coherent sheaf

Let $R$ be a ring with unity, and $X=\mathrm{Spec}R$ be its prime spectrum. Given an $R$-module $M$, one can define a presheaf on $X$ by defining its sections on an open set $U$ to be ${𝒪}_X(U){\otimes }_{R}M$. We call the sheafification of this $\tilde{M}$, and a sheaf of this form on $X$ is called quasi-coherent. If $M$ is a finitely generated module, then $\tilde{M}$ 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