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We know that the evaluation functor $\Gamma(u, -):Qcoh(X) \to \{\cal O}_X(u)-mod$ is a left exact functor preserving limits. We also know that the category of all quasi-coherent sheaves on scheme is locally finitely presented. So we may use "Adjoint Functor Theorem" and deduce that there is a left adjoint for the evaluation functor.

Is there any explicit description for this adjoint?

The answer is true if we replace $Qcoh(X)$ by the category of sheaves on $X$ (see section 2 of "Relative Homological Algebra in Categories of Representations of Infinite Quivers" by S. Estrada)

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  • $\begingroup$ This question is not well posed. You should make clear what $X,u$ etc. are. Besides, I think you don't mean $\mathcal{O}_X(u)$, but rather the category of modules on it. $\endgroup$ – Martin Brandenburg Mar 10 '12 at 9:32
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    $\begingroup$ Side remark: If X is a scheme, when Qcoh(X) can not be written as some category in section 2 of the paper you cite. Even if we fix the representation of rings O_X, we cannot write Qcoh(X) as a category of representations of modules on this. This is claimed in "Relative homological algebra in the category of quasi-coherent sheaves", but it is wrong because no compatibility data can be imposed on a quiver. Instead, one has to use categories (in this case the category of open affine subsets of X). This is rather unfortunate since many people cite this paper and take this claim for granted ... $\endgroup$ – Martin Brandenburg Mar 10 '12 at 9:41
  • $\begingroup$ Oh! I made a mistake. I mean the category of ${\mathcal O}_X(u)$-Mod by ${\cal O}_X(u)$. I think a misunderstanding has happened. Acctually the paper that I cited is different from "Relative Homological Algebra in Qco(X)". $\endgroup$ – Gholam Mar 11 '12 at 12:47
  • $\begingroup$ May you explain more on your claim (the one that expresses that we cannot consider Qco(X) as a category of representations)? $\endgroup$ – Gholam Mar 11 '12 at 12:50
  • $\begingroup$ In the paper that I cited there is an explicit description for the adjiont pair containing evaluation functor $\endgroup$ – Gholam Mar 11 '12 at 12:51
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It's not hard to give an explicit description of the left adjoint to $\def\O{\mathcal O}\def\mod{\textrm{-mod}}\Gamma(U,-):\O_X\mod\to\O_X(U)\mod$ by stringing together left adjoints. Perhaps a slight modification will do what you want.

For $j:U\hookrightarrow X$ an open immersion, the restriction functor $j^*:\O_X\mod\to \O_U\mod$ has a left adjoint $j_!$. The pushforward functor $f_*:\O_U\mod\to \O_{Spec(\O_U(U))}\mod$ has the left adjoint $f^*$. The functor $\Gamma(Spec A,-):\O_{Spec(A)}\mod\to A\mod$ has the left adjoint taking an $A$-module $M$ to the quasi-coherent sheaf $\widetilde M$. Stringing these together, we get that the functor sending an $\O_X(U)$-module $M$ to $j_!(f^*\widetilde M)$ is left adjoint to $\Gamma(U,-):\O_X\mod\to \O_X(U)\mod$.

I'm not sure how to modify $j_!$ to get a left adjoint to $j^*:QCoh(X)\to QCoh(U)$. If there is such a thing, it finishes the job.

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