## Colimit of locally finitely presented quasi-coherent modules

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Let $X$ be a quasicompact quasiseparated scheme. Consider the full subcategory $\text{Qcoh}_{fp}(X)$ of $\text{Qcoh}(X)$ which consists of the quasi-coherent modules which are locally of finite presentation.

Question Is every quasi-coherent module $M$ the colimit of the homomorphisms $N \to M$, where $N$ runs through $\text{Qcoh}_{fp}(X)$?

Note that this makes sense since $\text{Qcoh}_{fp}(X)$ is essentially small. The result is well-known if we allow quasi-coherent modules which are of finite type (in particular, everything is OK if $X$ is noetherian). If $X$ is affine, then the result is trivial.

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## 2 Answers

The answer is yes, at least if you believe Thomason-Trobaugh, Higher algebraic $K$-theory of schemes and of derived categories, which David Ben-Zvi already mentioned.

I quote from Appendix B.3 (p. 409f):

B.3. If $X$ is a quasi-compact and quasi-separated scheme, every sheaf in $\text{Qcoh}(X)$ is a direct colimit of its sub-`$\mathcal{O}_{X}$`-modules of finite type. Also, every sheaf in $\text{Qcoh}(X)$ is a filtering colimit of finitely presented `$\mathcal{O}_{X}$`- modules. ([EGA] I 6.9.9, 6.9.12.) In this case, the set of finitely presented $\mathcal{O}_{X}$-modules forms a set of generators for $\text{Qcoh}(X)$, which is then a Grothendieck abelian category and has enough injectives (cf. B.12.).

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Thank you. I knew this appendix and should have remembered it. The "new" EGA I covers more material on such things than the "old" EGA I. – Martin Brandenburg Jan 22 2011 at 23:59

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I don't know the answer, but take it as a good excuse to mention the wonderful theorem of Thomason-Trobaugh in the Grothendieck Festschrift that the analogous statement is true on the derived level --- namely for a quasicompact quasiseparated scheme, the quasicoherent derived category is compactly generated by perfect complexes.

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