nLab quasicoherent sheaf



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In higher category theory




A quasicoherent sheaf of modules (often just “quasicoherent sheaf”, for short) is a sheaf of modules over the structure sheaf of a ringed space that is locally presentable in that it is locally the cokernel of a morphism of free modules.

For comparison, by the Serre-Swan theorem a vector bundle on a suitable ringed space is equivalently encoded in its sheaf of sections which is even locally free and projective. In this sense quasicoherent sheaves of modules are a generalization of vector bundles. The category of vector bundles is too small to be closed under various natural operations like kernels, direct images and alike. In particular, it is not an abelian category. The category of all 𝒪\mathcal{O}-modules and especially its full subcategory of quasicoherent sheaves of 𝒪\mathcal{O}-modules are better behaved in that respect.

There are several different but equivalent ways to define and think of quasicoherent sheaves.

A very concrete definition characterizes quasicoherent sheaves as those that are, while not locally free, locally the cokernel of a morphism of free module sheaves. This is the definition given in the section

below. It makes very manifest how passing from vector bundles to quasicoherent sheaves adds in the cokernels that are missing in the category of vector bundles.

But it turns out that there is a more abstract, more sheaf theoretic reformulation of this definition: if we think of the underlying space as a (pre)sheaf (as motivated at motivation for sheaves, cohomology and higher stacks) we find that a quasicoherent sheaf on a space is given by an assignment of a module to each plot, such that the pullback of these modules is given, up to coherent isomorphism, by tensoring over the corresponding rings. This is described in the section

and in more details in the section

The tensoring operation appearing here is that defining the pullback operations in the stack that classifies the canonical bifibration ModCRingMod \to CRing of modules over rings. In view of this, one finds that this definition, in turn, is equivalent to a very fundamental definition:

with QC:=()Mod:RingCatQC := (-)Mod : Ring \to Cat the functor that sends a ring to its category of modules, one finds that the category of quasicoherent sheaves on a space XX is simply the hom-object

QC(X):=Hom(X,QC) QC(X) := Hom(X,QC)

in the corresponding 2-category of category-valued (pre)sheaves, i.e (pre)stacks. This is the perspective described in

below. By the equivalence between Grothendieck fibrations and pseudofunctors this in turn is directly equivalent to the identification of QC(X)QC(X) with the category of cartesian functors between the category of elements of XX and ModMod. This is described in

This definition, finally, provides a powerful nPOV on quasicoherent sheaves: all notions involved, sheaf, stack, morphism of stacks, have natural, immediate and well understood generalizations to higher category theory. Therefore this last definition immediately generalizes to a definition of quasicoherent \infty-sheaves or “derived” quasicoherent sheaves, such as they appear for instance in geometric ∞-function theory. This is discussed in the section


As locally presentable modules

Let (X,O X)(X,O_X) be a ringed space or, more generally, a ringed site.

A quasicoherent sheaf of O XO_X-modules on XX is a sheaf \mathcal{E} of O XO_X-modules that is locally a cokernel of a morphism of free modules.

This means: there is a cover {U α} αA\{U_\alpha\}_{\alpha\in A} of XX by open sets such that for every α\alpha there exist I αI_\alpha and J αJ_\alpha (not necessarily finite) and an exact sequence of sheaves of O XO_X-modules of the form

O X I α| U αO X J α| U α| U α0, O_X^{I_\alpha}|_{U_\alpha} \to O_X^{J_\alpha}|_{U_\alpha} \to \mathcal{E}|_{U_\alpha}\to 0,

This should be viewed as a local presentation of \mathcal{E}.

If I α,J αI_\alpha, J_\alpha can be chosen finite and \mathcal{E} is of finite type then the quasicoherent sheaf is a coherent sheaf. (See there for details.) However , coherent sheaves are ill-behaved for a general ringed space, and even general schemes; they behave well on Noetherian schemes.

