semi-strong
ø-covering) if the induced family of morphism of commutative graded rings {ß*(A*
*) -! ß*(Bi)}i2Iis
a ø-covering. This also defines a model topology øss on S - Aff.
Both the standard and semi-standard type model sites (and S-stacks over the*
*m, see Section 3)
could be of some interest in the study of geometry over even, periodic S-algebr*
*as (e.g. for elliptic
spectra as in [AHS ]).
3 S-stacks and geometric S-stacks
Let (M, ø) be a model site (i.e. a model category M endowed with a model topolo*
*gy ø in the sense
of [To-Ve 1]). Associated to it one has a model category of prestacks M^ and of*
* stacks M~,fi. For
details concerning these model categories we refer to [To-Ve 1, x4], and for th*
*e sake of brevity we only
recall the following facts.
oThe model category M^ is a left Bousfield localization of the model catego*
*ry SSetMop, of
simplicial presheaves on M together with the projective levelwise model st*
*ructure. The local
objects for this Bousfield localization are precisely the simplicial presh*
*eaves F : Mop -! SSet
which are equivalences preserving.
oThe model category M~,fiis a left Bousfield localization ([Hi, x3]) of the*
* model category of
prestacks M^ , and the localization (left Quillen) functor from M^ to M~,f*
*ipreserves (up to
equivalences) finite homotopy limits (i.e. homotopy pull-backs). The loc*
*al objects for this
Bousfield localization are the simplicial presheaves F : Mop- ! SSet which*
* satisfy the following
two conditions.
- The functor F preserves equivalences (i.e. is a local object in M^ ).
13
- For any ø-hypercover U* -! X in the model site (M, ø) ([To-Ve 1, x4.4*
*]), the induced
morphism
F (X) -! F (U*)
is an equivalence.
There is an associated stack functor a : Ho (M^ ) ! Ho (M~,fi) right adjoi*
*nt to the inclusion
Ho (M~,fi) ,! Ho (M^ ).
oThere is a homotopical variation 2
Rh_: Ho (M) ,! Ho (M^ )
of the Yoneda embedding ([To-Ve 1, x4.2]).
Specializing to our present situation, where M = S - Aff, we have one model*
* category S - Aff^
of prestacks and zounds of model categories stacks
S - Aff~,Zar, S - Aff~,'et, S - Aff~,thh-'et,
S - Aff~,fZar, S - Aff~,f'et, S - Aff~,fthh-'et,
s ~,'ets ~,ffqcs
S - Aff~,Zar, S - Aff , S - Aff ,
. .e.tc . . .
These model categories come with right Quillen functors (the morphism of change*
* of sites)
S - Aff~,'et__//_S - Aff~,thh-'et//_S -~Aff,Zar//_S -^Aff
S - Aff~,f'et_//_S - Aff~,fthh-'et//_S -~Aff,fZar//_S -^Aff
S - Aff~,'et__//S - Aff~,f'et//_S - Aff~,fZar//_S -^Aff
s___//_ ~,'et___//_ ~,Zar___//_ ^
S - Aff~,'et S - Aff S - Aff S - Aff
. .e.tc . . .
which allow to compare the various topologies on S - Aff.
Definition 3.0.2Let ø be a model topology on the model site S - Aff.
oThe model category of S-stacks for the topology ø is S - Aff~,fi.
oA simplicial presheaf F 2 SP r(S - Aff) is called an S-stack if it is a lo*
*cal object in S - Aff~,fi
(i.e. preserves equivalences and satisfies the descent property for ø-hype*
*rcovers).
oObjects in the homotopy category Ho (S - Aff~,fi) will simply be called S-*
*stacks (without refer-
ring, unless it is necessary, to the underlying topology).
2____________________________________
If x 2 M, Rh_(x) essentially sends y 2 M to the mapping space MapM(y, x).
