Lemma 53.5.2 (Riemann-Roch). Let $X$ be a proper scheme over a field $k$ which is Gorenstein and equidimensional of dimension $1$. Let $\omega _ X$ be as in Lemma 53.4.1. Then

$\omega _ X$ is an invertible $\mathcal{O}_ X$-module,

$\deg (\omega _ X) = -2\chi (X, \mathcal{O}_ X)$,

for a locally free $\mathcal{O}_ X$-module $\mathcal{E}$ of constant rank we have

\[ \chi (X, \mathcal{E}) = \deg (\mathcal{E}) - \textstyle {\frac{1}{2}} \text{rank}(\mathcal{E}) \deg (\omega _ X) \]and $\dim _ k(H^ i(X, \mathcal{E})) = \dim _ k(H^{1 - i}(X, \mathcal{E}^\vee \otimes _{\mathcal{O}_ X} \omega _ X))$ for all $i \in \mathbf{Z}$.

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