Replacing covers by open sets, by covers of a terminal object in a site, the definition extends to ringed sites with a terminal object; the restrictions of O XO_X-modules should be replaced by pullbacks. There are generalizations for algebraic stacks, ind-schemes, diagrams of schemes (for example configuration schemes of V. Lunts, obtained by gluing along closed embeddings of schemes; simplicial schemes) and so on.

As sheaves on Aff/XAff/X

There is an equivalent reformulation of the above in terms of sheaves of 𝒪\mathcal{O}-modules on the site Aff/XAff/X of affine schemes over XX.

This is the over category whose objects are morphism (of schemes) of the form SpecAXSpec A \to X and whose morphisms are commuting triangles

SpecA f SpecB a b X. \array{ Spec A &&\stackrel{f}{\to}&& Spec B \\ & {}_a\searrow && \swarrow_b \\ && X } \,.

Then: a quasicoherent sheaf on (X,𝒪 X)(X, \mathcal{O}_X) is a sheaf NN of 𝒪 X\mathcal{O}_X-modules on Aff/XAff/X such that for each morphism ff as above the restriction morphism

N(f):N(b)N(a) N(f) : N(b) \to N(a)

extends to an isomorphism

N(b) f *AN(a) N(b) \otimes_{f^*} A \stackrel{\simeq}{\to} N(a)

of AA-modules.

For a very explicit statement of this see for instance page 13 of

  • Paul Goerss, Topological modular forms (aftern Hopkins, Miller, and Lurie) (arXiv)

See also a very precise and detailed treatment in

  • Dmitri Orlov, Quasi-coherent sheaves in commutative and non-commutative geometry, Izv. RAN. Ser. Mat., 2003, 67:3, 119–138 (preprint versions: pdf dvi, ps)

Direct definition for presheaves of sets on Aff

Here is a more detailed way to say again what the above paragraph said.

Let Aff=CRing opAff = CRing^{op}; recall the fibered category ModCRing opMod\to CRing^{op} where for each f:ABf:A\to B in CRingCRing the inverse image functor is f *=B A: AMod BModf^*=B\otimes_A - :{}_A Mod\to {}_B Mod. Then the identity functor CRingCRingCRing\to CRing can be interpreted as the presheaf of rings and is denoted by OO (the “structure sheaf”). An OO-module is a presheaf of OO-modules. Usually some Grothendieck topology is given and one asks for sheaves in fact. We can Yoneda extend OO-modules to presheaves. We now define quasicoherent sheaves of OO-modules on an arbitrary presheaf XX on AffAff, viewed as a covariant functor on CRingCRing.

A quasicoherent sheaf of OO-modules on XX is a rule assigning to any ϕX(A)\phi\in X(A) an AA-module M ϕ=M A,ϕM_\phi = M_{A,\phi} and to any morphism f:ABf:A\to B in CRingCRing an isomorphism θ f,ϕ:f *(M ϕ)M X(f)(ϕ)\theta_{f,\phi}:f^*(M_\phi)\to M_{X(f)(\phi)} such that for any composable pair AfBgCA\stackrel{f}\to B\stackrel{g}\to C and any ϕX(A)\phi\in X(A) the cocycle condition

θ gf,ϕα fg=θ g,X(f)(ϕ)g *(θ f,ϕ):g *f *(M ϕ)M X(gf)(ϕ) \theta_{g\circ f,\phi}\circ \alpha_{fg} = \theta_{g,X(f)(\phi)}\circ g^*(\theta_{f,\phi})\colon g^* f^*(M_\phi)\to M_{X(g\circ f)(\phi)}

holds, where α fg:g *f *(M ϕ)(gf) *(M ϕ)\alpha_{fg}:g^* f^*(M_\phi)\to (g\circ f)^*(M_\phi) is the canonical isomorphism which is part of the data of the (covariant) pseudofunctor A AModA\to {}_A Mod, ff *f\mapsto f^*.