14
The category of S-stacks, being the homotopy category of a model category, *
*has all kind of ho-
motopy limits and colimits. Moreover, one can show that it has internal Hom's. *
*Actually, the model
category of S-stacks is a model topos in the sense of [To-Ve 1, x3.8] (see also*
* [To-Ve 4]), and therefore
behaves very much in the same way as a category sheaves (but in a homotopical s*
*ense). In practice
this is very useful as it allows to use a lot of usual properties of simplicial*
* sets in the context of S-stacks
(in the same way as a lot of usual properties of sets are true in any topos).
The Eilenberg-Mac Lane functor H from commutative rings to commutative S-al*
*gebras induces
left Quillen functors
H!: Aff~,fi0-! S - Aff~,fi,
where ø0 is one of the standard topologies on affine schemes (e.g. Zar, 'et, ff*
*qc,. . . ), and ø is one of its
possible extension to the model category S - Aff(e.g. Zar can be extended to Za*
*rsor to Zar, etc.).
Here, Aff~,fiis the usual model category of simplicial presheaves on the Grothe*
*ndieck site (Aff, ø)
(with the projective model structure [Bl]). By deriving on the left one gets a *
*functor
LH!: Ho (Aff~,fi0) -! Ho (S - Aff~,fi).
Therefore, our category of S-stacks receives a functor from the homotopy catego*
*ry of simplicial
presheaves. In particular, sheaves on affine schemes (and in particular the ca*
*tegory of schemes it-
self), and also 1-truncated simplicial presheaves (and in particular the homoto*
*py category of algebraic
stacks) can be all viewed as examples of S-stacks. However, one should be caref*
*ul that the functor
LH!has no reason to be fully faithful in general, though this is the case for a*
*ll the standard extensions
(but not semi-standard) described in x2.3 (the reason for this is that all cove*
*ring families of some HR
are in fact induced from covering families of affine schemes. In particular the*
* restriction functor from
S - Aff~,fi-! Aff~,fi0will preserve local equivalences.).
3.1 Some descent theory
With the notations above, one can compose the Yoneda embedding Rh_: Ho (S - Alg*
*)op! Ho (S - Aff^)
with the associated stack functor a : Ho (S - Aff^) ! Ho (S - Aff~,fi) and obta*
*in the derived Spec
functor
RSpec : Ho (S - Alg)op= Ho (S - Aff) -! Ho (S - Aff~,fi),
for any model topology ø on S - Aff.
Definition 3.1.1The topology ø is sub-canonical if the functor RSpec is fully f*
*aithful or, equivalently,
if for any A 2 S - Alg, Rh_Ais an S-stack.
Knowing whether a given model topology ø is sub-canonical or not is known a*
*s the descent problem
for ø, and in our opinion is a crucial question. Unfortunately, we do not know *
*if all the model topologies
presented in the previous Section are sub-canonical, and it might be that some *
*of them are not. The
following lemma gives examples of sub-canonical topologies.
Lemma 3.1.2 The (semi-)standard Zariski, Nisnevich, 'etale and flat model top*
*ologies of x2.3 are all
sub-canonical.
Sketch of proof: Let ø be one of these topologies, A be a commutative S-alg*
*ebra, and A -! B* be
a ø-hypercover ([To-Ve 1, x4.4]). Using that ß*(Bn) is flat over ß*(A) for any *
*n, one can check that
the cosimplicial ß*(A)-algebra ß*(B*) is again a ø-hypercover of commutative ri*
*ngs. By usual descent
15
theory for affine schemes this implies that the cohomology groups of the total *
*complex of [n] 7! ß*(Bn)
vanish except for H0(ß*(B*)) ' ß*(A). This implies that the spectral sequence f*
*or the holim
Hp([n] 7! ßq(Bn)) ) ßp-q(holimB*)
degenerates at E2 and that ß*(A) -! ß*(holimB*) is an isomorphism. *
* 2
Concerning the brave new Zariski topology one has the following partial res*
*ult.
