A coherent sheaf of modules over a sheaf of rings 
is a locally finitely generated sheaf 
such that the kernel
sheaf of every morphism 
is locally finitely
generated, for every open set 
and finite 
. It records finitely many local generators and relations with
compatible restrictions (The Stacks Project Authors 2026a).
On a locally Noetherian scheme, it is equivalently a quasi-coherent sheaf locally represented by a finitely generated module (The Stacks Project Authors 2026b). On such a scheme, locally free sheaves of finite rank and ideal sheaves of closed subschemes are examples. Coherent sheaves form an Abelian category on a Noetherian scheme and supply the objects used in its bounded derived category.
