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 Z _ {\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 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. Moreover, this category has enough injectives, and consequently one can and does define the sheaf cohomology H i ( X , − ) {\displaystyle \operatorname {H} ^{i}(X,-)} as the i-th right derived functor of the global section functor Γ ( X , − ) {\displaystyle \Gamma (X,-)} .
Examples 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 Ω X {\displaystyle \Omega _{X}} and the canonical sheaf ω X {\displaystyle \omega _{X}} is the n-th exterior power (determinant) of Ω X {\displaystyle \Omega _{X}} . A sheaf of algebras is a sheaf of modules that is also a sheaf of rings.
Operations Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by
F ⊗ O G {\displaystyle F\otimes _{O}G} or F ⊗ G {\displaystyle F\otimes G} , is the O-module that is the sheaf associated to the presheaf U ↦ F ( U ) ⊗ O ( U ) G ( U ) . {\displaystyle U\mapsto F(U)\otimes _{O(U)}G(U).} (To see that sheafification cannot be avoided, compute the global sections of O ( 1 ) ⊗ O ( − 1 ) = O {\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
H o m O ( F , G ) {\displaystyle {\mathcal {H}}om_{O}(F,G)}
denotes the O-module that is the sheaf U ↦ Hom O | U ( F | U , G | U ) {\displaystyle U\mapsto \operatorname {Hom} _{O|_{U}}(F|_{U},G|_{U})} . In particular, the O-module
H o m O ( F , O ) {\displaystyle {\mathcal {H}}om_{O}(F,O)}
is called the dual module of F and is denoted by F ˇ {\displaystyle {\check {F}}} . Note: for any O-modules E, F, there is a canonical homomorphism
E ˇ ⊗ F → H o m O ( E , F ) {\displaystyle {\check {E}}\otimes F\to {\mathcal {H}}om_{O}(E,F)} , which is an isomorphism if E is a locally free sheaf of finite rank. In particular, if L is locally free of rank one (such L is called an invertible sheaf or a line bundle), then this reads:
L ˇ ⊗ L ≃ O , {\displaystyle {\check {L}}\otimes L\simeq O,}
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