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A quasicoherent sheaf of modules (often just βquasicoherent sheafβ, 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 $\mathcal{O}$-modules and especially its full subcategory of quasicoherent sheaves of $\mathcal{O}$-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 $Mod \to CRing$ of modules over rings. In view of this, one finds that this definition, in turn, is equivalent to a very fundamental definition:
with $QC := (-)Mod : Ring \to Cat$ the functor that sends a ring to its category of modules, one finds that the category of quasicoherent sheaves on a space $X$ is simply the hom-object
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 $QC(X)$ with the category of cartesian functors between the category of elements of $X$ and $Mod$. 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 βderivedβ quasicoherent sheaves, such as they appear for instance in geometric β-function theory. This is discussed in the section
Let $(X,O_X)$ be a ringed space or, more generally, a ringed site.
A quasicoherent sheaf of $O_X$-modules on $X$ is a sheaf $\mathcal{E}$ of $O_X$-modules that is locally a cokernel of a morphism of free modules.
This means: there is a cover $\{U_\alpha\}_{\alpha\in A}$ of $X$ by open sets such that for every $\alpha$ there exist $I_\alpha$ and $J_\alpha$ (not necessarily finite) and an exact sequence of sheaves of $O_X$-modules of the form
This should be viewed as a local presentation of $\mathcal{E}$.
If $I_\alpha, J_\alpha$ 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 $O_X$-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.
There is an equivalent reformulation of the above in terms of sheaves of $\mathcal{O}$-modules on the site $Aff/X$ of affine schemes over $X$.
This is the over category whose objects are morphism (of schemes) of the form $Spec A \to X$ and whose morphisms are commuting triangles
Then: a quasicoherent sheaf on $(X, \mathcal{O}_X)$ is a sheaf $N$ of $\mathcal{O}_X$-modules on $Aff/X$ such that for each morphism $f$ as above the restriction morphism
extends to an isomorphism
of $A$-modules.
For a very explicit statement of this see for instance page 13 of
See also a very precise and detailed treatment in
Here is a more detailed way to say again what the above paragraph said.
Let $Aff = CRing^{op}$; recall the fibered category $Mod\to CRing^{op}$ where for each $f:A\to B$ in $CRing$ the inverse image functor is $f^*=B\otimes_A - :{}_A Mod\to {}_B Mod$. Then the identity functor $CRing\to CRing$ can be interpreted as the presheaf of rings and is denoted by $O$ (the βstructure sheafβ). An $O$-module is a presheaf of $O$-modules. Usually some Grothendieck topology is given and one asks for sheaves in fact. We can Yoneda extend $O$-modules to presheaves. We now define quasicoherent sheaves of $O$-modules on an arbitrary presheaf $X$ on $Aff$, viewed as a covariant functor on $CRing$.
A quasicoherent sheaf of $O$-modules on $X$ is a rule assigning to any $\phi\in X(A)$ an $A$-module $M_\phi = M_{A,\phi}$ and to any morphism $f:A\to B$ in $CRing$ an isomorphism $\theta_{f,\phi}:f^*(M_\phi)\to M_{X(f)(\phi)}$ such that for any composable pair $A\stackrel{f}\to B\stackrel{g}\to C$ and any $\phi\in X(A)$ the cocycle condition
holds, where $\alpha_{fg}:g^* f^*(M_\phi)\to (g\circ f)^*(M_\phi)$ is the canonical isomorphism which is part of the data of the (covariant) pseudofunctor $A\to {}_A Mod$, $f\mapsto f^*$.
Notice that if $X = h^C = h_{Spec C}$ is (co)representable presheaf, then $\phi\in [A,X]_{Pshv(Aff)}=[C,A]_{CRing}$ is the same as a morphism $\phi^{op}:C\to A$ of rings; restricting the quasicoherent sheaf to $Spec A$ along $\phi:Spec A\to X$ and taking the global sections over $A$, would give the $A$-module $M_\phi$.
Clearly, $Aff$ and $O$ can be much generalized. For example, rings may be noncommutative or one can take category opposite to the category of monads in $Set$ and an arbitrary (not identity) presheaf $D$ of monads in $Set$; the extension of scalars for monads gives an inverse image functor for Eilenberg-Moore categories. Durovβs construction of quasicoherent sheaves for monads in $Set$ is an example where commutative algebraic monads are used; the theory of quasicoherent sheaves of $D$-modules (β$O$-modules with integrable connectionβ) 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.
The above definition may be further re-interpreted as follows.
