Quasi-Coherent Sheaves
2.1 Sheaves on a Spectrally Ringed 8 -Topos
2.6.1 The Case of a Noetherian Commutative Ring . . . 247
2.6.2 The Case of a Commutative Ring . . . 250
2.6.3 The General Case . . . 252
2.7 Finiteness Properties of Modules . . . . 253
2.7.1 Finitelyn-Presented Modules . . . 256
2.7.2 Alternate Characterizations . . . 257
2.7.3 Extension of Scalars . . . 261
2.7.4 Fiberwise Connectivity Criterion . . . 264
2.8 Local Properties of Quasi-Coherent Sheaves . . . . 265
2.8.1 Etale-Local Properties of Spectral Deligne-Mumford Stacks . . . 265´
2.8.2 Flat Morphisms . . . 268
2.8.3 Fpqc-Local Properties of Spectral Deligne-Mumford Stacks . . . 271
2.8.4 Fpqc-Local Properties of Modules . . . 274
2.9 Vector Bundles and Invertible Sheaves . . . . 277
2.9.1 Locally Free Modules . . . 277
2.9.2 The Rank of a Locally Free Module . . . 278
2.9.3 Locally Free Sheaves . . . 280
2.9.4 Line Bundles . . . 282
2.9.5 Invertible Sheaves . . . 283
2.9.6 The Affine Case . . . 285
2.1 Sheaves on a Spectrally Ringed 8 -Topos
Let X be a topological space and let O be a sheaf of commutative rings on X. A sheaf of O-modulesis a sheaf of abelian groups F on X such that FpUq is equipped with the structure of a module over the commutative ringOpUq for every open subset U ĎX, depending functorially on U. Our goal in this section is to introduce an 8-categorical analogue of the theory of sheaves of modules. We will replace the topological spaceX with an arbitrary8-toposX, and O by an arbitrary sheaf ofE8-rings onX.
Definition 2.1.0.1. LetX be an8-topos and letO PShvCAlgpXq be a sheaf ofE8-rings on X. Recall thatO can be identified with a commutative algebra object of the symmetric monoidal 8-categoryShvSppXq of sheaves of spectra onX (see §??). We let ModO denote the 8-category ModOpShvSppXqq of O-module objects of ShvSppXq. Then ModO can be regarded as a symmetric monoidal8-category with respect to the relative tensor product bO (see §HA.3.4.4 ). We will refer to the objects of ModO as sheaves ofO-modules onX, or sometimes just as O-modules.
Warning 2.1.0.2. Let X be a topological space and letO be a sheaf of commutative rings onX. Then we can identify O with a sheaf of E8-rings on the 8-topos ShvpXq. In this case, Definition 2.1.0.1 does not recover the classical theory of sheaves ofO-modules onX, because we allow ourselves to consider sheaves of spectra rather than sheaves of abelian groups. However, we will prove below that the8-category ModO is stable and equipped with a natural t-structure (Proposition 2.1.1.1). The classical theory of sheaves ofO-modules can be recovered by taking the heart Mod♥O of the 8-category ModO. Moreover, the8-category ModO is closely related to the derived 8-category of its heart (see Corollary 2.1.2.3).
The next proposition summarizes some of the basic formal properties of Definition ??:
Proposition 2.1.0.3. Let X be an8-topos and O a sheaf ofE8-rings on X. Then:
p1q The8-categoryModO is stable.
p2q The 8-category ModO is presentable and the tensor product bO : ModOˆModO Ñ ModO preserves small colimits separately in each variable.
p3q The forgetful functor θ: ModO ÑShvSppXq is conservative and preserves small limits and colimits.
Proof. Assertionp1qfollows from Proposition HA.7.1.1.4 , assertionp2qfollows from Theorem HA.3.4.4.2 , and assertionp3q follows from Corollaries HA.3.4.3.2 and HA.3.4.4.6 .
Notation 2.1.0.4. Let pX,Oq be a spectrally ringed 8-topos, and suppose we are given objects F,F1 PModO. For every integern, we let ExtnOpF,F1q denote the abelian group ExtnMod
OpF,F1q of homotopy classes of maps fromF to ΣnF1 in ModO.
Remark 2.1.0.5. Let pX,Oq be a spectrally ringed 8-topos. Then the construction pU P Xq ÞÑ ModO|
U determines a functor from Xop into the 8-category yCat8 of (not necessarily small) 8-categories. Moreover, this functor preserves small limits.
To see this, consider the coCartesian fibration p : Funp∆1,Xq Ñ Funpt1u,Xq » X given by evaluation at t1u Ď ∆1. This coCartesian fibration is classified by a functor χ :X Ñ PrL, which assigns to each object U PX the 8-topos X{U. We claim that this functor preserves small colimits. To prove this, it suffices to show that the opposite functor χ : Xop Ñ PrLop » PrR preserves small limits; this functor classifies p as a Cartesian fibration, and is a limit diagram by virtue of Theorems HTT.6.1.3.9 and HTT.5.5.3.18 together Proposition HTT.5.5.3.13 . For any presentable 8-category C, we obtain a new functor given by the composition
X Ñχ PrLbÑCPrL,
2.1. SHEAVES ON A SPECTRALLY RINGED 8-TOPOS 191 which assigns to each object U PX the 8-categoryShvCpX{Uq (see Remark 1.3.1.6). The same reasoning yields a limit-preserving functor Xop ÑPrLop »PrR which, by virtue of Theorem HTT.5.5.3.18 , gives a limit-preserving functor χrCs:XopÑyCat8.
