{title2=$\widetilde M(D(f))=M_f$}

For an <affine variety> with <coordinate ring> $A$ and an $A$-module $M$, <localization> defines sections on <principal open subsets> by the displayed formula. Restrictions are the natural localization maps. Equivalently, a section is a family of <stalk> values which locally come from one fraction $m/f^r$. These data form a <quasi-coherent sheaf>, with <global sections> $M$ and stalk $M_{\mathfrak m_P}$ at $P$. Sheaf <tensor products> correspond to tensor products over $A$; <pullback of a sheaf of modules> corresponds to extension of scalars, and affine <direct image> to restriction of scalars.


 Affine module sheaf (source code)