On the site $Aff = CRing^{op}$, let
be the (pseudo)functor (stack) corresponding to the canonical Grothendieck fibration of modules $Mod \to CRing$. Its right Kan extension through the 2-Yoneda embedding $Y : CRing^{op} \hookrightarrow [CRing,Cat]$ is given on a presheaf $X : CRing \to Set$ by the hom-object
When $X$ is the functor represented by a scheme, then $QC(X)$ is the category of quasicoherent sheaves on $X$, as defined above.
We now explain the above statement in detail and thereby prove it.
Let $C =$Ring${}^{op}$ be the category of (commutative, unital) rings. For $R$ a ring write $Spec R$ for it regarded as an object of $C$. Write $Spec f = f^{op} : Spec(S) \to Spec(R)$ for the morphism in $Ring^{op}$ corresponding to the map $f : R \to S$ of commutative rings.
Consider the 2-category of (pre)stacks on $C$. The canonical module bifibration $p : Mod \to Ring$ of the category of modules over all rings is the bifibration whose fibered part corresponds to the (pre)stack $QC \in [C^{op},Cat]$ given on objects by
and on morphisms by
where on the right we have the functor that sends any $R$-module $N$ to the tensor product over $S$ with the $R$-$S$-bimodule $S = {}_S S_R$ with its canonical left $S$-action and with the right $R$-action induced by the ring homomorphism $f$.
One may think of this as the stack of generalized algebraic vector bundles:
the operation $S \otimes_{f} - : R Mod \to S Mod$ 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 $QC : CRing^{op} \to Cat$ through the Yoneda embedding $Ring^{op} \hookrightarrow [CRing,Cat]$ to get a definition of $QC$ on arbitrary presheaves.
Consider $X \in [C^{op},Set]$ any (pre)sheaf on $C$. This may be the presheaf represented by a scheme, but for the purposes of the definition of $QC$ 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
which using the Yoneda lemma is
This is the end-formula for the hom-object in an enriched functor category $[C^{op},Cat]$, hence this is nothing but the category of (pseudo)natural transformations between the 2-functor $X$ and the 2-functor $QC$.
We write for short
This definition of βgeneralized vector bundlesβ 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 $QC(X)$ is the category of quasicoherent sheaves on $X$ as defined by other means above, whenever that other definition applies to $X$.
To see this, straightforwardly unwrap the definition: an object $N$ in $QC(X) = [C^{op},Cat](X,QC)$ is a pseudonatural transformation of 2-functors $N : X \to QC$, where $X$ is regarded as a 2-functor by the canonical embedding $disc : Set \hookrightarrow Cat$ that regards a set as a discrete category.
The components of $N$ are
for each $Spec R \in Ring^{op}$ a functor $N|_{Spec R} : X(R) \to QC(R) = R Mod$:
this functor picks one $R$-module $N(r) \in R Mod$ for each plot $(r : Spec R \to X) \in X(Spec R)$;
for each morphism $f : Spec A \to Spec B$ a pseudonaturality square
in Cat (these are subject to coherence conditions). This unwraps to the following data:
the component functors $N|_{Spec A}$ provide an assignment $a \mapsto N(a)$ of modules $N(a)$ to each plot $(a : Spec A \to X) \in X(Spec A)$;
these assignments form a presheaf on the overcategory $Aff/X$ by taking the restriction morphism
to be that underlying the components of the natural isomorphism in the above diagram
i.e. the restriction of this morphism to $(n,1)$.
for each tuple of composable morphisms
a pseudo-naturality prism equation relating, $N(f)$, $N(g)$ and $N(g\circ g)$. 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 $N : X \to QC$ defines manifestly a quasicoherent sheaf on $Aff/X$ 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 $N : X \to QC$ under this prescription.
By the equivalence between pseudofunctors $Ring \to Cat$ and Grothendieck fibrations $F \to Ring^{op}$ 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 $Ring \to Cat$, whose morphisms are pseudonatural transformations, and whose 2-morphisms are modifications of these
on the other hand the bicategory whose objects are Grothendieck fibrations $F \to Ring^{op}$, whose morphism are cartesian functors
and whose 2-morphisms are natural transformations between these.
Recall furthermore that for $X : Ring \to Cat$ an ordinary presheaf, i.e. a pseudofunctor that factors through an ordinary functor $Ring \to Set$ via the inclusion $Set \to Cat$, the Grothendieck fibration associated with $X$ is the category of elements $Ring^{op}/X$ of $X$.
Recall furthermore that by definition, the pseudofunctor $QC : Ring Cat$ is the one corresponding to the Grothendieck fibration $Mod^{op} \to Ring^{op}$.
