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Concept# Sheaf of modules

Summary

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 that 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 . Moreover, this category has enough injectives, and consequently one can and does define the sheaf cohomology as the i-th right derived functor of the global section functor .
Given a ringed space (X, O), if F is an O-submodule of O, then it is called the sheaf of ideals or ideal sheaf of O, since for each open subset U of X, F(U) is an ideal of the ring O(U).
Let X be a smooth variety of dimension n. Then the tangent sheaf of X is the dual of the cotangent sheaf and the canonical sheaf is the n-th exterior power (determinant) of .
A sheaf of algebras is a sheaf of module that is also a sheaf of rings.
Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by
or ,
is the O-module that is the sheaf associated to the presheaf (To see that sheafification cannot be avoided, compute the global sections of 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 . In particular, the O-module
is called the dual module of F and is denoted by .

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