Notice that if X=h C=h SpecCX = h^C = h_{Spec C} is (co)representable presheaf, then ϕ[A,X] Pshv(Aff)=[C,A] CRing\phi\in [A,X]_{Pshv(Aff)}=[C,A]_{CRing} is the same as a morphism ϕ op:CA\phi^{op}:C\to A of rings; restricting the quasicoherent sheaf to SpecASpec A along ϕ:SpecAX\phi:Spec A\to X and taking the global sections over AA, would give the AA-module M ϕM_\phi.

Clearly, AffAff and OO can be much generalized. For example, rings may be noncommutative or one can take category opposite to the category of monads in SetSet and an arbitrary (not identity) presheaf DD of monads in SetSet; the extension of scalars for monads gives an inverse image functor for Eilenberg-Moore categories. Durov’s construction of quasicoherent sheaves for monads in SetSet is an example where commutative algebraic monads are used; the theory of quasicoherent sheaves of DD-modules (“OO-modules with integrable connection”) is another. Instead setups involving operads, higher operads and alike can be used as well; commutativity condition is useful if one wants a monoidal category of quasicoherent sheaves.

As homs into the stack of modules

The above definition may be further re-interpreted as follows.


On the site Aff=CRing opAff = CRing^{op}, let

QC:CRingCat QC : CRing \to Cat
(SpecSf opSpecR)(RModS fSMod) (Spec S \stackrel{f^{op}}{\to} Spec R) \mapsto (R Mod \stackrel{S \otimes_{f} -}{\to} S Mod)

be the (pseudo)functor (stack) corresponding to the canonical Grothendieck fibration of modules ModCRingMod \to CRing. Its right Kan extension through the 2-Yoneda embedding Y:CRing op[CRing,Cat]Y : CRing^{op} \hookrightarrow [CRing,Cat] is given on a presheaf X:CRingSetX : CRing \to Set by the hom-object

QC(X):=(Ran YQC)(X):=[CRing,Cat](X,QC). QC(X) := (Ran_Y QC)(X) := [CRing,Cat](X,QC) \,.

When XX is the functor represented by a scheme, then QC(X)QC(X) is the category of quasicoherent sheaves on XX, as defined above.

We now explain the above statement in detail and thereby prove it.

Let C=C = Ring op{}^{op} be the category of (commutative, unital) rings. For RR a ring write SpecRSpec R for it regarded as an object of CC. Write Specf=f op:Spec(S)Spec(R)Spec f = f^{op} : Spec(S) \to Spec(R) for the morphism in Ring opRing^{op} corresponding to the map f:RSf : R \to S of commutative rings.

Consider the 2-category of (pre)stacks on CC. The canonical module bifibration p:ModRingp : Mod \to Ring of the category of modules over all rings is the bifibration whose fibered part corresponds to the (pre)stack QC[C op,Cat]QC \in [C^{op},Cat] given on objects by

QC:SpecRRMod QC : Spec R \mapsto R Mod

and on morphisms by

QC:(SpecSf opSpecR)(RModS fSMod), QC : (Spec S \stackrel{f^{op}}{\to} Spec R) \mapsto (R Mod \stackrel{S\otimes_{f} }{\to} S Mod) \,,

where on the right we have the functor that sends any RR-module NN to the tensor product over SS with the RR-SS-bimodule S= SS RS = {}_S S_R with its canonical left SS-action and with the right RR-action induced by the ring homomorphism ff.

One may think of this as the stack of generalized algebraic vector bundles:

the operation S f:RModSModS \otimes_{f} - : R Mod \to S Mod corresponds to the pullback of bundles along a morphism of the underlying spaces. (See for instance the discussion of monadic descent at Sweedler coring for more on this.)

We may right Kan extend the 2-functor QC:CRing opCatQC : CRing^{op} \to Cat through the Yoneda embedding CRing op[CRing,Cat]CRing^{op} \hookrightarrow [CRing,Cat] to get a definition of QCQC on arbitrary presheaves.

