In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf of abelian groups F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times the restriction of s for any f in O(U) and s in F(U).

The standard case is when X is a scheme and O its structure sheaf. If O is the constant sheaf ${\displaystyle {\underline {\mathbf {Z} }}}$, then a sheaf of O-modules is the same as a sheaf of abelian groups (i.e., an abelian sheaf).

If X is the prime spectrum of a ring R, then any R-module defines an O_X-module (called an associated sheaf) in a natural way. Similarly, if R is a graded ring and X is the Proj of R, then any graded module defines an O_X-module in a natural way. O-modules arising in such a fashion are examples of quasi-coherent sheaves, and in fact, on affine or projective schemes, all quasi-coherent sheaves are obtained this way.

Sheaves of modules over a ringed space form an abelian category. Moreover, this category has enough injectives, and consequently one can and does define the sheaf cohomology ${\displaystyle \operatorname {H} ^{i}(X,-)}$ as the i-th right derived functor of the global section functor ${\displaystyle \Gamma (X,-)}$.

Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by

is the O-module that is the sheaf associated to the presheaf ${\displaystyle U\mapsto F(U)\otimes _{O(U)}G(U).}$ (To see that sheafification cannot be avoided, compute the global sections of ${\displaystyle O(1)\otimes O(-1)=O}$ where O(1) is Serre's twisting sheaf on a projective space.)

Similarly, if F and G are O-modules, then

denotes the O-module that is the sheaf ${\displaystyle U\mapsto \operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})}$. In particular, the O-module