coherent sheaf


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A coherent sheaf of modules is a geometric globalization of the notion of coherent module.


Over a ringed topos

Let (X,𝒪)(X,\mathcal{O}) be a ringed space or, more generally, a ringed site.

A sheaf \mathcal{E} on XX of 𝒪\mathcal{O}-modules is

Over a structured (,1)(\infty,1)-topos

Over a spectral Deligne-Mumford stack:

(Lurie QCoh, def. 2.6.20)


For a coherent sheaf \mathcal{E} over a ringed space, for every point yy in the base space XX there is a neighborhood VV such that the 𝒪 X(V)\mathcal{O}_X(V)-module (V)\mathcal{E}(V) of sections of \mathcal{E} over VV is finitely presented. On a noetherian scheme the notions of finitely presented and coherent sheaves of 𝒪\mathcal{O}-modules agree, but this is not true on a general scheme or general analytic space; sometimes even the structure sheaf 𝒪\mathcal{O} 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 𝒪\mathcal{O}-modules in a short exact sequence

00 0\to \mathcal{E}\to\mathcal{E}'\to \mathcal{E}''\to 0

are coherent then so is the third. All this holds even if 𝒪\mathcal{O} is a sheaf of noncommutative rings. For commutative 𝒪\mathcal{O}, the inner hom Hom 𝒪(,)Hom_{\mathcal{O}}(\mathcal{E},\mathcal{F}) in the category of sheaves of 𝒪\mathcal{O}-modules is coherent if ,\mathcal{E},\mathcal{F} are coherent.

A theorem of Serre says that the category of coherent sheaves over a projective variety of the form ProjRProj R where RR is a graded commutative Noetherian ring is equivalent to the localization of the category of finitely generated graded RR-modules modulo its (“torsion”) subcategory of (finitely generated graded) RR-modules of finite length.

Historical note and definition variants

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.

A. Vistoli commented at MathOverflow here that for some categorical purposes

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.


Categories of ind-coherent sheaves on schemes and stacks are studied in Dennis Gaitsgory, Notes on Geometric Langlands: ind-coherent sheaves, arxiv/1105.4857

Discussion in (∞,1)-topos theory