Therefore, by the above equivalence of 2-categories, we find that the category of functors $[Ring,Cat](X,QC)$ is equivalent to the category of cartesian functors over $Ring^{Op}$, $CartFunc_E(Ring^{op}/X,Mod^{op})$
In this form quasicoherent sheaves on $X$ 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 $\pi: \mathcal{F}\to\mathcal{B}$. Then the category of quasicoherent modules in this fibered category is the category opposite to the category of cartesian sections of $\pi$. This viewpoint is used by Rosenberg-Kontsevich in their preprint on noncommutative stacks (dvi, ps).
Given a category $\mathrm{Aff}$ of affine schemes (opposite to the category of rings) equipped with some subcanonical pretopology one considers the stack of $O$-modules over $\mathrm{Aff}$: the fiber over a ring $R$, it assigns the category $Qcoh(\mathrm{Spec}\,R)$. 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 β Zoran); this is in particular a source of a definition $Qcoh$ on an ind-scheme. On ind-schemes Beilinson and Drinfelβd in
consider two variants: a less important variant of quasicoherent $O_X^p$-modules (existing in bigger generality) and more delicate variant of quasicoherent $O^!_X$-modules defined for βreasonable ind-schemesβ; 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.
The last definition has a straightforward generalization to various higher geometry setups, such as derived schemes and other generalized schemes.
For instance the notion of quasicoherent sheaves generalized to derived stacks on the site of simplicial rings as described at geometric β-function theory is obtained, we claim, simply by taking $QC : SRing \to (\infty,1)Cat$ to be the functor that assigns the (β,1)-category for modules over a simplicial ring to any simplicial ring, and then setting for any derived stack $X$
Moreover, using the theorem described at tangent (β,1)-category, that the bifibration of modules over simplicial rings is nothing but the tangent (β,1)-category of $SRing$, one sees that all this is a special case of an even much more general abstract nonsense:
for any presentable (β,1)-category site $C$ whatsoever, we have the tangent (β,1)-category fibration $T_C \to C$. With the (β,1)-functor classifying it denoted $QC : C^{op} \to (\infty,1)Cat$ we may adopt for any β-stack $X : C^{op} \to \infty Grpd$ the definition
as a definition of generalized $\infty$-vector bundles on $X$. This general nonsense is considered further at β-vector bundle. Concrete realizations are discussed at quasicoherent β-stack.
In (LurieQC, section 2.2, section 2.3) the following definition is given.
Let $\mathcal{G}$ be a geometry (for structured (β,1)-toposes). Let
be the opposite of the full sub-(β,1)-category on the compact objects of the tangent (β,1)-category of its opposite (β,1)-category.
For instance for E-β geometry we have $\mathcal{G} = CRing_\infty$ is the (β,1)-category of E-β rings with etale morphisms as admissible maps.
(LurieQC, above Notation 2.2.4)
Then the canonical (β,1)-functor
is a morphism of discrete geometries.
For $\mathcal{X}$ an (β,1)-topos, a left exact (β,1)-functor
constitutes a $\mathcal{G}$-structure sheaf and makes $(\mathcal{X}, \mathcal{O})$ be a $\mathcal{G}$-structured (β,1)-topos. A left exact extension of this
exhibits a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules on $\mathcal{X}$.
Now if $(\mathcal{X},\mathcal{O})$ is locally representable structured (infinity,1)-topos then such an $\mathcal{O}$-module $\mathcal{F}$ is quasi-coherent if also $(\mathcal{X}, (\mathcal{O}, \mathcal{F}))$ is locally representable.
Given an affine scheme $X=\mathrm{Spec}\,R$ (where $R$ is a commutative unital ring), the affine Serre theorem establishes the equivalence of the category $Qcoh(\mathrm{Spec}\,R)$ of quasicoherent sheaves (in Zariski topology) and the category of $R$-modules. Similarly on a projective scheme of the type $Proj(A)$ where $A$ is a nonnegatively graded ring, the (projective) Serre theorem establishes the equivalence of $Qcoh(\mathrm{Proj}\,(A))$ and the localization of the category of graded $A$-modules by the subcategory of modules of finite length (and similarly, of coherent sheaves and graded $A$-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 the Orlovβs article quoted above.
In the case of general (commutative) schemes, every presheaf of $O_X$-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 $O_X$-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 $X$ is equivalently that of quasicoherent sheaves on the corresponding deRham space.
Quasicoherent sheaves in E-β geometry (on βSpectral Schemesβ over E-β 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