This paper introduces a new framework for associating modular model categories to small categories, especially schemes, enabling parametrization of model categories by various small categories, with applications to algebraic geometry.
Contribution
It defines a functorial construction of modular model categories parametrized by small categories, including schemes, and explores their applications in algebraic geometry and homotopy theory.
Findings
01
Constructs a functor $ ext{ModCat} o ext{Cat}$ associating modular model categories to small categories.
02
Provides a parametrization of model categories using schemes and compares different parametrizations.
03
Lays groundwork for applying modular model categories to algebraic geometry contexts.
Abstract
To any model category M, we associate a modular model category, a functor of points M[−]: Cat → Cat, that associates to any small category C a functor category M[C]=Funfes(C,M) of full and essentially surjective functors from C to M, providing parametrizations of a same model category M by different small categories. We are in particular interested in using schemes as parameters. We consider ZSm/k the category of linear combinations of smooth separated schemes of finite type over Spec(k), k a field, referred to as Z-schemes, and let C=Sh(ZSm/k,Nis). We contrast this with using the A1-homotopy category of Z-schemes as a parametrizing category.
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models · Algebraic Geometry and Number Theory
To any model category M, we associate a modular model category, a functor of point M[−]:Cat→Cat, that associates to any small category C a functor category M[C]=Funess.surj.full(C,M), providing parametrizations of a same model category M by different small categories. We are in particular interested in using schemes as parameters. We consider C=ZSm/k the category of linear combinations of smooth separated schemes of finite type over Spec(k), k a field, referred to as Z-schemes. We contrast this with using the A1-homotopy category of Z-schemes as a parametrizing category.
1 Introduction
Defining a model structure on a category M necessitates the introduction of a class of weak equivalences, among other things. From there one may consider different types of weak equivalence between fixed objects, each one being defined relative to some algebraic invariant. The idea here is to introduce some variability in the notion of weak equivalence we use. If the model structure on M is fixed however, so are the equivalences. The variety in weak equivalences can nevertheless be implemented with parametrizations. From there, one is led to considering parametrizations of M by small categories C, provided by Funess.surj.full(C,M)∈P(M), P(M) the category of parametrizations of M. If Catess.surj.full is the category of small categories and full, essentially surjective functors, we have a functor of points M[−]:Catess.surj.full→P(M), that associates to any small category C the functor category HomCatess.surj.full(C,M)=Funess.surj.full(C,M)=M[C], which we refer to as the modular model category associated to M. Note that this inscribes itself within the theory of parametrized homotopy theory ([MS]). We are in particular interested in using schemes as parameters. Simple categories were already parametrized by schemes in [Ma] for representationt theoretic purposes. Most recently, the topos M(X,I) was considered in [Ka], where M is a left proper combinatorial simplicial model category, X a site with interval I, with aim the construction of algebraic cobordism for motivic stacks, which is achieved by letting χ=Sm/k, I=A1 and M=SetΔ,∗. In the present work we use linear combinations of schemes as parameters, which we refer to as Z-schemes. This is implemented by first generalizing schemes to finite correspondences ([FSV]), and then to Z-schemes. Aside from providing a generalization, note that morphisms between schemes in SmCor(k) are finite correspondences, linear combinations of schemes. Thus by taking Z-schemes as objects, we place ourselves at any level in the ∞-category SmCor(k), an obvious generalization being ZSm/k, the ∞-category of Z-schemes. Using Yoneda we regard those as presheaves. We then consider full, essentially surjective functors from Sh(ZSm/k,Nis) into a given model category M. This is one object of P(M), the parametrization of M by Z-schemes. Next, we consider categories C endowed with an equivalence relation as an alternative to definning categories with an interval object. Define two morphisms ϕ:X→Y and ϕ′:X′→Y′ in C to be equivalent if X∼X′ and Y∼Y′ in C. Let C/∼ be the category of equivalence classes of objects of C with equivalence classes of morphisms between them. For F:C→M a functor, define FX∼FX′ if X∼X′ in C, and Fϕ∼Fϕ′ if ϕ∼ϕ′. Let M/∼ be the category M modulo those equivalence relations. We have an induced functor [F]:C/∼→M/∼. We apply this formalism to C=Sh(ZSm/k,Nis) in particular. For that purpose, we consider a notion of equivalence relation on ZSm/k, and one in particular we use is based on the Hochschild cohomology of schemes, generalized to Z-schemes. Probably the easiest way to define it is by HH⋅(X)=ExtOX×X⋅(Δ∗OX,Δ∗OX), where if X=∑mi[Xi] and Y=∑nj[Yj] are elements of ZSm/k, we say X∼Y if and only if the indexing sets are the same, mi=ni for all i, and HH⋅(Xi)≅HH⋅(Yi) for all i, resulting in HH⋅(X)≅HH⋅(Y) since HH⋅(X)=⊗miHH⋅(Xi). Of independent interest, we also define a notion of depth in the topology on ZSm/k for the sake of precision by defining the general notion of powered topology, which goes as follows: suppose we have two categories of objects of a same type,
XN={XN+1} and YN={YN+1} themselves objects of a category CN−1 with a functor FN:YN→XN. Suppose we have a Grothendieck topology τN−1 on CN−1 with FN a covering map, element of a covering family in K(XN). Define a loose pre-topology τN on XN by defining loose covering families in K(XN) to be families of morphisms YN+1→XN+1, satisfying the same defining properties as traditional covering families for traditional Grothendieck topologies. We obtain a layered morphism:
[TABLE]
If the top and bottom maps are (loose) covering maps, then such a square would define a covering map in K(XN,XN+1), thereby defining a notion of powered topology τN∘τN−1 on CN−1. This formalism has the obvious advantage of giving level-wise degrees of precision.
In Section 2, we introduce modular model categories. We consider parametrizations of model categories by schemes, so we introduce Z-schemes in Section 3. In Sections 4 and 5, we discuss sheaves on individual Z-schemes, and on the site (ZSm/k,Nis). In Section 6, we introduce layered morphisms and powered topologies, and in Section 7, we add equivalence relations to the picture. In Section 8 we take stock and define ZSm/k-parametrizations of model categories, M[Sh(ZSm/k,Nis)], which we compare with using the A1-homotopy theory of Z-schemes for base category of our parametrizations, thus we contrast using M[Sh(ZSm/k,[Nis])] with using M[ShA1(ZSm/k,Nis)].
Acknowledgments**.**
The author would like to thank the organizers of the 2019 Exchange of Mathematical Ideas conference, during which part of this work was presented.
