coherent sheaf

Let R be a ring with unity, and X=SpecR be its prime spectrum. Given an R-module M, one can define a presheafPlanetmathPlanetmathPlanetmath on X by defining its sectionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on an open set U to be 𝒪X(U)RM. We call the sheafificationPlanetmathPlanetmath of this M~, and a sheaf of this form on X is called quasi-coherent. If M is a finitely generated module, then M~ is called coherent. A sheaf on an arbitrary scheme X is called (quasi-)coherent if it is (quasi-)coherent on each open affine subset of X.

Title coherent sheaf
Canonical name CoherentSheaf
Date of creation 2013-03-22 13:51:27
Last modified on 2013-03-22 13:51:27
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 11
Author PrimeFan (13766)
Entry type Definition
Classification msc 14A15
Synonym quasi-coherent sheaf