The evident forgetful functor Mod Ñ CAlg determines a natural transformation of functorsχrMods ÑχrCAlgsfromXop to yCat8. Every sheafO ofE8-rings onX determines a natural transformation ˚ ÑχrCAlgs, where˚ denotes the constant functorXop ÑyCat8 taking the value ∆0. Forming a pullback diagram
φ //
χrMods
˚ //χrCAlgs,
we obtain a new limit-preserving functor φ:Xop Ñ yCat8. Unwinding the definitions, we see thatφ assigns to each objectU PX the 8-category ModO|U, and to every morphism f :U ÑV in X the associated pullback functor f˚ : ModO|
V Ñ ModO|
U. SinceχrCAlgs and χrMods preserve small limits, so doesφ.
2.1.1 The t-Structure on ModO
LetX be an8-topos and letO be a sheaf ofE8-rings onX. We will say that aO-module F is connective if it is connective when viewed as a sheaf of spectra on X: that is, if the homotopy sheaves πnF vanish for n ă 0. We let ModcnO denote the full subcategory of ModO spanned by the connective O-modules. This notion is primarily useful in the case where the sheafO is itself connective.
Proposition 2.1.1.1. LetX be an 8-topos and let O be a connective sheaf of E8-rings on X. Then:
paq The 8-category ModO admits a t-structure pModcnO,pModOqď0q, where pModOqď0 is the inverse image of ShvpSpqď0 under the forgetful functor θ: ModO ÑShvSppXq. pbq The t-structure on pModcnO,pModOqď0q is compatible with the symmetric monoidal
structure on ModO. In other words, the full subcategory pModcnO ĎModO contains the unit object of ModO and is closed under the relative tensor product bO.
pcq The t-structure pModcnO,pModOqď0q is right complete and compatible with filtered colimits (in other words, the full subcategorypModOqď0 is stable under filtered colimits in ModO).
Proof. We first provepaq. It follows immediately from the definitions that the full subcategory ModcnO ĎModOis closed under small colimits and extensions. Using Proposition HA.1.4.4.11 ,
we deduce the existence of an accessible t-structureppModcnO,Mod1Oqon ModO. To complete the proof, it will suffice to show that Mod1O “ pModOqď0. Suppose first that F PMod1O. Then the mapping space MapModOpG,Fq is discrete for every object G P pModOqě0. In particular, for every connective sheaf of spectra M P ShvSppXqě0, the mapping space MapMod
OpMbO,Fq » MapShv
SppXqpM, θpFqq is discrete, so that θpFq P ShvSppXqď0
and therefore F P pModOqď0.
Conversely, suppose that F P pModOqď0. We wish to prove that F P Mod1O. Let C denote the full subcategory of ModO spanned by those objects G P ModO for which the mapping space MapMod
OpG,Fq is discrete. We wish to prove that C contains ModcnO. Conditionp3q shows thatθ induces a functor ModcnO ÑShvSppXqě0 which is conservative and preserves small colimits; moreover, this functor has a left adjoint F, given informally by the formula FpMq »ObM. Using Proposition HA.4.7.3.14 , we conclude that ModcnO is generated under geometric realizations by the essential image of F. SinceC is stable under colimits, it will suffice to show that C contains the essential image of F. Unwinding the definitions, we are reduced to proving that the mapping space
MapMod
OpFpMq,Fq »MapShv
SppXqpM, θpFqq
is discrete for every connective sheaf of spectraMonX, which is equivalent to our assumption thatθpFq PShvSppXqď0. This completes the proof ofpaq.
We now prove pbq. The unit object of ModO is the sheaf O (regarded as a module over itself), which is connective by assumption. We claim that for every pair of objects F,G PModcnO, the relative tensor productFbOG is also connective. Note that, as a sheaf of spectra, the relative tensor prouctFbOG can be identified with the geometric realization of a simplicial object whose entires are iterated tensor productsFbOb ¨ ¨ ¨ bObG. Since F,G, andO are connective, the above tensor product is connective (Proposition 1.3.4.7);
because ModSppXqě0 is closed under colimits we conclude that FbOG is connective.
We now prove pcq. Since the forgetful functorθ: ModO ÑShvSppXq preserves filtered colimits (Proposition 2.1.0.3) and the full subcategory ShvSppXqď0 ĎShvSppXq is closed under filtered colimits (Proposition 1.3.2.7), it follows that the full subcategorypModOqď0Ď ModO is closed under filtered colimits. By virtue of Proposition HA.1.2.1.19 , to show that the t-structure pModcnO,pModOqď0q is right-complete, it is sufficient to show that it is right-separated: that is, that the intersection
č
pModOqď´n»θ´1pč
ShvSppXqď´nq
contains only zero objects of ModO. This follows from the conservativity of the functor θ, since the intersectionŞ
ShvSppXqď´n contains only zero objects of ShvSppXq (Proposition 1.3.2.7).