2 Modular Model Categories
Suppose one has a full, essentially surjective functor F:C→D from one category to another. Consider a morphism u:a→b in D. Then one can write u as F(ϕ):FX→FY for some X and Y in C such that FX=a and FY=b, with ϕ:X→Y. Suppose now we have another full, essentially surjective functor G:C→D. Then that same morphism u can be represented as G(ψ):GW→GZ, where W and Z are objects of C such that GW=a and GZ=b, with ψ:W→Z some morphism. Thus we have two different representations of a same morphism of D by morphisms of C. Hence the category Funess.surj.full(C,D) gives all the parametrizations of D relative to C. Define P(D) to be the set of elements of the form Funess.surj.full(C,D) for some small category C. We apply this formalism to model categories, so we let D=M be a model category in what follows. Let Catess.surj.full the category of small categories and full, essentially surjective morphisms between them. G:C→D in Catess.surj.full induces G∗:MD→MC,F↦F∘G. This defines a functor between different elements of P(M) making it into a category, which we call the category of parametrizations of M, and we have a functor:
[TABLE]
which we call a modular model category, namely the one associated with M.
Going back to the general case, consider a functor F:C→D from one small category to another, and suppose C is endowed with an equivalence relation. Define two morphisms ϕ:X→Y and ϕ′:X′→Y′ of C to be equivalent, ϕ∼ϕ′, if X∼X′ and Y∼Y′. Define FX∼FX′ in D, if X∼X′ in C. It follows that if X∼X′ and Y∼Y′, then on the one hand ϕ∼ϕ′, and on the other hand FX∼FX′ and FY∼FY′, from which Fϕ∼Fϕ′ by definition. F:C→D being given, having an equivalence relation on C induces:
[TABLE]
where we say we have a morphism [ϕ]:[A]→[B] if we can exhibit a representative ϕ:A→B, and we say [A]→[B] is of type Λ if there is a representative morphism of type Λ. Observe that if D=M is a model category, and Λ = C, W or F, meaning a cofibration, a weak equivalence or a fibration, then a same map [A]→[B] in M/∼ can be of different types simultaneously, so we lose in detail. By definition it is clear that M/∼ is still a model category, albeit a very weak one.
3 Z-schemes
In a first time, we will apply the above formalism to C=Sm/k, the category of smooth separated schemes of finite type over Spec(k), where k is a field. Following [FSV], we consider an extension of this category using finite correspondences as morphisms of schemes, giving rise to a category SmCor(k), whose objects are smooth schemes of finite type over k, and whose morphisms are finite correspondences, essentially linear combinations of integral schemes (see [FSV] for a complete definition).
There is a functor []:Sm/k→SmCor(k), that associates to any scheme X an object [X] of SmCor(k), and to any morphism f:X→Y in Sm/k its graph Γf⊂X×Y. For morphisms of schemes XfY and YgZ whose composition is defined, the corresponding composition in SmCor(k) reads [X]Γf[Y]Γg[Z], where the composition Γg∘Γf is defined as in [FSV]. We generalize this construction by defining ZSm/k, the category whose objects are linear combinations of elements of Sm/k(referred to as Z-schemes), and whose morphisms are appropriately chosen linear combinations of finite correspondences in a sense that we make precise presently.
Objects of ZSm/k are linear combinations of smooth separated schemes of finite type over Spec(k), they are of the form ∑mi[Xi]. In other terms ZSm/k is a free abelian group on Sm/k. We have an embedding ι(X)=1⋅[X] from Sm/k to ZSm/k. A map ϕ from Sm/k to ZSm/k naturally extends to a map Φ:ZSm/k→ZSm/k as follows: Φ(∑mi[Xi])=∑miϕ([Xi]). Indeed, there is a unique map ϕ∗ that makes the following diagram commutative for m∈Z:
[TABLE]
that is ϕ∗∘mι=mι∘ϕ, that is ϕ∗(m[X])=mϕ[X]. This can formally be presented as saying ϕ∗=mϕ, and this is the notation we will adopt. Of particular interest, observe that if we are looking at a morphism of schemes XϕY, then this means ϕ∗:m[X]→m[Y]. Indeed, if we start from a morphism ϕ:[X]→[Y], then mϕ:[X]→m[Y], so that ϕ∗:m[X]→m[Y]. Henceforth, we will drop the ∗ notation. We also consider morphisms of the form ∑mi[Xi]→∑nj[Yj]. Since each [Xi] may map to different [Yj]’s, mi will split as:
[TABLE]
where ∣i∣ is the number of [Yj]’s [Xi] is mapping to, and Ji is the subset of those indices j of J with a morphism [Xi]→[Yj]. It also means that:
[TABLE]
where Ij denotes the set of indices i for which we have a morphism [Xi]→[Yj]. If Ij is a singleton, then nj is simply not decomposed. Thus a morphism ϕ:∑i∈Imi[Xi]→∑j∈Jnj[Yj] will decompose as follows:
[TABLE]
If we denote by ϕji the restriction of ϕ to a map [Xi]→[Yj], then it is clear that we have:
[TABLE]
Observe that each morphism ϕji:[Xi]→[Yj] is in SmCor(k), i.e. it is really a finite correspondence. This makes ϕ a linear combination of finite correspondences, hence an element of ZSm/k itself.
Consider another morphism:
[TABLE]
with ∑j∈Jknjk=pk. Here ψ=∑njkψkj with ψkj:[Yj]→[Zk]. We will now define ψ∘ϕ. First consider:
[TABLE]
with ϕ:X→Y and ψ:Y→Z, then mψ∘mϕ=m(ψ∘ϕ). Now given k∈K, if j∈Jk we have a morphism:
[TABLE]
If in addition i∈Ij, then we have:
[TABLE]
where mijk is defined as follows. Write (ijk) for ijk for simplicity. We have pk=∑j∈Jknjk, and njk=∑i∈Ijm(ijk). Then we can define our composition:
[TABLE]
as:
[TABLE]
Thus defined, composition is clearly associative. In so doing, it helps to regard mijk as the coefficient of [Xi] in the decomposition of X for the composition [Xi]→[Yj]→[Zk], hence m(ijkl) is the coefficient of Xi in [Xi]→[Yj]→[Zk]→[Wl]. Indeed, consider the following composition:
[TABLE]
On the one hand:
[TABLE]
where:
[TABLE]
On the other hand:
[TABLE]
where we have n(jkl)=∑i∈Ijm(ijkl). However, we clearly have (γlk∘ψkj)∘ϕji=γlk∘(ψkj∘ϕji), hence associativity. About identity morphisms, if ϕ=∑j∈J,i∈Ijmijϕji:∑mij[Xi]→∑mij[Yj], then a right inverse is provided by ∑mijid[Yj], and a left inverse by ∑mijid[Xi].