Lemma 3.1.3 Let {A -! Ai}i2Ibe a finite Zariski covering family of commutativ*
*e S-algebras. Let
A -! B = _iAibe the coproduct morphism. Let A -! B* be the cosimplicial commuta*
*tive A-algebra
defined by
Bn := B ^LAB ^LA. .^.LAB
________-z_______"
(n+1) times
(i.e. homotopy co-nerve of the morphism A -! B). Then the induced morphism
A -! holimn2Bn
is an equivalence.
Sketch of proof: By definition of Zariski open immersion it is not hard to *
*see that the cosimplicial
commutative A-algebra B* is m-coskeletal, where m is the cardinality of I. This*
* means the following:
let im : m -! be the inclusion functor form the full sub-category of objec*
*ts [i] with i m.
Then, one has an equivalence of commutative A-algebras B* ' R(im )*i*m(B*) (her*
*e (i*m, R(im )*) is the
derived adjunction between -diagrams and m -diagrams). From this one deduces*
* easily that
holimn2 Bn ' holimn2 mBn.
In particular, holimn2 Bn is in fact a finite homotopy limit and therefore will*
* commute with the base
change from A to B, i.e.
(holimn2 Bn) ^LAB ' holimn2(Bn ^L B).
Now, as the functor Ho (A-Mod ) -! Ho (B -Mod ) is conservative (since the fa*
*mily {A -! Ai}i2I
is a Zariski covering), one can replace A by B and the Ai by B ^LAAi, and in pa*
*rticular one can
suppose that A -! B has a section. But, it is well known that any morphism A -!*
* B which has a
section is such that A ' holimnBn (the section can in fact be used in order to *
*construct a retraction). 2
Of course, Lemma 3.1.3 is not enough to check that the brave new Zariski to*
*pology is sub-canonical
as it only deals with very particular hypercovers, the ~Cech-hypercovers i.e. *
*the ones arising as nerves
of coverings. However, one can slightly modified our definition of S-stacks by*
* asking only descent
with respect to ~Cech-hypercovers instead of all hypercovers (this weaker notio*
*n of stacks is the one
used recently by J. Lurie in [Lu] and also appeared for stacks over Grothendiec*
*k sites in [DHI ]). Us-
ing this weaker notion of ~Cech S-stacks, Lemma 3.1.3 implies that the brave ne*
*w Zariski topology is
then sub-canonical. What will be said from here on, will also be correct for th*
*is weaker notion of stacks.
16
3.2 The S-stack of perfect modules
Let ø be a model topology on S - Aff. One defines the S-prestack Perf_of perfe*
*ct modules in the
following way. For any commutative S-algebra A, we consider the category Perf(A*
*), whose objects
are perfect and cofibrant A-modules, and whose morphisms are equivalences of A-*
*modules. The pull
back functors define a pseudo-functor
Perf: S - Alg -! Cat
A 7-! Perf(A)
(A ! B) 7-! (- ^A B : Perf(A) ! Perf(B)).
Making this pseudo-functor into a strict functor from S - Algto Cat ([May , Th.*
* 3.4]), and applying
the classifying space functor Cat ! SSet, we get a simplicial presheaf denoted *
*by Perf_.
The following theorem relies on the so called strictification theorem ([To-*
*Ve 1, A.3.2]), and its
proof will appear in [To-Ve 2].
Theorem 3.2.1 The object Perf_is an S-stack (i.e. satisfies the descent condit*
*ion for all ø-hypercovers)
iff the model topology ø is subcanonical.
Another way to state Theorem 3.2.1 is by saying that ø is subcanonical iff,*
* for any commutative
S-algebra A, the natural morphism
Hom__S-Aff~,ø(Spec A, Perf_) ' Perf_(A) -! RHom__S-Aff~,ø(Spec A, *
*Perf_)
is an equivalence of simplicial sets.
The S-stack of perfect complexes is a brave new analog of the stack of vect*
*or bundles, and is of
fundamental importance in brave new algebraic geometry.
3.3 Geometric S-stacks
In this paragraph we will work with a fixed sub-canonical model topology ø on t*
*he model site S - Aff.
