Sheaf of modules

In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf 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 , 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 OX-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 OX-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.[1] Moreover, this category has enough injectives,[2] and consequently one can and does define the sheaf cohomology as the i-th right derived functor of the global section functor .[3]

  1. ^ Vakil, Math 216: Foundations of algebraic geometry, 2.5.
  2. ^ Hartshorne, Ch. III, Proposition 2.2.
  3. ^ This cohomology functor coincides with the right derived functor of the global section functor in the category of abelian sheaves; cf. Hartshorne, Ch. III, Proposition 2.6.