4 Sheaves on Z-schemes
The motivation for considering presheaves of sets on ZSm/k is that the coproduct of schemes in Sm/k is not always well-defined, hence we have the same problem in ZSm/k as well. A convenient way to fix this problem is to formally add colimits in ZSm/k by considering presheaves of sets on ZSm/k, as already done in [V] for SmCor(k). We have a Yoneda embedding:
[TABLE]
Now in PreSh(ZSm/k), for the sake of doing homotopy, we have well-defined pushouts.
Now another problem surfaces. As pointed out in [V], if X=U∪V is a covering of a scheme X by two Zariski open subsets, the following diagram is a pushout in Sm/k, hence in ZSm/k as well:
[TABLE]
but the corresponding square of representable presheaves:
[TABLE]
is not necessarily a pushout in PreSh(Sm/k), hence not one in PreSh(ZSm/k) either. This can be remedied by considering sheaves. Recall that a presheaf of sets F:Cop→Set is a sheaf in a topology τ on C if for any covering family {fα:Uα→X} in this topology, the following sequence is exact:
[TABLE]
Thus it is clear that we will have to introduce a topology on ZSm/k. We will prove that sheaves, which are contravariant functors from ZSm/k to Set map certain pushout squares to cartesian squares. Those pushout squares are referred to as elementary distinguished squares in [V], [FSV], [ORV], [VM], and they are defined in terms of etale morphisms of Z-schemes. Those require the notion of sheaves on Z-schemes.
4.1 Sheaves on Z-schemes
For X=∑mi[Xi] an object of ZSm/k, a sheaf F on X decomposes as F=×iFi where each Fi is a sheaf on mi[Xi]. However for sheaves of abelian groups, addition is well-defined for a product of sheaves if we consider their tensor product instead, so F=⊗iFi. Now it suffices to define sheaves at objects of the form m[X] for m∈Z and X∈Sm/k.
For X∈ZSm/k of the form X=m[X] presently we adopt the notation X for [X] in such a manner that X=mX, and in the same fashion, if F is a sheaf of abelian groups on Sm/k, or even a presheaf to be more general, then one can define F=mF as a presheaf on ZSm/k according to the following commutative diagram:
[TABLE]
We now define sheaves of OX-modules on X=∑mi[Xi]∈ZSm/k. Let F be a sheaf of abelian groups on X, F=⊗Fi∣miXi=⊗imiFi∣Xi, with Fi sheaf on Xi, OX=⊗miOXi. A sheaf of OX-modules is a sheaf F on X such that for each open set U=∑i∈Imi[Ui], Ui open in Xi, the group F(U)=⊗miFi(Ui) is a ⊗miOXi(Ui) -module, that is Fi(Ui) is a OXi(Ui)-module for all i∈I, and for each inclusion of open sets V⊂U in ZSm/k, the indexed restriction homomorphisms are compatible with the module structure.
4.2 Flatness
Following [H] for the case of Sm/k, that we generalize to ZSm/k, let ϕ:X=∑mi[Xi]→Y=∑nj[Yj] be a morphism in ZSm/k, F a OX-module. Let x=∑mixi∈X with xi∈Xi for all i∈I. Since we have a decomposition ϕ=∑j∈J,i∈Ijmijϕji, with ϕji:[Xi]→[Yj], it follows that one can write x=∑j∈J,i∈Ijmijxi in such a manner that mijxi maps to some mijyji in Yj for all i∈Ij, that is ϕji(xi)=yji. Letting y=∑j∈J,i∈Ijmijyji in Y, one says F is flat over Y at x if Fx=⊗j∈J⊗i∈Ijmij(Fi)xi is a flat Oy=⊗j∈J⊗i∈IjmijOyji-module, where we consider (Fi)xi a Oyji-module via the natural maps ϕji#:Oyji→Oxi. Then one says F is flat over Y if it is flat at every point of X and one says X is flat over Y if OX is. Observe that if Y=Spec(k), if each Xi is flat, so is X=∑mi[Xi].
4.3 Sheaf of relative differentials
Remember from [H] that the sheaf of relative differentials ΩX/Y for a morphism of schemes f:X→Y is defined as Δ∗(I/I2) where Δ:X→X×YX is the diagonal map and I is the sheaf of ideals of ΔX on W, open subset of X×YX, ΔX closed subscheme thereof. For ϕ:X=∑mi[Xi]→Y=∑nj[Yj], we have the usual decomposition ϕ=∑j∈J,i∈Ijmijϕji, so that:
[TABLE]
with Δ(X)=∑j∈J,i∈IjmijΔji[Xi], where Δji:[Xi]→[Xi]×[Yj][Xi] is the diagonal map, in such a manner that if Iji is the sheaf of ideals of Δji[Xi] on W∩[Xi]×[Yj][Xi], then we have:
[TABLE]
so that:
[TABLE]
Now recall ([H]) that for f:X→Y a morphism of schemes, for G a sheaf on Y, f∗G=f−1G⊗f−1OYOX is a sheaf of OX-modules, where f−1G is the sheaf associated to the presheaf U↦limV⊇f(U)G(V). Here Δ∗(I/I2)=Δ−1I/I2⊗Δ−1OX×YXOX, where Δ−1I/I2 is the sheaf associated with U=∑mi[Ui]↦limV⊇ΔUI/I2(V), with ΔU=∑mijΔji[Ui]⊂∑mijVji=V, and Vji is open in [Xi]×[Yj][Xi]. We also have:
[TABLE]
Indeed recall that if F=mF, F(m[X])=mF[X]. Note that once X is fixed, so are the coefficients mi, hence so are the mij’s as well once Y is fixed, so the above decomposition does not depend on V. It follows:
[TABLE]
so that Δ−1I/I2=⊗mijΔji−1Iji/Iji2, as well as Δ−1OX×YX=Δ−1(⊗mijO[Xi]×[Yj][Xi])=⊗mijΔji−1O[Xi]×[Yj][Xi], so that:
[TABLE]
where in going from the first line to the second, we used mF⊗mOYmG=mF⊗OYG since mF⊗mG=m(F⊗G) is a sheaf of mOY-modules.
