Coherent sheaf

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.