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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    $\begingroup$ (i) is definitely the "wrong" condition, and (ii) and (iii) turn out to be equivalent (either by your quote of Murfet's notes, but it also follows directly from the results in EGA I (1970), Section 6.9. One really needs concentrated schemes here: In general (ii) => (iii), but (iii) => (ii) is equivalent(!) to being concentrated. // You should make your question more precise. What do you want to know about f.p. qc modules? Work with (ii) and and use the affine case. $\endgroup$ Apr 25, 2012 at 7:42
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    $\begingroup$ Unless $X$ satisfies further properties, there are few finitely presented objects. The category of quasi-coherent sheaves is a Grothendieck category. It is locally presentable, but usually not locally finitely presentable. $\endgroup$ Oct 1, 2012 at 21:28

1 Answer 1


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.

  • $\begingroup$ excuse me, I couldn't find any key related to comments. $\endgroup$
    – gholam
    Apr 25, 2012 at 10:21
  • $\begingroup$ If you know German, I can send you diploma thesis (you can find my mail in the profile). There is a section about finiteness conditions. $\endgroup$ May 13, 2012 at 9:08
  • $\begingroup$ Thank you Martin, for your comments. Unfortunately I don't know German. $\endgroup$
    – gholam
    May 25, 2012 at 15:32

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