4.4 Etale maps
Recall that an etale map f:X→Y is a smooth map of relative dimension zero, which in Sm/k means f flat, for any irreducible components X′⊂X and Y′⊂Y such that f(X′)⊂Y′, we have dim(X′)=dim(Y′), and dimk(x)(ΩX/Y⊗k(x))=0 for any point x∈X. We generalize these notions to ZSm/k. We first define what it means to be a morphism of finite type in ZSm/k, since smooth of relative dimension zero subsumes of finite type. We simply define a morphism ϕ:∑mi[Xi]→∑nj[Yj] in ZSm/k to be of finite type if in the decomposition ϕ=∑j∈J,i∈Ijmijϕji, each of the morphisms ϕji:[Xi]→[Yj] is of finite type itself, for j∈J and i∈Jj. We define ϕ:X→Y in ZSm/k to be etale in exactly the same fashion that it was defined in Sm/k, namely ϕ flat, for any irreducible components X′ of X and Y′ of Y, if ϕ(X′)⊂Y′, then dimX′=dimY′, and for any x∈X, dim(ΩX/Y⊗k(x))=0. Flatness has been defined in section 4.2. For the dimensional statement on irreducible components, consider X′=∑mi[Xi′]⊂X and Y′=∑nj[Yj′]⊂Y. We consider a morphism ϕ:X→Y which has the usual decomposition ϕ=∑mijϕji so that we can actually write X′=∑j∈J,i∈Ijmij[Xi′] and Y′=∑j∈J,i∈Jimij[Yj′]. However for irreducible components, we just consider individual such terms, X′ is of the form [Xi′] for some i∈Ij, and Y′=[Yj′]. Then the condition ϕ[Xi′]⊂[Yj′] reads ϕji[Xi′]⊂[Yj′]. Then dim(Xi′)=dim(Yj′) for all such choices if the ϕji’s satisfy this dimensional statement on irreducible components i.e. ϕ=∑mijϕji satisfies the dimensional statement on irreducible components if all of the ϕji do. Finally we generalize the dimensional statement involving the sheaf of relative differentials. The local ring Ox=⊗miOxi has mx=⊗mimxi for maximal ideal, with residue field k(x)=⊗miOxi/mimxi=⊗miOxi/mxi. Now since we consider a morphism ϕ:X=∑mi[Xi]→Y=∑nj[Yj] then we have to consider the decomposition Ox=⊗j∈J,i∈IjmijOxi, from which it follows k(x)=⊗j∈J,i∈IjmijOxi/mxi. Then we have:
[TABLE]
Hence:
[TABLE]
if each summand is zero. We have shown:
Lemma 4.4.1**.**
ϕ=∑mijϕji is etale if and only if each ϕji is etale.
4.5 Elementary distinguished squares
Following [V], [VM], [ORV], we define an elementary distinguished square in ZSm/k to be a square of the form:
[TABLE]
where p is an etale morphism of Z-schemes, ψ is an open embedding, and p−1(X−U)≅X−U. Observe that ψ being an open embedding implies that if X=∑mi[Xi], U≅∑mi[Ui], where Ui is an open subset of Xi.
We define elementary distinguished squares in this section, since they deal with morphisms of Z-schemes. However it is in the definition of sheaves on ZSm/k that such squares are important, and we define those next.
5 Sheaves on ZSm/k
5.1 Sheaves
Since representables presheaves generate presheaves [MML], we will deduce properties of sheaves on ZSm/k from those of representable presheaves. We first consider F=Hom(−,∑nj[Yj]). We have:
[TABLE]
hence for sheaves F on ZSm/k, we also have:
[TABLE]
A presheaf F is a sheaf for a Grothendieck topology on ZSm/k if for any covering {fα:Uα→X} in ZSm/k for this topology, we have an equalizer:
[TABLE]
5.2 Nisnevich topology on ZSm/k
We define a Nisnevich covering in ZSm/k as in [V], [VM], [ORV], to be a finite family of etale morphisms {fα:Uα→X} such that for every x∈X, there is some α, there is some u∈Uα mapping to x such that k(u)≅k(x). On ZSm/k this reads as follows. {fα:Uα→X} is a Nisnevich covering if for any x=∑mjxj in X with xj∈Xj, there is some Uα=∑i∈Iαμαi[Uαi]=∑j∈J,i∈Iαjμαij[Uαi], there are uαi∈Uαi for all i∈Iαj, with fαji:uαi→xj (subject to ∑i∈Iαjμαij=mj), such that k(uαi)≅k(xj), and this for all j∈J. The morphisms fα being etale means that each fαji:Uαi→Xj is etale.
Lemma 5.2.1**.**
{fα:Uα→X,α∈A} is a Nisnevich covering if for all j∈J, {fα:Uα→mj[Xj],α∈Aj} is a Nisnevich covering, and A=∩j∈JAj.
Proof.
To have a Nisnevich covering over mj[Xj] means for any xj∈Xj, there is some α∈Aj, there is some uα∈Uα mapping to mjxj, such that k(uα)≅k(mjxj). Precisely, this means there is some uα=∑i∈Iαjμαijuαi∈Uα=∑i∈Iαjμαij[Uαi], with each uαi mapping to xj for all i∈Iαj, with k(uαi)≅k(xj), subject to ∑i∈Iαjμαij=mj. Now if x=∑mjxj∈X, assuming the hypothesis of the lemma, there is some α∈∩j∈JAj (after possible reindexing of the Uαi’s), there is some uαi∈Uαi for any i∈Iαj with uαi→xj, giving the decompositions:
[TABLE]
with uα=∑μαijuαi. Thus we see that k(x)≅k(uα) if k(xj)≅k(uαi) for each j∈J and i∈Iαj.In other terms {fα:Uα→X,α∈A} is a Nisnevich covering if for all j∈J, {fα:Uα→mj[Xj],α∈Aj} is a Nisnevich covering, and A=∩j∈JAj.
∎
For later purposes, denote by (a1X×mX⋯×mXanX)Σ the limit a1X×mX⋯×mXanX, subject to ∑ai=m, and denote by (a1X×mX⋯×mXanX)Δ the diagonal of the limit a1X×mX⋯×mXanX. We can then represent a Nisnevich covering over mj[Xj] as a Nisnevich covering {∏μαijfαji:∏μαijUαi→(μαi1Xj×mjXj⋯×mjXjμαikXj)ΣΔ} where k=∣Iαj∣.
We denote by Sh(ZSm/k,Nis) the category of sheaves on ZSm/k equipped with the Nisnevich topology.
5.3 Elementary distinguished squares
We now prove a generalization of Proposition 3.1.4 of [VM], which in the present situation would read as follows:
Proposition 5.3.1**.**
A presheaf on ZSm/k is a sheaf if and and only if it maps every elementary distinguished square in ZSm/k to a cartesian square.
The proof is identical in form to [VM], and differs only in the fact that we work in ZSm/k, not Sm/k, hence we have to deal with hybrid/local indexed terms, which does not make the proof any different in spirit, but there are technicalities we have to keep track of.
