Serre duality

Remarks

The idea of the proof: first treat $\mathbb{P}^N_k$, where the cohomology of the sheaves $\mathcal{O}(d)$ is computed explicitly by Čech cohomology and the pairing is visible on monomials. For $X \subseteq \mathbb{P}^N_k$ of codimension $r$, the dualizing sheaf of $X$ is $\mathcal{E}xt^r(\mathcal{O}_X, \omega_{\mathbb{P}^N})$, and for smooth $X$ this sheaf is $\omega_X$.

Example: for a smooth projective curve $X$ of genus $g$ and $\mathcal{E} = \mathcal{O}_X$, duality gives $h^1(X, \mathcal{O}_X) = h^0(X, \omega_X) = g$. Combined with Riemann–Roch it gives $\deg \omega_X = 2g - 2$.

Properness is necessary: for $X = \mathbb{A}^1_k$, $H^0(X, \mathcal{O}_X) = k[t]$ is infinite-dimensional while $H^1(X, \omega_X) = 0$ by affine vanishing. More generally the statement holds for projective Cohen–Macaulay schemes of pure dimension $n$, with $\omega_X$ replaced by the dualizing sheaf.

Going deeper

For a student

Serre duality is the algebraic analogue of Poincaré duality for compact oriented manifolds. It is used constantly with the Riemann–Roch theorem: on a curve, $h^1(L) = h^0(K - L)$, which turns Riemann–Roch into a statement about global sections only. Its general form, valid for proper morphisms and complexes of sheaves, is Grothendieck duality.