Locally free sheaf - Knowlpedia

Definition

Let $(X,\mathcal O_X)$ be a ringed space. An $\mathcal O_X$ -module sheaf $\mathcal E$ is locally free of rank $r$ if every $x\in X$ has an open neighborhood $U$ with an isomorphism

\mathcal E|_U\cong\mathcal O_X|_U^{\oplus r}.

Such an isomorphism is a local frame. A finite-rank locally free sheaf allows a finite rank that is locally constant, and hence constant on each connected component.

Transition matrices

For a locally free sheaf of constant rank $r$ , two local frames over $U_i$ and $U_j$ differ on the overlap by an invertible matrix

g_{ij}\in GL_r(\mathcal O_X(U_i\cap U_j)).

These matrices satisfy the cocycle identities and glue the local free modules. Conversely, compatible invertible transition matrices construct a locally free sheaf.

Relation to vector bundles

For a smooth real or complex vector bundle, the sheaf of smooth sections is finite-rank locally free over $C^\infty_M$ . Conversely, a finite-rank locally free $C^\infty_M$ -module sheaf glues trivial bundles to recover a smooth vector bundle.

The adjective “locally free” applies to the sheaf. Its global section module need not be free: a global basis would be a global frame and would trivialize the associated bundle.

References

The Stacks Project Authors, The Stacks Project. Tag 01C5. Relevant: finite locally free modules and locally free sheaves.
Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: Chapter 3, transition functions and vector bundles.