That a sheaf on ZSm/k maps elementary distinguished squares to cartesian squares follows from the original proof of [VM] due to its formality. Vice-versa, suppose now a presheaf F on ZSm/k maps elementary distinguished squares to cartesian squares. We aim to show it is a sheaf. In other terms if U={fα:Uα→X} is a Nisnevich covering, we want:
[TABLE]
to be exact. To do so we define a splitting sequence for U in exactly the same manner that it was initially introduced in [VM], but obviously adapted to our setting. We first need to prove that if U is a Nisnevich covering, it admits a splitting sequence. This means we have to first define rational sections in ZSm/k.
5.4 Rational sections of ZSm/k
We say X=∑mi[Xi] is Noetherian if each Xi is Noetherian in Sm/k for all i. We generalize to ZSm/k the definition of rational maps as initially introduced in [G]. Let X=∑mi[Xi] and Y=∑nj[Yj] be two objects of ZSm/k. Let U=∑mi[Ui] and V=∑mi[Vi] be two open subsets of X. Then f:U→Y and g:V→Y are said to be equivalent if they coincide in an open dense subset of U∩V, of the form ∑mi[Wi], with Wi an open dense subset of Ui∩Vi, i.e. if f∣Ui and g∣Vi agree on Wi for all i. Then one defines a rational map X→Y in ZSm/k to be an equivalence class of morphisms of dense open subsets X′=∑mi[Xi′] of X into Y, with Xi′ dense open subset of Xi for each i. To be specific, a rational map from X=∑mi[Xi] to Uα=∑μαj[Uαj] has for representation:
[TABLE]
where Xi′ is a dense open subset of Xi for all i∈I and ∑i∈Ijmij=μαj, mi=∑j∈Jαimij. The above map is given on each Xi′ by ∑j∈Jαimij[Xi′]→∑j∈Jαimij[Uαj], so a rational map is of the form ∑i∈I,j∈Jαimijrαji where all rαji:[Xi]→[Uαj] for j∈Jαi are rational maps, simultaneously over the same open set Xi′ for i fixed, that is giving a rational map on [Xi] is equivalent to giving a map:
[TABLE]
Now a rational section of Uα→X is a rational map X→Uα=∑i∈I,j∈Jαiμαij[Uαj] which is also a section, i.e. a map ∑i∈I,j∈Jαimijσαji where each σαji is a rational map, and a section, so that ∑i∈I,j∈Jαimijpαij∘σαji=idX, with pαij:[Uαj]→[Xi].
5.5 Construction of rational sections of Nisnevich covers
Observe that in the initial Nisnevich covering of X=∑mj[Xj], we have morphisms fα:Uα→X, with each Uα=∑j∈J,i∈Iαjμαij[Uαi]. If we write U=∐αUα, we have:
[TABLE]
and the collection of morphisms ∐α∑i∈Iαjμαij[Uαi]→mj[Xj] forms a Nisnevich covering of mj[Xj]. Indeed, fα=∑ijμαijfαji etale implies ∑iμαijfαji etale for α∈A, so ∐α∑iμαijfαji is etale by Lemma 4.4.1. It follows {fα:Uα→X} is a Nisnevich covering implies {∐α∑iμαijUαi→mj[Xj]} is a Nisnevich covering. We just drop the index j and call the above coproduct U. Observe, as pointed out in [VM], that to give a rational map from mX to U is equivalent to giving one on each irreducible component of X, so we might as well assume X to be irreducible. Now we apply Lemma 1.5 of [Ho] to U=∐α∑i∈Iαμαi[Uαi]→mX. Note that we can write this coproduct as ∑α,i∈Iαμαi[Uαi]. Let x be the generic point of X, let α be an index such that there is some u∈Uα over x. After reindexing, write Ui=μαi[Uαi]∐ possible other schemes, none of which is of the form μαj[Uαj] for j=i. Let Iα′ be the indexing set for those i’s. Let pαi:Uαi→X. Then each pi=μαipαi∐⋯:Ui→μαiX is etale, of finite type, completely decomposed in the sense of Hoyois. It follows it has a rational section σi for all i, hence so does ∏pi:∏Ui→μα1X×⋯×μα∣Iα′∣X. For our rational section we take:
[TABLE]
as representative, X′ dense open subset of X, where σi=μαiσ[pαi], as constructed in the previous subsection.
5.6 Existence of splitting sequences for Nisnevich covers
We now construct a splitting sequence for U over mX. We have argued p=∑pi:U=∑Ui→mX has a rational section ∑i∈Iα′μαiσ[pαi], so there is some dense open subset X′ of X such that we have a map σ:mX′→U, section of p−1(mX′)→mX′, σ=∑i∈Iα′μαiσ[pαi], p=∑α,i∈Iα′μαipαi, p∘σ=midX′. The rest of the construction is identical to that of [VM] or [Ho]. This proves that we have a splitting sequence for U. With this in hand, we can now finish the proof of Proposition 5.3.1:
Lemma 5.6.1**.**
If a presheaf on the Nisnevich site ZSm/k maps elementary distinguished squares to cartesian squares, it is a sheaf.
Proof.
Let U={Ui→mX} be a Nisnevich covering of mX. The reasoning will be the same as in [VM], or [Ho]. The only addition we bring here is the index notation to keep track of the components. Let mX=mZ0,⋯,mZn+1=∅ a splitting sequence of minimal length for U, which exists as we have just shown. Choose a splitting for the morphism p−1(mZn)→mZn, whose existence is guaranteed by the previous subsection. This means picking a rational section σ=∑i∈Iα′σi. Since each pi:Ui→μαiX is etale, we have a decomposition pi−1(μαiZn)=Im(σi)∐Yi, with σi=μαiσ[pαi], with Yi a closed subset of Ui. Then as in cite [VM], we let W=X−Zn, Vi=Ui−Yi, V=∏i∈Iα′Vi. We then claim that mW and V form elementary distinguished squares over mX, and that U×mXmW→mW is a Nisnevich covering of mW with a splitting sequence of length n−1. For the first claim, we have the following elementary distinguished square as a classical result:
[TABLE]
with pi etale, ψi open immersion. This is an elementary distinguished square as argued in [VM]. It follows that the following square is also an elementary distinguished square:
[TABLE]
About the second point, {pi:Ui→X} is a Nisnevich covering, so by [VM], U×μαiXμαiW→μαiW is a Nisnevich covering, from which it follows that U×mXmW→mW is a Nisnevich covering.
