Remark 53.4.3. Let $X$ be a proper scheme of dimension $\leq 1$ over a field $k$. Let $\omega _ X^\bullet$ and $\omega _ X$ be as in Lemma 53.4.1. If $\mathcal{E}$ is a finite locally free $\mathcal{O}_ X$-module with dual $\mathcal{E}^\vee$ then we have canonical isomorphisms

$\mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, \mathcal{E}), k) = H^ i(X, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _ X^\bullet )$

This follows from the lemma and Cohomology, Lemma 20.48.5. If $X$ is Cohen-Macaulay and equidimensional of dimension $1$, then we have canonical isomorphisms

$\mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, \mathcal{E}), k) = H^{1 + i}(X, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \omega _ X)$

by Lemma 53.4.2. In particular if $\mathcal{L}$ is an invertible $\mathcal{O}_ X$-module, then we have

$\dim _ k H^0(X, \mathcal{L}) = \dim _ k H^1(X, \mathcal{L}^{\otimes -1} \otimes _{\mathcal{O}_ X} \omega _ X)$

and

$\dim _ k H^1(X, \mathcal{L}) = \dim _ k H^0(X, \mathcal{L}^{\otimes -1} \otimes _{\mathcal{O}_ X} \omega _ X)$

Comment #7163 by Xuande Liu on

I think the second equation should be $Hom_k(H^i(X,E),k) = H^{1-i}(X,E^\vee\otimes_{O_X}\omega_X)$.

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