CRing op QC Cat Y Ran YQC [CRing,Cat]. \array{ CRing^{op} &\stackrel{QC}{\to}& Cat \\ {}^{Y}\downarrow & \nearrow_{\mathrlap{Ran_Y QC}} \\ [CRing,Cat] } \,.

Consider X[C op,Set]X \in [C^{op},Set] any (pre)sheaf on CC. This may be the presheaf represented by a scheme, but for the purposes of the definition of QCQC it may be much more generally any presheaf.

By the general formula for Kan extension in terms of a weighted limit given by an end we have

Ran YQC:X RRing([Ring,Cat] op(X,Y(R))) QC(R) Ran_Y QC : X \mapsto \int_{R \in Ring} ([Ring,Cat]^{op}(X,Y(R)))^{QC(R)}

which using the Yoneda lemma is

= RCRing[CRing,Cat](X(R),QC(R)). \cdots = \int_{R \in CRing} [CRing,Cat](X(R), QC(R)) \,.

This is the end-formula for the hom-object in an enriched functor category [C op,Cat][C^{op},Cat], hence this is nothing but the category of (pseudo)natural transformations between the 2-functor XX and the 2-functor QCQC.

We write for short

QC(X):=(Ran YQC)(X):=[C op,Cat](X,QC). QC(X) := (Ran_Y QC)(X) := [C^{op},Cat](X,QC) \,.

This definition of “generalized vector bundles” on arbitrary presheaves is entirely analogous to the definition of differential forms on arbitrary presheaves, that is discussed in some detail for instance in the entry on rational homotopy theory.

We claim that the category QC(X)QC(X) is the category of quasicoherent sheaves on XX as defined by other means above, whenever that other definition applies to XX.

To see this, straightforwardly unwrap the definition: an object NN in QC(X)=[C op,Cat](X,QC)QC(X) = [C^{op},Cat](X,QC) is a pseudonatural transformation of 2-functors N:XQCN : X \to QC, where XX is regarded as a 2-functor by the canonical embedding disc:SetCatdisc : Set \hookrightarrow Cat that regards a set as a discrete category.

The components of NN are

  • for each SpecRRing opSpec R \in Ring^{op} a functor N| SpecR:X(R)QC(R)=RModN|_{Spec R} : X(R) \to QC(R) = R Mod:

    this functor picks one RR-module N(r)RModN(r) \in R Mod for each plot (r:SpecRX)X(SpecR)(r : Spec R \to X) \in X(Spec R);

  • for each morphism f:SpecASpecBf : Spec A \to Spec B a pseudonaturality square

    X(SpecB) X(f) X(SpecA) N| SpecB Γ(f) N| SpecA BMod f *A AMod \array{ X(Spec B) &\stackrel{X(f)}{\to}& X(Spec A) \\ {}^{\mathllap{N|_{Spec B}}}\downarrow &{}^{\Gamma(f)}\swArrow_{\simeq}& \downarrow^{\mathrlap{N|_{Spec A}}} \\ B Mod &\underset{- \otimes_{f^*} A}{\to}& A Mod }

    in Cat (these are subject to coherence conditions). This unwraps to the following data:

    • the component functors N| SpecAN|_{Spec A} provide an assignment aN(a)a \mapsto N(a) of modules N(a)N(a) to each plot (a:SpecAX)X(SpecA)(a : Spec A \to X) \in X(Spec A);

    • these assignments form a presheaf on the overcategory Aff/XAff/X by taking the restriction morphism

      N(f):N(b)N(a) N(f) : N(b) \to N(a)

      to be that underlying the components of the natural isomorphism in the above diagram

      N(b) f *AN(a), N(b)\otimes_{f^*} A \stackrel{\simeq}{\to} N(a) \,,

      i.e. the restriction of this morphism to (n,1)(n,1).