We will define the notion of geometric S-stack, which roughly speaking are quot*
*ients of affine S-stacks
by a smooth affine groupoid. They will be brave new generalizations of Artin a*
*lgebraic stacks (see
[La-Mo ]). In order to state the precise definition, one first needs a notion o*
*f smoothness for morphisms
of commutative S-algebras.
For any perfect S-module M one has the (derived) free commutative S-algebra*
* over M, S -!
LFS(M). For any commutative S-algebra A, one gets a morphism
A -! A ^LSLFS(M) ' LFA(A ^LSM).
Any morphism A -! B in Ho (A - Alg) which is isomorphic to such a morphism will*
* be called a
perfect morphism of commutative S-algebras (and we will also say that B is a pe*
*rfect commutative
A-algebra).
Definition 3.3.1A morphism of commutative S-algebras f : A -! B is called smoot*
*h if it satisfies
the following two conditions.
oThe A-algebra B is finitely presented.
17
oThere exists an 'etale covering family {vi: B ! B0i}i2Iand, for any i 2 I,*
* a homotopy commu-
tative square of commutative S-algebras
f
A _____//B
u|| vi||
|fflffl fflffl|
A0_f0_//_B0i,
i
where f0iis a perfect morphism, and u is an 'etale morphism.
One checks easily that smooth morphisms are stable by compositions and homo*
*topy base changes.
Furthermore, any 'etale morphism is smooth, and therefore so is any Zariski ope*
*n immersion.
Assumption: At this point we will assume that the notion of smooth morphism*
*s is local with
respect to the chosen model topology ø.
This assumption will insure that the notion of geometric S-stack, to be def*
*ined below, behaves well.
Some terminology:
oLet us come back to our homotopy category Ho (S - Aff~,fi) of S-stacks, an*
*d the Yoneda em-
bedding (or derived Spec)
RSpec : Ho (S - Alg)op-! Ho (S - Aff~,fi).
The essential image of RSpec is called the category of affine S-stacks, wh*
*ich is therefore anti-
equivalent to the homotopy category of commutative S-algebras. We will als*
*o call affine S-stack
any object in S - Aff~,fiwhose image in Ho (S - Aff~,fi) is an affine S-st*
*ack. Clearly, affine
S-stacks are stable by homotopy limits (indeed holimi(RSpec Ai) ' RSpec (h*
*ocolimiAi)).
oA morphism of affine S-stacks is called smooth (over S) (resp. 'etale, a Z*
*ariski open immersions
. . . ) if the corresponding one in Ho (S - Alg) is so.
oA Segal groupoid object in S - Aff~,fiis a simplicial object
X* : op-! S - Aff~,fi
which satisfies the following two conditions.
- For any n 1, the n-th Segal morphism
Xn -! X1xhX0X1xhX0. .X.1
_________-z________"
n times
is an equivalence (in the model category S - Aff~,fiof S-stacks). Wh*
*en this condition
is satisfied, it is well known that one can define a composition law *
*(well defined up to
homotopy)
~ : X1xhX0X1 -! X1.
- The induced morphism
(~, pr2) : X1xhX0X1 -! X1xhX0X1
is an equivalence (i.e. the composition law is invertible up to homot*
*opy).
18
oFor any simplicial object X* : op-! S - Aff~,fi, we will denote by |X*| t*
*he homotopy colimit
of X* in the model category S - Aff~,fi.
We are now ready to define geometric S-stacks.
Definition 3.3.2An S-stack F is called geometric if it is equivalent to some |X*
**|, where X* is a
Segal groupoid in S - Aff~,fisatisfying the following two additional conditions.
oThe S-stacks X0 and X1 are affine S-stacks.
oThe morphism d0 : X1 -! X0 is a smooth morphism of affine S-stacks.
The theory of geometric S-stacks can then be pursued along the same lines a*
*s the theory of al-
gebraic stacks (as done in [La-Mo ]). For example, one can define the notions o*
*f quasi-coherent and
perfect modules on a geometric S-stack, K-theory of a geometric S-stack (using *
*perfect modules on
it), higher geometric S-stacks (such as 2-geometric S-stacks), etc. We refer th*
*e reader to [To-Ve 2] for
details.
