# 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 $\mathcal{O}_{X}(U)\otimes_{R}M$. 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 CoherentSheaf 2013-03-22 13:51:27 2013-03-22 13:51:27 PrimeFan (13766) PrimeFan (13766) 11 PrimeFan (13766) Definition msc 14A15 quasi-coherent sheaf