  • for each tuple of composable morphisms

    SpecB f g SpecA SpecC \array{ && Spec B \\ & {}^f\nearrow && \searrow^g \\ Spec A &&\to&& Spec C }

    a pseudo-naturality prism equation relating, N(f)N(f), N(g)N(g) and N(gg)N(g\circ g). The present author is too lazy to write out the diagram in detail, but it is of precisely the kind described in great detail for instance in the entry on group cohomology. Under the above identification, this yields the cocycle condition mentioned in the above definitions.

This way, the transformation N:XQCN : X \to QC defines manifestly a quasicoherent sheaf on Aff/XAff/X in the sense of the definition in the above section As sheaves on Aff/X. Conversely, every quasicoherent sheaf according to that definition gives rise to a transformation N:XQCN : X \to QC under this prescription.

As cartesian morphisms of fibrations

By the equivalence between pseudofunctors RingCatRing \to Cat and Grothendieck fibrations FRing opF \to Ring^{op} induced by the Grothendieck construction, the above may equivalently be reformulated as follows.

Recall from the discussion at Grothendieck fibration that the equivalence in question is between the following two bicategories:

  • on the one hand the bicategory whose objects are pseudofunctors RingCatRing \to Cat, whose morphisms are pseudonatural transformations, and whose 2-morphisms are modifications of these

  • on the other hand the bicategory whose objects are Grothendieck fibrations FRing opF \to Ring^{op}, whose morphism are cartesian functors

    F 1 F 2 Ring op \array{ F_1 &&\to&& F_2 \\ & \searrow && \swarrow \\ && Ring^{op} }

    and whose 2-morphisms are natural transformations between these.

Recall furthermore that for X:RingCatX : Ring \to Cat an ordinary presheaf, i.e. a pseudofunctor that factors through an ordinary functor RingSetRing \to Set via the inclusion SetCatSet \to Cat, the Grothendieck fibration associated with XX is the category of elements Ring op/XRing^{op}/X of XX.

Recall furthermore that by definition, the pseudofunctor QC:RingCatQC : Ring Cat is the one corresponding to the Grothendieck fibration Mod opRing opMod^{op} \to Ring^{op}.

Therefore, by the above equivalence of 2-categories, we find that the category of functors [Ring,Cat](X,QC)[Ring,Cat](X,QC) is equivalent to the category of cartesian functors over Ring OpRing^{Op}, CartFunc E(Ring op/X,Mod op)CartFunc_E(Ring^{op}/X,Mod^{op})

QC(X)[Ring,Cat](X,QX)CartFunc(Ring op/X,Mod op). QC(X) \simeq [Ring,Cat](X,QX) \simeq CartFunc(Ring^{op}/X, Mod^{op}) \,.

In this form quasicoherent sheaves on XX are conceived for instance in paragraph 1.1.5 of

Here, as in the above discussion, the fibered category of modules can be replaced by a more general fibered category π:\pi: \mathcal{F}\to\mathcal{B}. Then the category of quasicoherent modules in this fibered category is the category opposite to the category of cartesian sections of π\pi. This viewpoint is used by Rosenberg-Kontsevich in their preprint on noncommutative stacks (dvi, ps).

Given a category Aff\mathrm{Aff} of affine schemes (opposite to the category of rings) equipped with some subcanonical pretopology one considers the stack of OO-modules over Aff\mathrm{Aff}: the fiber over a ring RR, it assigns the category Qcoh(SpecR)Qcoh(\mathrm{Spec}\,R). Now given any stack on a subcanonical site, one defines the fiber over a sheaf on it so that the fiber over a representable sheaf is equivalent to the fiber over its representing object. There is a canonical way to do this (will write later about it – Zoran); this is in particular a source of a definition QcohQcoh on an ind-scheme. On ind-schemes Beilinson and Drinfel’d in

  • A. Beilinson, V. Drinfel’d, Quantization of Hitchin’s integrable system and Hecke eigensheaves on Hitchin system, preliminary version (pdf)

consider two variants: a less important variant of quasicoherent O X pO_X^p-modules (existing in bigger generality) and more delicate variant of quasicoherent O X !O^!_X-modules defined for “reasonable ind-schemes”; one of the differences is which functors play the role of pullbacks. In particular, these notions apply for a rather general variant of the category of formal schemes.

