A coherent sheaf of modules is a geometric globalization of the notion of coherent module.
finitely generated, or of finite type , if every point has an open neighbourhood such that there is a surjective morphism
from a free module to , where is finite.
coherent if it is
for every open in the base space (resp. every object in the base site), every finite and every morphism
of -modules has a finitely generated kernel.
finitely presented if there is an exact sequence of the form
with and finite.
Every finitely presented -module is finitely generated.
Over a spectral Deligne-Mumford stack:
For a coherent sheaf over a ringed space, for every point in the base space there is a neighborhood such that the -module of sections of over is finitely presented. On a noetherian scheme the notions of finitely presented and coherent sheaves of -modules agree, but this is not true on a general scheme or general analytic space; sometimes even the structure sheaf itself is a counterexample (not coherent while finitely presented).
The notion of coherent sheaf behaves well on the category of noetherian schemes. On a general topological space, by a basic result of Serre, if two of the sheaves of -modules in a short exact sequence
are coherent then so is the third. All this holds even if is a sheaf of noncommutative rings. For commutative , the inner hom in the category of sheaves of -modules is coherent if are coherent.
A theorem of Serre says that the category of coherent sheaves over a projective variety of the form where is a graded commutative Noetherian ring is equivalent to the localization of the category of finitely generated graded -modules modulo its (“torsion”) subcategory of (finitely generated graded) -modules of finite length.
First works on coherent sheaves in complex analytic geometry. Serre adapted their work to algebraic framework in his famous article FAC. Hartshorne’s definitions are changed/adapted to the special setup of Noetherian schemes with the excuse that the coherence does not behave that well otherwise; thus they differ from the definitions in EGA and FAC.
one should interpret “coherent” as meaning “quasi-coherent of finite presentation”. The notion of coherent sheaf, as defined in EGA, is not functorial, that is, pullbacks of coherent sheaves are not necessarily coherent. Hartshorne’s book defines “coherent” as “quasi-coherent and finitely generated”, but this is a useless notion when working with non-noetherian schemes.
H. Grauert, R. Remmert, Coherent analytic sheaves, Grundlehren der Math. Wissenschaften 265, Springer 1984. xviii+249 pp.
D. O. Orlov, Производные категории когерентных пучков и эквивалентности между ними (pdf, Russian) Uspekhi Mat. Nauk 58 (2003), no. 3(351), 89–172; Engl. transl. Derived categories of coherent sheaves and equivalences between them, Russian Math. Surveys 58 (2003), no. 3, 511–591.
V. D. Golovin, Homology of analytic sheaves and duality theorems, Contemporary Soviet Mathematics (1989) viii+210 pp. transl. from Russian original Гомологии аналитических пучков и теоремы двойственности, Moskva, Nauka 1986. (192 pp.)
Qing Liu, Algebraic geometry and arithmetic curves, 5.1.3
Discussion in (∞,1)-topos theory