= Locally free module
{title2=$P_{f_i}\cong A_{f_i}^r$}
A <locally free module> is one whose associated sheaf is locally a <free module>. In the finite constant-rank case, it is a <finite projective module> whose <localization at a prime ideal> has rank $r$ everywhere. Equivalently, it becomes $A_{f_i}^r$ on a finite cover by <principal open subsets>. Its associated sheaf is a <locally free sheaf> and defines an algebraic <vector bundle>.
A <surjection> onto such a module splits. If its source is also a <finite projective module>, the kernel is finite projective as a <direct summand>. A source hypothesis is essential: the projection $\mathbb Z\oplus\mathbb Z/2\mathbb Z\to\mathbb Z$ splits and has a free target, but its kernel $\mathbb Z/2\mathbb Z$ is not projective.
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