A left adjoint for the evaluation functor \Gamma(, U)

2012-03-10T00:46:55Z

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)

Answer by Anton Geraschenko for A left adjoint for the evaluation functor \Gamma(, U)

2012-03-13T21:42:15Z

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.

A left adjoint for the evaluation functor \Gamma(, U) - MathOverflow

A left adjoint for the evaluation functor \Gamma(, U)

Answer by Anton Geraschenko for A left adjoint for the evaluation functor \Gamma(, U)