Let (
X,
O) be a ringed space. If
F and
G are
O-modules, then their tensor product, denoted by :F \otimes_O G or F \otimes G, is the
O-module that is the sheaf associated to the presheaf U \mapsto F(U) \otimes_{O(U)} G(U). (To see that
sheafification cannot be avoided, compute the global sections of 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 :\mathcal{H}om_O(F, G) denotes the
O-module that is the sheaf U \mapsto \operatorname{Hom}_{O|_U}(F|_U, G|_U). In particular, the
O-module :\mathcal{H}om_O(F, O) is called the
dual module of
F and is denoted by \check F. Note: for any
O-modules
E,
F, there is a canonical homomorphism :\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: :\check{L} \otimes L \simeq O, implying the isomorphism classes of invertible sheaves form a group. This group is called the
Picard group of
X and is canonically identified with the first cohomology group \operatorname{H}^1(X, \mathcal{O}^*) (by the standard argument with
Čech cohomology). If
E is a locally free sheaf of finite rank, then there is an
O-linear map \check{E} \otimes E \simeq \operatorname{End}_O(E) \to O given by the pairing; it is called the
trace map of
E. For any
O-module
F, the
tensor algebra,
exterior algebra and
symmetric algebra of
F are defined in the same way. For example, the
k-th exterior power :\bigwedge^k F is the sheaf associated to the presheaf U \mapsto \bigwedge^k_{O(U)} F(U). If
F is locally free of rank
n, then \bigwedge^n F is called the
determinant line bundle (though technically
invertible sheaf) of
F, denoted by det(
F). There is a natural perfect pairing: :\bigwedge^r F \otimes \bigwedge^{n-r} F \to \det(F). Let
f: (
X,
O) →(
X,
O) be a morphism of ringed spaces. If
F is an
O-module, then the
direct image sheaf f_* F is an
O-module through the natural map
O →
f*
O (such a natural map is part of the data of a morphism of ringed spaces.) If
G is an
O-module, then the module inverse image f^* G of
G is the
O-module given as the tensor product of modules: :f^{-1} G \otimes_{f^{-1} O'} O where f^{-1} G is the
inverse image sheaf of
G and f^{-1} O' \to O is obtained from O' \to f_* O by
adjuction. There is an adjoint relation between f_* and f^*: for any
O-module
F and
O'-module
G, :\operatorname{Hom}_{O}(f^* G, F) \simeq \operatorname{Hom}_{O'}(G, f_*F) as abelian group. There is also the
projection formula: for an
O-module
F and a locally free
O'-module
E of finite rank, :f_*(F \otimes f^*E) \simeq f_* F \otimes E. == Properties ==