∎
6 Powered topologies
We now define a notion of layered morphism, and a corresponding notion of layered (or powered) topology. We define this iteratively. Let CN−1 be a category with objects χN of some type ΛN. With this terminology, C=C0 is our initial category, with objects X1 of type Λ1. Suppose each object XN of CN−1 has some internal structure, and can be regarded as being made up of objects XN+1 of type ΛN+1. Categorify each such object XN in such a manner that its objects as a category are its constituting elements, and its morphisms are maps XN+1′→XN+1 between objects of XN, if such maps exist. A layered morphism is any commutative diagram of the form:
[TABLE]
with possible additional lower layers defined as in:
[TABLE]
where in (1), XN and YN are categories, YN+1 is an object of YN, XN+1 is an object of XN and we have well-defined maps XN→YN and XN+1→YN+1. Define C[N,N+p] as the category with objects of the form (XN+p↪⋯↪XN), with morphisms layered morphisms such as the one above. Identity and composition are obvious. A presheaf on C[N,N+p] is a functor F:C[N,N+p]op→PoSet that maps XN+p→⋯→XN to FN+pXN+p←⋯←FNXN. More generally, a functor from C[N,N+p] to D[N,N+p] is a map F:C[N,N+p]→D[N,N+p] with F(XN+p↪⋯↪XN)=FN+pXN+p↪⋯FNXN. Strictly speaking, FN+p is a functor on XN+p−1, which could be different from a functor on YN+p−1, for which we still use the notation FN+pYN+p, but categories are assumed by construction to be levelwise of a same type. Thus functors in this setting are understood levelwise not as functors from one category to another, but from one type of categories to another type of categories. Hence:
[TABLE]
is mapped to:
[TABLE]
under F. We have F(idXN+p↪⋯↪XN)=idF(XN+p↪⋯↪XN), since each Fi is a functor on types, and F(g∘f)=F(g)∘F(f) as well, represented as F((g∘f)N+p,⋯,(g∘f)N)=(F(g)N+p∘F(f)N+p,⋯,F(g)N∘F(f)N).
Once that is defined, we can define what we call a layered topology. If N=1, maps X1→Y1 are in C0. If one categorifies X1 and Y1, X1={X2} and Y1={Y2}, in writing:
[TABLE]
the bottom map is no longer in Y1. Hence if we regard such a bottom map as an element of a covering of Y2, necessarily coverings, hence layered topologies, must be interpreted in a looser sense. We formalize this: suppose our categories admit pullbacks. Define a basis for a loose topology on YN for N≥1 to be given by a function K which assigns to each object YN+1 of YN a collection K(YN+1) of families of morphisms codomain YN+1, but with domains categories that are possibly different from YN+1, satisfying the same conditions as those of covering families for classical Grothendieck topologies.
Suppose now we have a basis for a loose topology on Xn−1={XN} , with Yn→Xn a covering map in K(Xn), and write Xn={Xn+1} and Yn={Yn+1}, categorified. Suppose XN has a basis for a loose topology as well, with Yn+1→Xn+1 a covering map in K(Xn+1). Then a covering map, element of a covering family in K(Xn+1,Xn), is defined to be a layered morphism:
[TABLE]
where the top map is in K(Xn), and the bottom one is in K(Xn+1). Hence loose covering maps in C[N,N+p] are layered morphisms that are levelwise loose covering maps, hence also follow the same defining properties of covering maps for traditional Grothendieck topologies. Indeed, if in a diagram such as the one above, the top and bottom maps are isomorphisms, the whole diagram is itself in K(Xn+1,Xn) by definition. It is also clear compositions are stable; if
[TABLE]
is a covering map in K(X′,X), and if:
[TABLE]
is a covering map in K(Xi′,Xi), then the composition:
[TABLE]
is in K(X′,X). Now let:
[TABLE]
be elements of K(X′,X), indexed by i, and consider any layered morphism:
[TABLE]
we have:
[TABLE]
is isomorphic to:
[TABLE]
Now the top map of the back face is in K(Z′), the top map of the front face is in K(Z), which means exactly that the following map:
[TABLE]
is in K(Z′,Z).
7 Blurry topologies
Objects of a given type come with a notion of weak equivalence (possibly trivial). Consider the accompanying equivalence relation (generated by the relation of weak equivalence), thereby defining equivalence classes of objects of some given type. Later we will define two schemes X and Y to be equivalent if they have isomorphic Hochschild cohomology, thereby bypassing the need to introduce a notion of weak equivalence, and working directly with an equivalence relation. We will show if X=∑mi[Xi], then the Hochschild cohomology of X is defined by HH⋅(X)=⊗miHH⋅(Xi), hence if Y=∑nj[Yj], then Xi∼Yi and ni=mi for all i implies X∼Y. In particular, if X=X1×X2, Y=Y1×Y2, then if Xi∼Yi for i=1,2 we have X∼Y. This means Y1×Y2∈[X1]×[X2], implies Y1×Y2∼X1×X2∈[X1×X2]. We are led to defining categories of type Γ to be those for which their objects satisfy [A]×[B]⊂[A×B].
Start with C=X0 a category, which we suppose admits pullbacks. Objects of X0 are of type Λ1, X0 itself is of type Λ0. Objects of type Λ1 are assumed to come with a notion of weak equivalence. Take the equivalence relation generated by it, and consider its corresponding equivalence classes. One can then write X0=∐[X1]. Assume X0 comes with a Grothendieck topology already. A basis for a blurry topology on X0 is a function K that assigns to each equivalence class [X1] a collection K[X1] of morphisms of X0 with codomain [X1]. We say {[Y]→[X]} is in K[X] if {Y→X} is in K(X) (or equivalently if a representative morphism is in K(X)). This defines a topology on X0=∐[X1], or a blurry topology on X0={X1}. Indeed, if [Y]→[X] is an isomorphism, this means we have an isomorphism Y→X, hence {Y→X} is in K(X), that is {[Y]→[X]} is in K[X]. Now if {ϕi:[Xi]→[X]} is in K[X], if [Y]→[X] is any morphism, we show {π2:[Xi]×[X][Y]→[Y]} is in K[Y]. First {ϕi:Xi→X} is in K(X), so {π2:Xi×XY→Y} is in K(Y), i.e. {[Xi×XY]→[Y]} is in K[Y].We limit ourselves to categories X0 of type Γ:Λ0, hence [Xi]×[X][cY]⊂[Xi×XY]. It follows {[Xi]×[X][Y]→[Y]} is in K[Y]. Finally for composition if {[Xi]→[X]} is in K[X] and {[Xij]→[Xi]} is in K[Xi], then {Xi→X} is in K(X), and {Xij→Xi} is in K(Xi), from which it follows that {Xij→Xi→X} is in K(X), which means that {[Xij]→[Xi]→[X]} is in K[X] since Xij→Xi→X is a representative morphism. Thus from a Grothendieck topology on an ordinary category of type Γ, one can derive a blurry topology.
