A locally free sheaf on a ringed space 
is a sheaf 
of 
-modules such that each point 
has an open
neighborhood 
and an index set 
with an isomorphism of sheaves of 
-modules
 |
The direct sum may have an arbitrary index set. The sheaf is finite locally free if these index
sets can all be chosen finite. It is finite locally
free of rank 
if they can all have size 
, for a fixed nonnegative
integer 
(Stacks Project Authors 2026c).
On a scheme, locally free sheaves of rank one are line bundles, equivalently invertible sheaves (Stacks Project Authors 2026b). On a locally Noetherian scheme, every finite locally free sheaf is a coherent sheaf (Stacks Project Authors 2026a).
