I would like to know a reference for Grothendieck duality in a resolution of singularities. More precisely, let $Y$ be a normal, Gorenstein variety with finite quotient singularities, and suppose that $f\colon X \to Y$ is a crepant resolution of singularities, meaning that $f^*\omega_Y \cong \omega_X$. In particular, since $Y$ is normal, we know that $Rf_*\mathcal{O}_X \cong \mathcal{O}_Y$.

By general results valid for any projective morphism, there is a functor $f^!\colon D^b(Y) \to D^b(X)$ between the derived categories of $X$ and $Y$ which gives an isomorphism

$$ Rf_*R\mathcal{Hom}(F, f^! G ) \cong R\mathcal{Hom}(Rf_*F,G) $$ for any $F \in D^b(X)$ and $G \in D^b(Y)$

My question is: is there nice description of $f^{!}$ in this situation?

I guess it should be related to $Lf^*$ and the relative canonical, but I don't know much about this topic.


1 Answer 1


In general, $f^! = R Hom_X(L f^*R Hom_Y(\__, \omega_Y^\bullet), \omega_X^\bullet)$, so given that both $X$ and $Y$ are Gorenstein, this simplifies quite a bit. In particular, if $G$ is a locally free sheaf, then $f^!G \simeq f^*G$.


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