Now let’s see what happens if we have layered morphisms. Suppose we have a blurry topology on X0, and X1 and Y1 are objects of X0, both of type Γ:Λ1, categorified, with a notion of weak equivalence on their respective objects and corresponding equivalence classes, so that we can write X1=∐[X2] and Y1=∐[Y2]. Suppose both have a loose topology defined on them. Define a blurry loose topology by just generalizing the notion of blurry topology on X0: {[Y2]→[X2]} is in K[X2] if {Y2→X2} is in K(X2). It is not difficult to see that this also defines a loose topology on X1=∐[X2].
Now a diagram such as:
[TABLE]
where the top horizontal map is a covering map for a blurry topology [τ0] on X0, with τ0 a Grothendieck topology on X0, and the bottom map is a covering map for a blurry loose topology [τ1] on X1, where τ1 is a loose Grothendieck topology on X1, together define a layered, or powered blurry topology[τ1]∘[τ0] on X0. This can of course be generalized iteratively. Applying this to functors of types F:C[N,N+p]→D[N,N+p], if we have towers of equivalences in C[N,N+p], this induces level-wise quotient maps [F]=([FN+p],⋯,[FN]).
8 Z-schemes-parametrized model categories
8.1 Parametrizations of model categories by Z-schemes
In this section we consider full, essentially surjective functors ξ:PreSh(ZSm/k)→M where M is any model category. For the sake of having a good notion of space parametrizing morphisms of M, we consider functors of the form Sh(ZSm/k,Nis)→M instead. Morphisms of ZSm/k, which are elements of ZSm/k themselves, map to morphisms of M. In this manner we have morphisms of M being parametrized by Z-schemes. We now suppose we have a notion of equivalence on Z-schemes. In the next subsection we define one example of equivalence relation on such objects. Having such a notion of equivalence on Z-schemes, on which we also have a Nisnevich topology, produces a blurry Nisnevich topology [Nis].
Recall that a Nisnevich covering on ZSm/k is a finite family of etale morphisms {fα:Uα→X} in ZSm/k, such that for all x∈X, there is a α, there is some u∈Uα with fα(u)=x and k(u)≅k(x). We generalize this notion to that of a blurry Nisnevich topology, whose coverings are given by finite families of morphisms in ZSm/k, {[fα]:[Uα]→[X]}, such that for any x∈X∈[X], there is some α, there is some u∈Uα∈[Uα] with fα(u)=x, and k(u)≅k(x), fα etale morphism in ZSm/k. Let Sh(ZSm/k,[Nis]) be the category of sheaves on ZSm/k for that topology. If we have an object F∈M[Sh(ZSm/k,Nis)]=Funess.surj.full(Sh(ZSm/k,Nis),M), we have an induced morphism [F]:Sh(ZSm/k,[Nis])→M/∼. This is the first stage with τ0=Nis. Now one could stop there or use powered topologies. If that is the case, we categorify each object X=∑mi[Xi] of ZSm/k, defining it as a category with objects [Xi], with morphisms those morphisms in Sm/k. Z-schemes X are of type Λ1. Now covering families on X are finite families of etale morphisms {fij:Yj→Xi} such that for any xi∈Xi, there is some yj∈Yj such that fij(yj)=xi and k(yj)≅k(xi), where those Yj′s originate from some Y=∑nj[Yj], with a preexisting morphism Y→X in ZSm/k. Here in the covering family we have not used brackets for objects of SmCor(k) to avoid confusion with equivalence classes of schemes. This gives us a loose topology τ1 on X. We can consider the associated blurry loose topology [τ1], resulting in a powered blurry topology [τ1]∘[τ0] on ZSm/k. Moving forward we can further categorify each Xi, smooth over k, so of finite type, hence it can be covered by finitely many affine schemes (SpecRik,Oik), objects, with morphisms morphisms of affine schemes. If we have a morphism from Yj to Xi we do likewise for Yj, covered by (SpecSjl,Ojl), and coverings are finite families of etale morphisms of ringed spaces SpecSjl→SpecRik, giving us a loose topology τ2, with an associated blurry loose topology [τ2], yielding a powered blurry topology [τ2]∘[τ1]∘[τ0] on ZSm/k. We can pursue in this manner as many times as needed, provided subsequent topologies can be defined. This gives rise to Sh(ZSm/k,[τ2]∘[τ1]∘[τ0]). If we have a functor F:Sh(ZSm/k,τ2∘τ1∘τ0)→M, this induces [F]=([F2],[F1],[F0]):Sh(ZSm/k,[τ2]∘[τ1]∘[τ0])→M/∼. Another alternative consists in not having a notion of equivalence on ZSm/k, but to have an interval object I instead on the site (ZSm/k,Nis), such as the affine line A1, and this is the point of view adopted in [Ka]. What is studied in [Ka] is the topos M(Sm/k,Nis,A1), for M a left proper, combinatorial simplicial model category. In the present paper we put no restriction on our model categories M for the simple reason that we do not take a Bousfield localization of our topos Fun(Sh(ZSm/k,Nis),M). Nevertheless we will come back later to a generalization of the work done in [Ka] to contrast this with using equivalences on schemes.
8.2 Equivalence relations on ZSm/k
For our notion of equivalence, we will use Hochschild cohomology on ZSm/k, which we will define as a generalization of the usual Hochschild cohomology of schemes as developed in [GS] and [S] in particular, but where some relevant treatments can also be found in [K], [Ku]. The idea of using Hochschild cohomology is derived from the fact that since one has functors from PreSh(ZSm/k) to M, one would want equivalent Z-schemes to map to the same object. If we regard functors as representations, one would think in algebraic terms about Morita equivalent algebras, which is trivial for commutative rings. From Morita theory one can easily think of Hochschild cohomology. The latter is not trivial on Sm/k however. Recall, from [GS] and [S], that for X a separated scheme of finite type over k, F a sheaf of OX-modules, one can define the Hochschild cohomology of X with coefficients in F by:
[TABLE]
where F is regarded as a sheaf of OX×X-modules via the diagonal functor. Define the Hochschild cohomology of a scheme X as Hn(X)=Hn(OX,OX), and we define two schemes X and Y to be equivalent if H⋅(X)≅H⋅(Y). We now generalize this Z-schemes. Let X=∑mi[Xi] and Y=∑nj[Yj]. To be explicit, in Sm/k we have H⋅(X)=ExtOX×X⋅(Δ∗OX,Δ∗OX). We have X×X=∑miXi×Xi, so that OX×X=⊗OmiXi×Xi=⊗miOXi×Xi, and it also follows Δ∗(OX)=Δ∗(⊗miOXi)=⊗miΔ∗(OXi). Now:
[TABLE]
thus we can define two objects X=∑i∈Imi[Xi] and Y=∑j∈Jnj[Yj] of ZSm/k to be equivalent if the indexing sets I=J, mi=ni for all i∈I, and H⋅(Xi)≅H⋅(Yi) for all i∈I. This then defines a notion of Hochschild equivalence on ZSm/k.
