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As stated on the wikipedia page, Grothendieck generalized Serre duality by stating that there exists a right adjoint functor $f^!$ to the functor $Rf_!$ when one works within the correct category.

Serre Duality [Hartshorne]: Let $X$ be a projective scheme of dimension $n$ over an algebraically closed field $k$. Let $\omega_X^\circ$ be the dualizing sheaf on $X$. Then for any coherent sheaf $\mathcal{F}$ on $X$, there are natural functorial maps $$\theta^i : \operatorname{Ext}(\mathcal{F}, \omega_X^\circ) \rightarrow H^{n-i}(X, \mathcal{F})'.$$

How can one recast the above theorem in terms of an adjoint functor?

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up vote 4 down vote accepted

In the derived category of coherent sheaves on a smooth projective scheme $X$ of dimension $n$, Serre duality in the general form $\mathrm{Ext}^i(F,G \otimes \omega) \cong \mathrm{Ext}^{n-i}(G,F)^*$ becomes $\hom(F,G \otimes \omega[i]) \cong \hom(G,F[n-i])^*$, or simply $\hom(F,G \otimes \omega[n]) \cong \hom(G,F)^*$. This means that tensoring with $\omega[n]$ is a Serre functor. As soon as we have a Serre functor, it is abstract nonsense that every functor which has a left adjoint, also has a right adjoint. Roughly, it is given by twisting the left adjoint with the Serre functors. For example, if $f : X \to Y$ is a morphism between smooth projective schemes, this means that $R f_*$ has a right adjoint, given by $$R f^! := L f^* (- \otimes \omega_Y^{-1} [-\dim(Y)]) \otimes \omega_X[\dim(X)]$$

All this is very nicely explained in Lecture 4 of Caldararu's "Derived categories of sheaves: a skimming", online.

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Thanks for the summary and the reference to Caldararu's notes, they helped a lot. However I am still slightly confused on how this went down historically. It is my impression (from the notes and wikipedia) that (i) Grothendieck recasted Serre duality as a search for a right adjoint, (ii) Saw that no right adjoint existed, and (iii) moved to derived categories. However your explanation (as well as the one in the notes) does (iii) and then (i). So did (iii) really come before (i)? Or is there a way to see Serre duality as a search for a right adjoint without moving to derived categories? – RghtHndSd Nov 5 '13 at 1:52

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