nLab
coherent sheaf

Idea

A coherent sheaf of modules is a geometric globalization of the notion of coherent module.

Definition

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

A sheaf $ℰ$ on $X$ of $𝒪$-modules is

• finitely generated, or of finite type , if every point $x\in X$ has an open neighbourhood such that there is a surjective morphism

  $${𝒪}^n{\mid }_U\to ℰ{\mid }_U$$\mathcal{O}^n|_U \to \mathcal{E}|_U

  from a free module to $E{\mid }_U$, where $n$ is finite.

• coherent if it is

  1. finitely generated

  2. for every open $U$ in the base space (resp. every object $U$ in the base site), every finite $p\in \mathbb{N}$ and every morphism

     $${𝒪}^p{\mid }_U\to {ℰ}_U$$\mathcal{O}^p|_U\to \mathcal{E}_U

     of $𝒪{\mid }_U$-modules has a finitely generated kernel.

• finitely presented if there is an exact sequence of the form

  $${𝒪}^p\to {𝒪}^n\to ℰ\to 0$$\mathcal{O}^p\to\mathcal{O}^n\to\mathcal{E}\to 0

  with $p$ and $n$ finite.

  Every finitely presented $𝒪$-module is finitely generated.

• quasi coherent if it is locally – on a cover $\{U_i\}$ – presentable, i.e. for each $i$ there is an exact sequences

  $${𝒪}^{I_i}{\mid }_{U_i}\to {𝒪}^{J_i}{\mid }_{U_i}\to ℰ{\mid }_{U_i}\to 0,$$\mathcal{O}^{I_i}|_{U_i} \to \mathcal{O}^{J_i}|_{U_i} \to \mathcal{E}|_{U_i} \to 0\,,

  where $I_i$ and $J_i$ may be infinite, i.e. of $ℰ$ is locally the cokernel of free modules. For more see quasicoherent sheaf.

Properties

For a coherent sheaf $ℰ$ over a ringed space, for every point $y$ in the base space $X$ there is a neighborhood $V$ such that the ${𝒪}_X(V)$-module $ℰ(V)$ of sections of $ℰ$ over $V$ is finitely presented. On a noetherian scheme the notions of finitely presented and coherent sheaves of $𝒪$-modules agree, but this is not true on a general scheme or general analytic space; sometimes even the structure sheaf $𝒪$ itself is a counterexample (not coherent while finitely presented).

The notion of coherent sheaf behaves well on the category of noetherian schemes. On a general topological space, by a basic result of Serre, if two of the sheaves of $𝒪$-modules in a short exact sequence

$$0\to ℰ\to ℰ\prime \to ℰ″\to 0$$0\to \mathcal{E}\to\mathcal{E}'\to \mathcal{E}''\to 0

are coherent then so is the third. All this holds even if $𝒪$ is a sheaf of noncommutative rings. For commutative $𝒪$, the inner hom ${\mathrm{Hom}}_{𝒪}(ℰ,ℱ)$ in the category of sheaves of $𝒪$-modules is coherent if $ℰ,ℱ$ are coherent.

A theorem of Serre says that the category of coherent sheaves over a projective variety of the form $\mathrm{Proj}R$ where $R$ is a graded commutative Noetherian ring is equivalent to the localization of the category of finitely generated graded $R$-modules modulo its (“torsion”) subcategory of (finitely generated graded) $R$-modules of finite length.

References

• J-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2) 61, (1955) 197–278, doi.

• H. Grauert, R. Reinhold, Coherent analytic sheaves, Grundlehren der Math. Wissenschaften 265, Springer 1984. xviii+249 pp.

• M. M. Kapranov, On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), no. 3, 479–508, doi.

• D. O. Orlov, Производные категории когерентных пучков и эквивалентности между ними (pdf, Russian) Uspekhi Mat. Nauk 58 (2003), no. 3(351), 89–172; Engl. transl. Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys 58 (2003), no. 3, 511–591.

• V. D. Golovin, Homology of analytic sheaves and duality theorems, Contemporary Soviet Mathematics (1989) viii+210 pp. transl. from Russian original Гомологии аналитических пучков и теоремы двойствености, Moskva, Nauka 1986. (192 pp.)

• EGA 0.5.3.1