Grothendieck duality

Remarks

Grothendieck's theory of duality was first written up in Hartshorne's book Residues and Duality. In a later approach by Neeman, a right adjoint of $Rf_*$ on the unbounded category $D_{qc}(X)$ exists by Brown representability. The main content is then the explicit form of $f^!$ and of the trace map $Rf_* f^! \to \mathrm{id}$.

Example: let $Y = \operatorname{Spec} k$ and $X$ a smooth projective variety of dimension $n$ over $k$. Then $f^! k = \omega_X[n]$, and taking $\mathcal{F}$ to be a vector bundle placed in degree $-i$ gives Serre duality $H^{n-i}(X, \mathcal{E}^\vee \otimes \omega_X) \cong H^i(X, \mathcal{E})^\vee$.

For a finite morphism $f$, the right adjoint is $f^! \mathcal{G} = R\mathcal{H}om_{\mathcal{O}_Y}(f_* \mathcal{O}_X, \mathcal{G})$, viewed as a complex of $\mathcal{O}_X$-modules.

Going deeper

For a student

The formula $f^! = Lf^* \otimes \omega_{X/Y}[n]$ for smooth $f$ and the description of $f^!$ for finite $f$ together compute $f^!$ for projective morphisms, by factoring through a closed immersion into projective space. The complex $f^! \mathcal{O}_Y$ is the relative dualizing complex; for flat $f$ of finite type it is concentrated in one degree exactly when the fibers of $f$ are Cohen–Macaulay. Grothendieck duality underlies the theory of dualizing complexes, local duality and the Grothendieck–Riemann–Roch formalism of pushforwards.