Semicoherence

Semicoherence is a finiteness condition for sheaves of modules on a ringed space, used mainly in complex analytic geometry and algebraic geometry. It serves as a relaxation of coherence, allowing certain infinite-type sheaves to retain enough finiteness control for practical work.

One standard way to formulate semicoherence is this: a sheaf F of OX-modules is semi-coherent if, for every open set U and every finite subset S of F(U), there exists an open covering {Ui} of U such that the sections in S lie inside a finitely presented OX(Ui)-submodule of F(Ui) for each i. Equivalently, F is a directed union of finitely presented submodules. In this view, semicoherent sheaves are built from finitely presented pieces, but without requiring a single global finite presentation.

Relations to other notions are straightforward. Every coherent sheaf is semi-coherent, since coherence is a stronger condition. Conversely, on Noetherian spaces (or under suitable finiteness hypotheses on the ringed space), semicoherence and coherence coincide: every finitely generated or finitely presented substructure behaves well enough to meet the stricter definition.

Examples and behavior: the structure sheaf OX is semi-coherent, and, in many settings, coherent sheaves are semi-coherent as well. Semicoherence is stable under finite direct sums and under forming subobjects that arise from locally finitely presented data; under appropriate hypotheses it is preserved by pushforward along certain proper maps in analytic geometry.

See also: coherent sheaf, quasi-coherent sheaf, coherence theorems in analytic and algebraic geometry.