a coherent sheaf

A quasi-coherent sheaf can be enormous, built from infinitely-generated modules. For most of the deep theorems of algebraic geometry — finite-dimensional cohomology, Riemann-Roch, Serre duality — you need the finiteness that makes the sheaf 'as big as a finite-rank bundle, no bigger'. A coherent sheaf is a quasi-coherent sheaf with exactly this finiteness built in: locally it is generated by finitely many sections and the relations among those generators are themselves finitely generated. These are the sheaves on which all the powerful cohomological machinery works.

Precisely, on a Noetherian scheme X (the setting where the definition is cleanest), an O_X-module F is coherent if it is quasi-coherent and locally finitely generated — on a suitable affine cover Spec R_i, F restricts to M_i-tilde for FINITELY GENERATED R_i-modules M_i. (Over a Noetherian ring 'finitely generated' automatically gives 'finitely presented', so the relations are finite too; on non-Noetherian schemes one must demand finite presentation and a finiteness on submodules explicitly, and coherence of O_X itself is no longer automatic.) Coherent sheaves form an abelian category — kernels, cokernels, images, and extensions of coherent sheaves are coherent — and they are stable under tensor product and pullback by finite-type morphisms. Locally free coherent sheaves of finite rank are precisely the (algebraic) vector bundles, and an invertible sheaf (line bundle) is the rank-one case.

Coherent sheaves are where the finiteness theorems live: on a projective scheme over a field, every coherent sheaf has finite-dimensional cohomology groups H^i, all vanishing above the dimension (Serre, Grothendieck), and Euler characteristics like h^0 - h^1 are computable invariants — this is the engine behind Riemann-Roch and Serre duality. Honest cautions. First, coherent is strictly stronger than quasi-coherent: an infinite direct sum of structure sheaves is quasi-coherent but not coherent, and structure sheaves of non-Noetherian or badly infinite schemes can fail to be coherent over themselves. Second, coherent does NOT mean locally free: a coherent sheaf can have torsion and jumping fiber dimension (the structure sheaf of a singular point, or an ideal sheaf, is coherent but not a vector bundle), so 'coherent sheaf' is much more flexible than 'vector bundle' — that flexibility, allowing kernels and cokernels to stay in the category, is precisely its advantage over working only with bundles.