Another definition of Hochschild cohomology of Z-schemes we can use is the Grothendieck-Loday definition of such, as presented in [S] for Sm/k. Recall that if A is an algebra over a field k, letting Ae=A⊗kA, we can define the bar complex by B⋅(A)=A⊗kA⊗⋅⊗kA. Then we define C⋅(A)=A⊗AeB⋅(A)=A⊗kA⊗⋅. If X is a smooth scheme over k, we define a presheaf on X by letting C⋅(U)=C⋅(Γ(U,OX)). We denote by aC⋅ the associated sheaf, where a is the sheafification functor. It is a sheaf of OX-modules. Now if F⋅ is a chain complex of sheaves of OX-modules, if G is a OX-module with an injective resolution 0→G→I⋅, then we define the hyperext by:
[TABLE]
In ZSm/k, I⋅(⊗miGi)=⊗miI⋅Gi.
With this in hand we can define the Grothendieck-Loday type definition of Hochschild cohomology of schemes with values in F by HH⋅(X,F)=ExtOX⋅(aC⋅,F), and the Hochschild cohomology of schemes by HH⋅(X)=HH⋅(X,OX)=ExtOX⋅(aC⋅,OX). As usual, we say X∼Y in Sm/k if and only if HH⋅(X)≅HH⋅(Y). We now generalize this definition to ZSm/k. First Γ(−,OX)=⊗Γi(−,OmiXi)=⊗miΓi(−,OXi), where Γi is the section functor on Xi. Since C⋅(U)=C⋅(Γ(U,OX)) defines a presheaf on X, it follows C⋅,mX=C⋅(Γ(U,OmX))=C⋅(mΓ(U,OX)) is also equal to mC⋅,X, from which it follows that if X=∑mi[Xi]∈ZSm/k:
[TABLE]
hence we define, again, X=∑i∈Imi[Xi] and Y=∑j∈Jnj[Yj] in ZSm/k to be equivalent if I=J, mi=ni for all i∈I, and HH⋅(Xi)≅HH⋅(Yi) for all i∈I.
8.3 A1-homotopy category of Z-schemes as parameter space
As pointed out above, an alternative to using a notion of equivalence on Z-schemes consists in using an interval object I on (ZSm/k,Nis). Naturally one would take I=A1, as done for the homotopy theory of schemes ([VM]), developed from presheaves of simplicial sets. We will use a variant of such a construction, not regarding A1 as an interval object, but just as a presheaf, we will localize with respect to A1-local maps, and then use the Nisnevich topology on such a localization. The construction is fairly transparent.
Recall that in this work, we consider presheaves of sets. In this section in particular, we consider objects of PreSh(ZSm/k,Nis,A1). In A1-homotopy theory of schemes however, we work with simplicial sheaves. Thus we regard presheaves of sets as constant simplicial presheaves. Consider HomZSm/k(−,A1), the representable presheaf associated with A1, that we still denote by A1. Consider the functor category Fun((ZSm/k)op,SetΔ). From there we essentially follow [Hi]. Recall that if M is a model category, S is a class of maps in M, we can define a model category structure on the underlying category of M, denoted LSM, for which weak equivalences are S-local equivalences in M, cofibrations are those of M, and fibrations are those maps that have the right lifting property with respect to cofibrations that are also S-local equivalences. Recall also what those are: an object W of M is said to be S-local if it is fibrant, and if for any f:A→B in S, the induced map of homotopy function complexes f∗:map(B,W)→map(A,W) is a weak equivalence of simplicial sets. A map g:X→Y in M is a S-local equivalence if for any S-local object W, the induced map of homotopy function complexes g∗:map(Y,W)→map(X,W) is a weak equivalence of simplicial sets. It is a fact that if M is a left proper, cellular model category, S a set of maps in M, then the left Bousfield localization LSM of M exists. Now SetΔ is a left proper cellular model category, ZSm/k is a small category, hence SetΔZSm/kop is also left proper cellular ([Hi]). Denote it by M, let S be the set of projection maps {F×A1→F} for F∈M. An object G of M is S-local, or A1-local, if it is fibrant, and for any p:F×A1→F in S, the induced map of homotopy function complexes p∗:map(F,G)→map(F×A1,G) is a weak equivalence in SetΔ. Then α:F⇒H is an A1-local equivalence if for any A1-local object G, the induced map α∗:map(H,G)→map(F,G) is a weak equivalence in SetΔ. We consider LA1SetΔ(ZSm/k)op, the left Bousfied localization of SetΔZSm/kop with respect to S={F×A1→F}, then use the Nisnevich topology on ZSm/k, giving rise to Ξ=LA1SetΔ(ZSm/kop,Nis), and hence to Λ=ShA1(ZSm/k,Nis)⊂Ξ, and finally functors Λ→M for M any model category can be regarded as providing parametrizations of M by A1-homotopic Z-schemes, where now we consider A1-local objects instead of Hochschild equivalent Z-schemes.
Bibliography16
The reference list from the paper itself. Each links out to its DOI / PubMed record.
1[FSV] E.M. Friedlander, A. Suslin and V. Voevodsky, Cycles, Transfers, and Motivic Homology Theories , Annals of Mathematics Studies, 143 , Princeton University Press, Princeton, NJ, 2000.
2[GS] M. Gerstenhaber and D. Schack, Algebraic Cohomology and Deformation Theory , Deformation Theory of Algebras and Structures and Applications, (Il Ciocco, 1986), Vol. 247 , NATO Adv. Sci. Inst. Ser. C Mathematical Physics, Kluwer Acad. Publ., Dordrecht, 1988, pp 11-264.
3[G] A. Grothendieck, Elements de Geometrie Algebrique , Publications Mathematiques de l’IHES, tome 4 (1960), p.5-228.
4[H] R. Hartshorne, Algebraic Geometry , Graduate Texts in Mathematics, Springer-Verlag (1977), New-York.
5[Hi] P.S. Hirschhorn, Model Categories and their Localizations , Math. Surveys and Monographs Series, Vol. 99 , AMS, Providence, 2003.
6[Ho] M. Hoyois, Nisnevich Topology , available at www-bcf.usc.edu/ hoyois/papers/nisnevich.pdf.
7[Ka] Y. Kato, Motivic Model Categories and Motivic Derived Algebraic Geometry , ar Xiv:1703.02849 v 3 [math.CT].
8[K] M. Kontsevich, Homological Algebra of Mirror Symmetry , Proceedings of the International Congress of Mathematicians, Vol. 0,2 , (Zurich, 1994).