Locally free module ($P_{f_i}\cong A_{f_i}^r$)

A locally free module is one whose associated sheaf is locally a free module. In the finite constant-rank case, it becomes free of rank $r$ on a principal open subset cover of $\mathrm{Spec}A$. Equivalently it is a finite projective module of constant rank $r$. A surjection onto it splits, so the kernel is also finite projective. Its associated sheaf is a locally free sheaf and defines an algebraic vector bundle.

// Target: algebraic-geometry.bigb