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.



Grothendieck duality for resolution of singularities
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 
$f^!G \simeq f^\*G$. 