We will finish this paragraph with the definition of the tangent S-stack an*
*d its main properties.
First of all, one defines a commutative S-algebra S["] := S _ S, which is t*
*he trivial extension of
S by S. The S-algebra S["] can be thought as the brave new algebra of dual numb*
*ers, i.e. the analog
of Z["]. For any commutative S-algebra A, one has A["] := A ^LSS["] ' A _ A, t*
*he commutative
A-algebra of dual numbers over A.
For any S-stack F 2 S - Aff~,fi, one defines the tangent S-stack of F as
T F : S - Alg -! SSet
A 7! F (A["]).
The tangent S-stack T F comes equipped with a natural projection p : T F -!*
* F . One first notice
that if F is a geometric S-stack (over any base A), then so is T F . Furthermor*
*e, the homotopy fibers
of the projection p are linear S-stacks in the following sense. Let A be a comm*
*utative S-algebra and
x : RSpec A -! F be a morphism of S-stacks, i.e. an A-point of F . One consider*
*s the homotopy pull
back
Fx ______//_T F
| |
| |
fflffl| fflffl|
RSpec A__x__//F.
Then, one can show that there exists an A-module M, such that Fx is equivalent *
*(as a stack over
RSpec A) to RSpec(LFA(M)). In other words, one has a natural equivalence
Fx(B) ' RHom__A-Mod(M, B)
for any commutative A-algebra B. The A-module M is called the cotangent complex*
* of F at the point
x, and denoted by L F,x. Its derived dual A-module D(L F,x) is called the tange*
*nt space of F at x.
4 Derived moduli spaces in algebraic topology as S-stacks
In this last Section we present an example of a geometric S-stack that arises f*
*rom a classification
problem in Algebraic Topology. This example shows that moduli spaces in Algebra*
*ic Topology are
19
not only discrete homotopy types (as e.g. in [B-D-G ]), but might also have som*
*e additional rich geo-
metric structures very similar to the moduli spaces one studies in Algebraic Ge*
*ometry. The example
presented below seems to us the simplest one, and several more involved and int*
*eresting moduli prob-
lems can also be constructed and studied as geometric S-stacks.
We will work with a fixed subcanonical model topology ø on S - Aff.
4.1 The brave new group scheme RAut_(M)
We fix a perfect S-module M, and we are going to define a group S-stack RAut_(M*
*), of auto-
equivalences of M. This group S-stack will be a generalization of the group sc*
*heme GLn, since it
will be shown to be an affine and smooth group S-stack. Like many algebraic st*
*acks in Algebraic
Geometry are quotients of affine schemes by GLn, our example of a geometric S-s*
*tack in x4.2 will be
a quotient of an affine S-stack by RAut_(M) for some S-module M.
For any commutative S-algebra A, one first defines
REnd_(M)(A) := RHom__A-Mod(A ^L M, A ^L M).
Using for example the Dwyer-Kan simplicial localization techniques ([D-K1 , D-K*
*2]), one can make
A 7! REnd_(M)(A) into a functor from S - Algto the category SMon of simplicial*
* monoids
REnd_(M) : S - Alg -! SMon
A 7-! REnd_(M)(A).
This defines REnd_(M) as a monoid object in S - Aff~,fi. As its underlying *
*object in S - Aff~,fi
is an S-stack (for example using Theorem 3.2.1), we will say that REnd_(M) is a*
* monoid S-stack.
Lemma 4.1.1 The S-stack REnd_(M) is affine and the structural morphism REnd_(*
*M) -! RSpec S
is perfect (hence smooth).
Proof: This is clear as
REnd_(M) ' RSpec (LFS(M ^LSD(M))).
