Quasi-coherent sheaf

A sheaf of modules on a scheme is quasi-coherent when every affine chart $\mathrm{Spec}A$ restricts it to the sheaf associated with an $A$-module. On an affine scheme it is determined by its module of global sections.

On an affine scheme, a morphism between quasi-coherent sheaves is determined by the induced map on global sections, because its maps on principal open subsets are their localizations. Exactness of localization gives the displayed identity. Thus the kernel sheaf of a morphism between quasi-coherent sheaves is quasi-coherent on an arbitrary scheme, by applying the affine argument locally. No finite generation is needed.

For an affine variety with coordinate ring $A$ and an $A$-module $M$, localization defines sections on principal open subsets by the displayed formula. Restrictions are the natural localization maps. Equivalently, a section is a family of stalk values which locally come from one fraction $m/f^r$. These data form a quasi-coherent sheaf, with global sections $M$ and stalk $M_{{\mathfrak{m}}_P}$ at $P$. Sheaf tensor products correspond to tensor products over $A$; pullback of a sheaf of modules corresponds to extension of scalars, and affine direct image to restriction of scalars.

For a Noetherian ring $A$, a finitely generated module $M$ has a finite presentation of a module, and kernels of maps from finite free modules to $M$ are finitely generated. Its affine module sheaf is therefore a coherent sheaf. Conversely, if $\tilde{M}$ is coherent, choose a finite principal open subset cover on which its section modules are finitely generated. Lift their finitely many generators to numerators in $M$. The submodule generated by these numerators becomes all of $M$ on every chart, and hence is all of $M$: its quotient has zero localization at every point. This proves finite generation without assuming that global sections preserve arbitrary sheaf surjections.

In $0\to \mathcal{F}\to \mathcal{G}\to \mathcal{E}\to 0$, assume the outer sheaves are quasi-coherent. On an affine open, any section of $\mathcal{E}$ has local lifts to $\mathcal{G}$ on a finite principal cover. Their differences are a Čech cocycle in the module defining $\mathcal{F}$. Exactness of the unit-ideal localization Čech complex makes this cocycle a coboundary, so adjusted lifts glue. Thus sections are exact on the affine open and on every principal open in it. Comparing the resulting sequences after localization, the outer isomorphisms force the middle localization map to be an isomorphism. Hence $\mathcal{G}$ is associated with its global-section module and is quasi-coherent.

Choose a finite cover of an affine scheme by principal opens on which the quasi-coherent sheaf is associated with a module. The sheaf gluing axiom expresses global sections as the kernel of the difference map from the finite product of chart sections to the finite product of overlap sections. Localization commutes with kernels and finite products. Localizing at $a$ therefore gives the same kernel for the cover intersected with $D(a)$, proving the formula. The principal-open formulas identify the sheaf with that associated with its global-section module; no finite-generation hypothesis on the module is needed.

On an affine variety or affine scheme with coordinate ring $A$, the global section functor and $M\mapsto \tilde{M}$ are inverse equivalences between modules and quasi-coherent sheaves. On every principal open subset, sections are the appropriate localization of a module. Localization detects exactness at maximal ideals, so these functors are exact even for modules which are not finitely generated.

On an integral scheme, a quasi-coherent sheaf is torsion-free when its module on every affine chart is a torsion-free module over that chart's integral domain. It embeds into its generic fibre. Therefore a torsion-free sheaf with zero generic fibre is zero; locally free sheaves and their subsheaves are torsion-free.

Every quasi-coherent sheaf on an affine scheme has zero higher sheaf cohomology. Together with the equivalence between modules and quasi-coherent sheaves on an affine scheme, this makes affine varieties the basic acyclic pieces for computing cohomology. An affine open cover of a separated scheme has affine finite intersections, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology.

On a locally Noetherian scheme, a coherent sheaf is a quasi-coherent sheaf that is locally associated with a finitely generated module. Kernels, cokernels, and finite direct sums of coherent sheaves remain coherent.

A sheaf of modules is globally generated when the evaluation map from its global sections is surjective. Equivalently, the germs of global sections span each stalk as a module over the local ring. For a coherent sheaf on a projective variety, every sufficiently positive twist is globally generated: embed the variety in projective space, present its direct-image sheaf as a quotient of a finite sum of $\mathcal{O}(-a)$, twist until those line bundles are globally generated, and restrict their generating sections to the variety.

On ${\mathbb{P}}_k^1$, $\mathcal{O}$ and $\mathcal{O}(-1)$ are line bundles, so each stalk of either is a free rank-one module over the local ring. Nevertheless their spaces of global sections are $k$ and zero. Indeed a section of $\mathcal{O}(-1)$ on the two standard charts would require a polynomial $a(t)$ to equal $t^{-1}b(t^{-1})$, which has only strictly negative powers. Only the zero section can satisfy this. Pointwise existence of unrelated stalk isomorphisms does not supply a morphism of sheaves.

An ideal sheaf of a closed subscheme that is a coherent sheaf as a module over the structure sheaf. On a Noetherian scheme, every quasi-coherent ideal sheaf is coherent.

A coherent sheaf on ${\mathbb{P}}_k^n$ is the sheaf associated with a graded module for a finite graded $k[t_0,\dots ,t_n]$-module. Choose a finite graded free resolution by the Hilbert syzygy theorem and sheafify it; exactness of localization preserves exactness. All terms are finite sums of twisting sheaves on projective space. Twisting sufficiently positively makes every term acyclic in positive degree, and dimension shifting in long exact sequences in sheaf cohomology proves Serre vanishing. The graded-module description is supplied by Stacks Project, Section 30.15.

The support dimension is the dimension of a scheme of the closed support of a coherent sheaf. Tensoring with a line bundle leaves its support unchanged. Quotienting by a section that avoids its associated points reduces the support dimension by at least one.

An associated point of a coherent sheaf is a point whose corresponding prime is the annihilator of an element of the module on an affine chart. There are finitely many on a Noetherian scheme. A section of a line bundle avoiding them acts injectively on the sheaf, even when the scheme is nonreduced. Over an infinite field a sufficiently ample linear system supplies such a section.

If $i:Z\hookrightarrow X$ is a closed immersion of Noetherian schemes and $\mathcal{F}$ is a coherent sheaf on $Z$, then $i_{\ast }\mathcal{F}$ is coherent on $X$. Affine-locally this regards a finitely generated $A/I$-module as a finitely generated $A$-module.