The Stacks project

Lemma 53.4.2. Let $X$ be a proper scheme over a field $k$ which is Cohen-Macaulay and equidimensional of dimension $1$. The module $\omega _ X$ of Lemma 53.4.1 has the following properties

  1. $\omega _ X$ is a dualizing module on $X$ (Duality for Schemes, Section 48.22),

  2. $\omega _ X$ is a coherent Cohen-Macaulay module whose support is $X$,

  3. there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^ i_ X(K, \omega _ X[1]) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, K), k)$ compatible with shifts for $K \in D_\mathit{QCoh}(X)$,

  4. there are functorial isomorphisms $\mathop{\mathrm{Ext}}\nolimits ^{1 + i}(\mathcal{F}, \omega _ X) = \mathop{\mathrm{Hom}}\nolimits _ k(H^{-i}(X, \mathcal{F}), k)$ for $\mathcal{F}$ quasi-coherent on $X$.

Proof. Recall from the proof of Lemma 53.4.1 that $\omega _ X$ is as in Duality for Schemes, Example 48.22.1 and hence is a dualizing module. The other statements follow from Lemma 53.4.1 and the fact that $\omega _ X^\bullet = \omega _ X[1]$ as $X$ is Cohen-Macualay (Duality for Schemes, Lemma 48.23.1). $\square$

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