*
* 2
For any commutative S-algebra A, one defines RAut_(M)(A) to be the sub-mono*
*id of REnd_(M)(A)
consisting of auto-equivalences. In other words, RAut_(M)(A) is defined by the *
*following homotopy
pull-back diagram in SSet
RAut_(M)(A)________//_REnd_(M)(A)
| |
| |
fflffl| " fflffl|
[M ^L A, M ^L A]0Ø__//_[M ^L A, M ^L A]
where [-, -]0is the subset of isomorphisms in Ho (SSet). This defines a functor
RAut_(M) : S - Alg -! SMon
A 7-! RAut_(M)(A).
Once again the underlying object in S - Aff~,fiis an S-stack, and therefore RAu*
*t_(M) is a monoid
S-stack. Furthermore, the monoid law on RAut_(M) is invertible up to homotopy, *
*and we will therefore
say that RAut_(M) is a group S-stack.
20
Lemma 4.1.2 The S-stack RAut_(M) is affine and the structural morphism RAut_(*
*M) -! RSpec S
is smooth. In other words, RAut_(M) is an affine and smooth group S-stack.
Proof: The following proof is inspired by the proof of [EGA-I , I.9.6.4]. L*
*et B be the commutative
S-algebra LFS(M ^LSD(M)) corresponding to the affine S-stack REnd_(M). There ex*
*ists a universal
endomorphism of B-modules
u : M ^LSB -! M ^LSB
such that for any commutative B-algebra C, the endomorphism
u ^LBidC : M ^LSC -! M ^LSC
is equal (in Ho (B - Mod )) to the corresponding point in
REnd_(M)(C) ' RHom__S-Alg(B, C).
Consider now the homotopy cofiber K 2 Ho (B -Mod ) of the universal endomo*
*rphism u. Clearly,
K is a perfect B-module, and one can therefore consider the open Zariski immers*
*ion B -! BK
(Lemma 2.1.6). It is easy to check by construction that
RAut_(M) ' RSpec BK ,
which proves that RAut_(M) is an affine S-stack. Finally, as the morphism RSpec*
* BK -! RSpec B is
smooth (being a Zariski open immersion), one sees (using that RSpec B is perfec*
*t hence smooth) that
RAut_____(M) -! RSpec S is also smooth. *
* 2
4.2 Moduli of algebra structures
In this paragraph, we fix a perfect S-module M. We will define an S-stack Ass_M*
*, classifying associative
and unital algebras whose underlying module is M.
For any commutative S-algebra A, we have the category A - Ass, of associati*
*ve and unital A-
algebras (i.e. associative monoids in the monoidal category (A - Mod , ^A)); th*
*ese are new versions
of the old A1 -ring spectra. The category A - Asshas a model category structure*
* for which fibrations
and equivalences are detected on the underlying objects in A - Mod . We denote *
*by A - AsscofMthe
subcategory of A - Ass whose objects are cofibrant objects B such that there ex*
*ists a ø-covering
family {A -! Ai}i2Isuch that each Ai-module B ^LAAi is equivalent to M ^LAAi (w*
*e say that the
underlying A-module of B is ø-locally equivalent to M), and whose morphisms are*
* equivalences of
A-algebras. The base change functors define a lax functor
AssM : S - Alg - ! Cat
A 7! A - AsscofM
(A ! B) 7! - ^A B.
Strictifying this functor (([May , Th. 3.4])) and then applying the classifying*
* space functor, one gets
a simplicial presheaf
Ass_M: S - Alg -! SSet
A 7! B(A - AsscofM).
For the following theorem, let us recall that for any commutative S-algebra*
* A, any associative and
unital A-algebra B and any B-bimodule M, one has an A-module of A-derivations D*
*erA(B, M) from
B to M. This can be derived on the left (in the model category of associative a*
*nd unital A-algebras
!) to LDerA(B, M).
21
Theorem 4.2.1 1. The object Ass_M2 S - Aff~,fiis an S-stack.
2.The S-stack Ass_Mis geometric.