2.1. SHEAVES ON A SPECTRALLY RINGED 8-TOPOS 193 Warning 2.1.1.2. The t-structure of Proposition 2.1.1.1 is generally not left complete or even left separated. However, it is left complete (left separated) whenever the t-structure pModSppXqě0,ModSppXqď0q is left complete (left separated). For example, ifX is hyper-complete, then ModO is left separated; if Postnikov towers inX are convergent, then ModO is left complete.
2.1.2 The Derived 8-Category of Mod♥O
Let X be an 8-topos and let O be a connective sheaf of E8-rings on X. We let Mod♥O ĎModO denote the heart of the t-structure described in Proposition 2.1.1.1 (that is, the intersection ModcnO XpModOqď0).
Remark 2.1.2.1. Unwinding the definitions, we can identifyπ0O as a commutative ring object in the underlying toposX♥, and Mod♥O with the abelian category ofpπ0Oq-module objects ofX♥.
According to Remark HA.1.3.5.23 , the inclusionι: Mod♥O ãÑModO admits an essentially unique extension to a t-exact functorDpMod♥Oqă8 Ñ ModO, where DpMod♥Oqă8 denotes the derived 8-category of Mod♥O (see §HA.1.3.2 ). If the 8-topos X is hypercomplete, then the t-structure on ModO is left separated (Warning 2.1.1.2), so Theorem C.5.4.9 implies that ιadmits an essentially unique extension to a colimit-preserving t-exact functor ρ:DpMod♥Oq ÑModO.
Theorem 2.1.2.2. Let pX,Oq be a spectrally ringed 8-topos satisfying the following condi-tions:
paq The structure sheafO is discrete.
pbq For each object X PX, there exists an effective epimorphism U ÑX where U is a discrete object ofX.
pcq The8-topos X is hypercomplete.
Then the functor ρ:DpMod♥Oq ÑModO supplied by Theorem C.5.4.9 is an equivalence of 8-categories.
Before giving the proof of Theorem 2.1.2.2, let us note some of its consequences.
Corollary 2.1.2.3. Let pX,Oq be a spectrally ringed 8-topos satisfying the following con-ditions:
paq The structure sheafO is discrete.
pbq For each object X PX, there exists an effective epimorphism U ÑX where U is a discrete object ofX.
Then the inclusion Mod♥O ãÑ ModO extends to a fully faithful embedding ι:DpMod♥Oq ãÑ ModO, whose essential image is the full subcategory of hypercomplete O-module objects of ShvSppXq.
Proof. Let f˚ :X ÑXhyp be a left adjoint to the inclusion. Then the spectrally ringed 8-topospXhyp, f˚Oqsatisfies the hypotheses of Theorem 2.1.2.2, so that we have an equivalence of 8-categories
DpMod♥Oq »DpMod♥f˚Oq »Modf˚OpShvSppXhypqq.
We now define ιto be the composition of this equivalence with the pushforward functor f˚ : Modf˚OpShvSppXhypqq ÑModOpShvSppXqq “ModO
(which is a fully faithful embedding whose essential image is spanned by the hypercomplete O-module objects of ShvSppXq).
Corollary 2.1.2.4. Let pX,Oq be a spectrally ringed 8-topos satisfying the following con-ditions:
paq The structure sheafO is discrete.
pbq For each object X PX, there exists an effective epimorphism U ÑX where U is a discrete object ofX.
Then the functorι:DpMod♥Oqă8ÑModO supplied by Remark HA.1.3.5.23 is a fully faithful embedding, whose essential image is the unionŤ
ně0pModOqďn.
We begin by proving Theorem 2.1.2.2 in the special case where X is a presheaf8-topos.
Proposition 2.1.2.5. Let C be a category, let O P FunpCop,CAlg♥q be a presheaf of commutative rings on C, and let ModO “ ModOpFunpCop,Spqq be the 8-category of O -modules on the hypercomplete8-toposFunpCop,Sq. Then the canonical mapρ:DpMod♥Oq Ñ ModO (supplied by Corollary ??) is an equivalence of 8-categories.
Proof. For each object C P C, let hC : Cop Ñ Set Ă S be the functor represented by C (given on objects by the formula hCpDq “HomCpD, Cq) and letFC PModO be the tensor
product ObΣ8`phCq, given on objects by the formulaFCpDq »À
η:DÑCOpDq. Note that FC belongs to the heart of ModO.
For any object G PModO, we have a canonical homotopy equivalence MapMod
OpFC,Gq » MapFunpCop,SpqpΣ8`hC,Gq
» MapFunpCop,SqphC,Ω8Gq
» Ω8GpCq.
2.1. SHEAVES ON A SPECTRALLY RINGED 8-TOPOS 195