(Locally free)
\(\mcf\) is free if \(\mcf \cong \bigoplus_{i \in I} \OO_X\) for some index set \(I\), whose cardinality is its rank; it is free of rank \(n\) if \(\mcf \cong \OO_X\sumpower{n}\). \(\mcf\) is locally free if \(X\) has an open cover \(\ts{U_j}\) with each \(\ro{\mcf}{U_j}\) free, and locally free of rank \(n\) if each \(\ro{\mcf}{U_j}\) is free of rank \(n\). The rank of a locally free sheaf is constant on each connected component of \(X\). An invertible sheaf is a locally free sheaf of rank \(1\).
(Bundles and sheaves)
Taking a rank-\(n\) vector bundle \(\pi: E \to X\) to its sheaf of sections \(U \mapsto \ts{s: U \to E \mid \pi s = \id}\) is an equivalence between rank-\(n\) vector bundles on \(X\) and locally free \(\OO_X\)-modules of rank \(n\).
Both objects are glued from trivialisations on a cover and transition functions in \(\GL_n(\OO_X(U_{ij}))\) satisfying the cocycle condition; the equivalence identifies these two constructions. The sheaf side is the one to compute with, because it lives in an abelian category where kernels, cokernels and cohomology exist; the bundle side is the one that makes geometric statements about fibres legible.
A locally free sheaf is quasicoherent, and on a locally Noetherian scheme it is coherent when the rank is finite; local freeness additionally requires the transition maps to lie in \(\GL_n\) on a trivializing cover. The fibre \(\mcf \tensor \kappa(x)\) recovers the bundle’s fibre, and a coherent sheaf on a reduced locally Noetherian scheme is locally free exactly when the fibre dimension \(x \mapsto \dim_{\kappa(x)} \mcf \tensor \kappa(x)\) is locally constant.
Dependencies
D-MODOXSheaves of modules, and the operations on themuses
Backlinks
FE-MODZOOSeparating quasicoherent, coherent, and locally freeusesD-MODPICThe Picard groupusesT-BIRKGROTHVector bundles on the projective line splituses