Quasicoherent modules in higher/derived geometry

The last definition has a straightforward generalization to various higher geometry setups, such as derived schemes and other generalized schemes.

By maps into the stack QCohQCoh

For instance the notion of quasicoherent sheaves generalized to derived stacks on the site of simplicial rings as described at geometric ∞-function theory is obtained, we claim, simply by taking QC:SRing(,1)CatQC : SRing \to (\infty,1)Cat to be the functor that assigns the (∞,1)-category for modules over a simplicial ring to any simplicial ring, and then setting for any derived stack XX

QC(X):=Hom(X,QC). QC(X) := Hom(X,QC) \,.

Moreover, using the theorem described at tangent (∞,1)-category, that the bifibration of modules over simplicial rings is nothing but the tangent (∞,1)-category of SRingSRing, one sees that all this is a special case of an even much more general abstract nonsense:

for any presentable (∞,1)-category site CC whatsoever, we have the tangent (∞,1)-category fibration T CCT_C \to C. With the (∞,1)-functor classifying it denoted QC:C op(,1)CatQC : C^{op} \to (\infty,1)Cat we may adopt for any ∞-stack X:C opGrpdX : C^{op} \to \infty Grpd the definition

QX(X):=[C op,(,1)Cat](X,QC) QX(X) := [C^{op},(\infty,1)Cat](X,QC)

as a definition of generalized \infty-vector bundles on XX. This general nonsense is considered further at ∞-vector bundle. Concrete realizations are discussed at quasicoherent ∞-stack.

As extensions of the structure sheaf

In (LurieQC, section 2.2, section 2.3) the following definition is given.

Let 𝒢\mathcal{G} be a geometry (for structured (∞,1)-toposes). Let

𝒢 mod(T(𝒢 op)) cpt op \mathcal{G}^{mod} \coloneqq (T (\mathcal{G}^{op}))_{cpt}^{op}

be the opposite of the full sub-(∞,1)-category on the compact objects of the tangent (∞,1)-category of its opposite (∞,1)-category.

For instance for E-∞ geometry we have 𝒢=CRing \mathcal{G} = CRing_\infty is the (∞,1)-category of E-∞ rings with etale morphisms as admissible maps.

(LurieQC, above Notation 2.2.4)

Then the canonical (∞,1)-functor

𝒢𝒢 mod \mathcal{G} \longrightarrow \mathcal{G}^{mod}

is a morphism of discrete geometries.

For 𝒳\mathcal{X} an (∞,1)-topos, a left exact (∞,1)-functor

𝒪:𝒢𝒳 \mathcal{O} \colon \mathcal{G} \longrightarrow \mathcal{X}

constitutes a 𝒢\mathcal{G}-structure sheaf and makes (𝒳,𝒪)(\mathcal{X}, \mathcal{O}) be a 𝒢\mathcal{G}-structured (∞,1)-topos. A left exact extension of this

𝒢 𝒪 𝒳 (𝒪,) 𝒢 mod \array{ \mathcal{G} &\stackrel{\mathcal{O}}{\longrightarrow}& \mathcal{X} \\ \downarrow & \nearrow_{\mathrlap{(\mathcal{O}, \mathcal{F})}} \\ \mathcal{G}^{mod} }

exhibits a sheaf \mathcal{F} of 𝒪\mathcal{O}-modules on 𝒳\mathcal{X}.

(LurieQC, notation 2.2.4)

Now if (𝒳,𝒪)(\mathcal{X},\mathcal{O}) is locally representable structured (infinity,1)-topos then such an 𝒪\mathcal{O}-module \mathcal{F} is quasi-coherent if also (𝒳,(𝒪,))(\mathcal{X}, (\mathcal{O}, \mathcal{F})) is locally representable.

