Complex Analytic and Differential Geometry 2024, 52: Coherent Sheaves

For a sheaf of rings A on a topological space X, a sheaf S of A-modules is locally free of rank r over A, if it's locally isomorphic to the outer sum of A on a neighborhood of every point. S is then locally free for a covering by open sets on which S has free generators. The idea of locally free sheafs is closely related to vector bundles in that the sheaf of rings A is a subsheaf of the sheaf C of continuous functions with values in whatever field it exists in, containing the sheaf of locally constant functions from X into the field. There exists an evaluation map with a kernel that's a maximal ideal, so that the modulo is the field again. To each point in X, there is an associated vector space, the joined set of which is equipped with a natural projection back onto X, where the spaces are analogous to fibers.

If a sheaf of local rings S, S' has an A-morphism between locally free sheafs of A-modules of rank r, r' respectively. If the rank of the r' ⨉ r matrix for the map is constant for all points in X, then the kernel of the matrix and its image are locally free subsheaves of S, S', and its cokernel is locally free.

A sheaf of rings on a topological space X and a sheaf of modules S over A-modules, then S is locally finitely generated if for all x in X there is a neighborhood U and sections in S(U) so that for all u in U, the stalk is generated by said sections as an A-module. A locally finitely generated sheaf S of A-modules on X and sections {G} in S(U) that generate S at some points specific points in U. G then generates S near that point. If U is an open subset of X, the "restriction of S to U" is the union of all stalks for all x in U. The "sheaf of relations" between sections that make up the kernel of the sheaf homomorphism is

A sheaf of A-modules S on X is coherent, if S is locally finitely generated, and if for any open subset U of X and any set of sections {F} in S(U), the sheaf of relations is locally finitely generated. An A-morphism of coherent sheaves has coherent image and kernel. If two of three sheaves of A-modules in a short exact sequence are coherent, so is the third. Two coherent subsheaves of a coherent analytic sheaf have a coherent intersection. A sheaf or rings is coherent, if it's coherent as a module over itself. Then, if A is a coherent sheaf or rings, any of its locally finitely generated subsheafs is coherent.

For an n-dim. complex analytic manifold M and a sheaf of germs of analytic functions on M O, an analytic sheaf over M is a sheaf of modules over O. The sheaf of rings is coherent for any complex manifold M (Coherence Theorem, Oka). A point on the subdisk with a module R is generated by its elements whose components are germs of analtic polynomials in O[z] with degree in z at most equal to the maximum of degrees. A coherent analytic sheaf F on complex manifold M with an increasing sequence of coherent subsheaves {G}, then the sequence (G) is stationary on every compact subset of M.