3.Let A be a commutative S-algebra, B be an associative A-algebra whose unde*
*rlying A-module
is ø-locally equivalent to M ^LSA and x : RSpec A -! Ass_M the correspondi*
*ng point. Then,
the tangent space of Ass_M at the point x is equivalent (as an A-module, s*
*ee the end of x3.3)
to the suspension LDerA(B, B)[1] of the A-module of derived A-derivations *
*of the associative
A-algebra B into the B-bimodule B.
Sketch of proof: Point (1) can be proved with the same techniques used in T*
*heorem 3.2.1 and will
not be proved here. We refer to [To-Ve 2] for details.
Point (3) seems to be a well-known fact. We again refer to [To-Ve 2] for de*
*tails.
Let us prove part (2) which is in fact a corollary of one of the main resul*
*t of C. Rezk thesis [Re].
Let us start by considering the full sub-S-stack of Perf_(see x3.2) consist*
*ing of perfect modules
which are ø-locally equivalent to M. By the result of Dwyer and Kan [D-K3 , 2.*
*3], this S-stack is
clearly equivalent as an object in S - Aff~,fito BRAut_(M), the classifying sim*
*plicial presheaf of the
group S-stack RAut_(M). Forgetting the algebra structure gives a morphism of S-*
*stacks
f : Ass_M-! BRAut_(M).
Using the techniques of equivariant stacks developed in [Ka-Pa-To1] (or more pr*
*ecisely their straight-
forward extensions to the present context of S-stacks), one sees that the S-sta*
*ck Ass_Mis equivalent
to the quotient S-stack
[X=RAut_(M)],
where X is the homotopy fiber of the morphism f and RAut_(M) acts on X. By Lemm*
*a 4.1.2, RAut_(M)
is an affine smooth group S-stack, so we only need to show that X is an affine *
*S-stack (because
the classifying Segal groupoid for the action of RAut_(M) on X will then satisf*
*ies the conditions of
Definition 3.3.2).
Using [Re, Thm. 1.1.5], one sees that the homotopy fiber X is equivalent to*
* the S-stack
RHom__Oper(ASS , End_(M)) : A 7-! RHom__Oper(ASS , End(M ^LSA)),
where RHom__Oper(ASS , End(M ^LSA)) is the derived Hom (or mapping space) of un*
*ital operad mor-
phisms from the final operad ASS (classifying associative and unital algebras) *
*to the endomorphisms
operad End(M ^LSA) of the A-module M ^LSA (here operads are in the symmetric mo*
*noidal category
S of S-modules). This means that, for any commutative S-algebra A, there is an *
*equivalence
X(A) ' RHom__Oper(ASS , End_(M))(A),
functorial in A.
Now, writing the operad ASS as a homotopy colimit
ASS ' hocolimn2 opOn,
where each On is a free operad, one sees that
X ' holimn2RHom__Oper(On, End_(M)).
22
Since affine S-stacks are stable under homotopy limits, it is therefore enough *
*to check that the S-stack
RHom______Oper(O, End_(M)) is affine for any free operad O. But, saying that an*
* operad O is free means
that there is a family {Pm }m>0of S-modules, and functorial (in A 2 S - Alg) eq*
*uivalences
Y L
RHom__Oper(O, End_(M))(A) ' RHom__S-Mod(Pm ^LSM^ m ^LSD(M), A),
m
where the funny notation M^Lm stands for the derived smash product M ^L . .^.LM*
* of M with itself
m times. So it is enough to show that for any S-module P , the (pre)stack
Lm L
A 7-! RHom__S-Mod(P ^LSM^ ^S D(M), A)
is affine. But this is clear since this stack is equivalent to RSpec B where B*
* is the derived free
commutative S-algebra
Ln L
B := LFS(P ^LSM^ ^S D(M)).
This implies that X is an affine S-stack and completes the proof. *
* 2
Theorem 4.2.1 has also generalizations when one consider algebra structures*
* over a given operad
(for example commutative algebra structures). It can also be enhanced by consi*
*dering categorical
structures such as A1 -categorical structures, as explained in [To-Ve 3]; the c*
*orresponding moduli
space gives an example of a 2-geometric S-stack.
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