The Stacks project

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

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

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

  3. 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 $i \in \mathbf{Z}$.

Proof. Recall that Gorenstein schemes are Cohen-Macaulay (Duality for Schemes, Lemma 48.24.2) and hence $\omega _ X$ is a dualizing module on $X$, see Lemma 53.4.2. It follows more or less from the definition of the Gorenstein property that the dualizing sheaf is invertible, see Duality for Schemes, Section 48.24. By ( applied to $\omega _ X$ we have

\[ \chi (X, \omega _ X) = \deg (c_1(\omega _ X) \cap [X]_1) + \chi (X, \mathcal{O}_ X) \]

Combined with Lemma 53.5.1 this gives

\[ 2\chi (X, \mathcal{O}_ X) = - \deg (c_1(\omega _ X) \cap [X]_1) = - \deg (\omega _ X) \]

the second equality by ( Putting this back into ( for $\mathcal{E}$ gives the displayed formula of the lemma. The symmetry in dimensions is a consequence of duality for $X$, see Remark 53.4.3. $\square$

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