Let (
X,
OX) be a ringed space. Isomorphism classes of sheaves of
OX-modules form a
monoid under the operation of tensor product of
OX-modules. The
identity element for this operation is
OX itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if
L is a sheaf of
OX-modules, then
L is called
invertible if it satisfies any of the following equivalent conditions: • There exists a sheaf
M such that L \otimes_{\mathcal{O}_X} M \cong \mathcal{O}_X. • The natural homomorphism L \otimes_{\mathcal{O}_X} L^\vee \to \mathcal{O}_X is an isomorphism, where L^\vee denotes the dual sheaf \underline{\operatorname{Hom}}(L, \mathcal{O}_X). • The functor from
OX-modules to
OX-modules defined by F \mapsto F \otimes_{\mathcal{O}_X} L is an
equivalence of categories. Every
locally free sheaf of rank one is invertible. If
X is a locally ringed space, then
L is invertible if and only if it is locally free of rank one. Because of this fact, invertible sheaves are closely related to
line bundles, to the point where the two are sometimes conflated. ==Examples==