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 -modules and especially its full subcategory of quasicoherent sheaves of -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
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 of modules over rings. In view of this, one finds that this definition, in turn, is equivalent to a very fundamental definition:
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 with the category of cartesian functors between the category of elements of and . 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 -sheaves or “derived” quasicoherent sheaves, such as they appear for instance in geometric ∞-function theory. This is discussed in the section
This should be viewed as a local presentation of .
If can be chosen finite and 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 -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.
extends to an isomorphism
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 ; recall the fibered category where for each in the inverse image functor is . Then the identity functor can be interpreted as the presheaf of rings and is denoted by (the “structure sheaf”). An -module is a presheaf of -modules. Usually some Grothendieck topology is given and one asks for sheaves in fact. We can Yoneda extend -modules to presheaves. We now define quasicoherent sheaves of -modules on an arbitrary presheaf on , viewed as a covariant functor on .
A quasicoherent sheaf of -modules on is a rule assigning to any an -module and to any morphism in an isomorphism such that for any composable pair and any the cocycle condition
holds, where is the canonical isomorphism which is part of the data of the (covariant) pseudofunctor , .
Notice that if is (co)representable presheaf, then is the same as a morphism of rings; restricting the quasicoherent sheaf to along and taking the global sections over , would give the -module .
Clearly, and can be much generalized. For example, rings may be noncommutative or one can take category opposite to the category of monads in and an arbitrary (not identity) presheaf of monads in ; the extension of scalars for monads gives an inverse image functor for Eilenberg-Moore categories. Durov’s construction of quasicoherent sheaves for monads in is an example where commutative algebraic monads are used; the theory of quasicoherent sheaves of -modules (“-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.
The above definition may be further re-interpreted as follows.
On the site , let
When is the functor represented by a scheme, then is the category of quasicoherent sheaves on , as defined above.
We now explain the above statement in detail and thereby prove it.
Consider the 2-category of (pre)stacks on . The canonical module bifibration of the category of modules over all rings is the bifibration whose fibered part corresponds to the (pre)stack given on objects by
and on morphisms by
where on the right we have the functor that sends any -module to the tensor product over with the --bimodule with its canonical left -action and with the right -action induced by the ring homomorphism .
One may think of this as the stack of generalized algebraic vector bundles:
which using the Yoneda lemma is
We write for short
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 is the category of quasicoherent sheaves on as defined by other means above, whenever that other definition applies to .
To see this, straightforwardly unwrap the definition: an object in is a pseudonatural transformation of 2-functors , where is regarded as a 2-functor by the canonical embedding that regards a set as a discrete category.
The components of are
for each a functor :
this functor picks one -module for each plot ;
for each morphism a pseudonaturality square
in Cat (these are subject to coherence conditions). This unwraps to the following data:
the component functors provide an assignment of modules to each plot ;
these assignments form a presheaf on the overcategory 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 .
for each tuple of composable morphisms
a pseudo-naturality prism equation relating, , and . 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 defines manifestly a quasicoherent sheaf on 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 under this prescription.
on the one hand the bicategory whose objects are pseudofunctors , whose morphisms are pseudonatural transformations, and whose 2-morphisms are modifications of these
on the other hand the bicategory whose objects are Grothendieck fibrations , whose morphism are cartesian functors
and whose 2-morphisms are natural transformations between these.
Recall furthermore that for an ordinary presheaf, i.e. a pseudofunctor that factors through an ordinary functor via the inclusion , the Grothendieck fibration associated with is the category of elements of .
Recall furthermore that by definition, the pseudofunctor is the one corresponding to the Grothendieck fibration .
Therefore, by the above equivalence of 2-categories, we find that the category of functors is equivalent to the category of cartesian functors over ,
In this form quasicoherent sheaves on 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 . Then the category of quasicoherent modules in this fibered category is the category opposite to the category of cartesian sections of . This viewpoint is used by Rosenberg-Kontsevich in their preprint on noncommutative stacks (dvi, ps).
Given a category of affine schemes (opposite to the category of rings) equipped with some subcanonical pretopology one considers the stack of -modules over : the fiber over a ring , it assigns the category . 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 on an ind-scheme. On ind-schemes Beilinson and Drinfel’d in
consider two variants: a less important variant of quasicoherent -modules (existing in bigger generality) and more delicate variant of quasicoherent -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.
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 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
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 , one sees that all this is a special case of an even much more general abstract nonsense:
In (LurieQC, section 2.2, section 2.3) the following definition is given.
Let be a geometry (for structured (∞,1)-toposes). Let
Then the canonical (∞,1)-functor
is a morphism of discrete geometries.
exhibits a sheaf of -modules on .
Now if is locally representable structured (infinity,1)-topos then such an -module is quasi-coherent if also is locally representable.
Let be a scheme. Recall that the big Zariski topos of is the topos of sheaves over (or, more precisely, the affine schemes over which are locally of finite presentation). In this topos, there is a local ring , the sheaf mapping an -scheme to .
A sheaf of modules over is quasicoherent if and only if, from the internal point of view of the big Zariski topos, the canonical map
is bijective for all finitely presented -algebras . The outer Hom set is the set of all maps from the set to the (underlying set of) .
This characterization has a geometric interpretation. The set deserves the name “spectrum of ”, since it consists of what classically is known as the (-)rational points of . Furthermore, if is induced from a sheaf of -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 is therefore the set of all -valued functions on the spectrum of . An element of gives rise to such a function: associate to a pure tensor the function .
In a synthetic/algebraic context, there should be no more functions than those which result from this construction. This is what the characterization expresses.
Given an affine scheme (where is a commutative unital ring), the affine Serre theorem establishes the equivalence of the category of quasicoherent sheaves (in Zariski topology) and the category of -modules. Similarly on a projective scheme of the type where is a nonnegatively graded ring, the (projective) Serre theorem establishes the equivalence of and the localization of the category of graded -modules by the subcategory of modules of finite length (and similarly, of coherent sheaves and graded -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.
In the case of general (commutative) schemes, every presheaf of -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 -modules over any quasicompact quasiseparated scheme is a Grothendieck category and in particular has enough injective objects.
Their descent properties are discussed in