I am looking for a text on finitely presented objects in Qco(x), the category of quasi coherent sheaves on a scheme X. Is there some good properties for these sheaves like for finitely presented modules over a ring. For a finitely presented module F we know that if we have an epic morphism $N\to F\to 0$, hen there is a finitely generated submodule N’ of N and an induced morphism $N’\to F\to$. Can we say the same thing for quasi-coherent sheaves? Also one cn prove that the submodule A of a module B is pure iff the induced map $ Hom(F,B)\to Hom(F,B/A)$ be surjective for all finitely presented R-modules F.
One can consider three different definitions of a finitely presented quasi-coherent sheaf.
(i) A globally finitely presented sheaf is one that can be written as the cokernel of a morphism
of two finite coproducts of the structure sheaf.
(ii) A locally finitely presented sheaf is one that can be locally written as such a cokernel.
(iii) A finitely presented sheaf is one satisfying the categorical property. F is f.p. if $ Hom(F,-)$
Preserves direct limits.. This makes sense in both Qco(X) and Mod(X) so we will specify the ambient category.
And we know that
Proposition 75. Let X be a concentrated scheme and F a quasi-coherent sheaf. Then F is
finitely presented in Qco(X) if and only if it is locally finitely presented.[Murfet Homepage, Modules over a scheme].
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Acctually I want to find a good reference in finiteness conditions of quasi coherent sheaves, firstly. Also I am going to know what is the relation between finitely presented objects and pure sequences.