(LurieQC, def. 2.3.6)

Synthetic definition using the internal language

Let XX be a scheme. Recall that the big Zariski topos of XX is the topos of sheaves over Sch/XSch/X (or, more precisely, the affine schemes over XX which are locally of finite presentation). In this topos, there is a local ring 𝔸 1\mathbb{A}^1, the sheaf mapping an XX-scheme TT to 𝒪 T(T)\mathcal{O}_T(T).

A sheaf NN of modules over 𝔸 1\mathbb{A}^1 is quasicoherent if and only if, from the internal point of view of the big Zariski topos, the canonical map

N 𝔸 1AHom(Hom 𝔸 1Alg(A,𝔸 1),N) N \otimes_{\mathbb{A}^1} A \longrightarrow Hom(Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1), N)

is bijective for all finitely presented 𝔸 1\mathbb{A}^1-algebras AA. The outer Hom set is the set of all maps from the set Hom 𝔸 1Alg(A,𝔸 1)Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1) to the (underlying set of) NN.

This characterization has a geometric interpretation. The set Hom 𝔸 1Alg(A,𝔸 1)Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1) deserves the name “spectrum of AA”, since it consists of what classically is known as the (𝔸 1\mathbb{A}^1-)rational points of AA. Furthermore, if AA is induced from a sheaf of 𝒪 X\mathcal{O}_X-modules, then the object of the Zariski topos which is described by this set-theoretical expression coincides with the functor of points of the relative spectrum of that sheaf.

The set Hom(Hom 𝔸 1Alg(A,𝔸 1),N)Hom(Hom_{\mathbb{A}^1-Alg}(A, \mathbb{A}^1), N) is therefore the set of all NN-valued functions on the spectrum of AA. An element of N 𝔸 1AN \otimes_{\mathbb{A}^1} A gives rise to such a function: associate to a pure tensor xfx \otimes f the function φf(φ)x\varphi \mapsto f(\varphi) x.

In a synthetic/algebraic context, there should be no more functions than those which result from this construction. This is what the characterization expresses.


Quasicoherent sheaves over affine schemes

Given an affine scheme X=SpecRX=\mathrm{Spec}\,R (where RR is a commutative unital ring), the affine Serre theorem establishes the equivalence of the category Qcoh(SpecR)Qcoh(\mathrm{Spec}\,R) of quasicoherent sheaves (in Zariski topology) and the category of RR-modules. Similarly on a projective scheme of the type Proj(A)Proj(A) where AA is a nonnegatively graded ring, the (projective) Serre theorem establishes the equivalence of Qcoh(Proj(A))Qcoh(\mathrm{Proj}\,(A)) and the localization of the category of graded AA-modules by the subcategory of modules of finite length (and similarly, of coherent sheaves and graded AA-modules of finite type modulo finite-length). These theorems are among basic motivating theorems for noncommutative algebraic geometry. An interesting in-depth comparison of the notions of quasi-coherent sheaves in commutative and noncommutative context are also in Orlov’s article quoted above.

The category of quasicoherent sheaves

In the case of general (commutative) schemes, every presheaf of O XO_X-modules which is quasicoherent in the sense of having local presentation as above, is in fact a sheaf. It is known that the category of quasicoherent sheaves of O XO_X-modules over any quasicompact quasiseparated scheme is a Grothendieck category and in particular has enough injective objects.



The category of D-modules on a space XX is equivalently that of quasicoherent sheaves on the corresponding deRham space.


Quasicoherent sheaves in E-∞ geometry (on “Spectral Schemes” over E-∞ rings) are discussed in

Their descent properties are discussed in

and a Grothendieck existence theorem for coherent sheaves in this higher context is discussed in

On a geometric stack

  • Leovigildo Alonso Tarrío, Ana Jeremías López, Marta Pérez Rodríguez, María J. Vale Gonsalves, A functorial formalism for quasi-coherent sheaves on a geometric stack, Expositiones Mathematicae 33:4 (2015) 452-501 doi

See also

Last revised on April 10, 2024 at 19:57:22. See the history of this page for a list of all contributions to it.