The Stacks project

Proof. If $H^1(X, \mathcal{I}\mathcal{L})$ is nonzero, then there is a nonzero map $\varphi : \mathcal{I}\mathcal{L} \to \omega _ X$, see Lemma 53.4.2. Let $Y \subset X$ be the union of the irreducible components $C$ of $X$ such that $\varphi $ is nonzero in the generic point of $C$. Then $Y$ is a reduced closed subscheme. Let $\mathcal{J} \subset \mathcal{O}_ X$ be the ideal sheaf of $Y$. Since $\mathcal{J}\mathcal{I}\mathcal{L}$ has no embedded associated points (as a submodule of $\mathcal{L}$) and as $\varphi $ is zero in the generic points of the support of $\mathcal{J}$ (by choice of $Y$ and as $X$ is reduced), we find that $\varphi $ factors as

We can view $\mathcal{I}\mathcal{L}/\mathcal{J}\mathcal{I}\mathcal{L}$ as the pushforward of a coherent sheaf on $Y$ which by abuse of notation we indicate with the same symbol. Since $\omega _ Y = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{O}_ Y, \omega _ X)$ by Lemma 53.4.5 we find a map

of $\mathcal{O}_ Y$-modules which is injective in the generic points of $Y$. Let $\mathcal{I}' \subset \mathcal{O}_ Y$ be the ideal sheaf of $Z \cap Y$. There is a map $\mathcal{I}\mathcal{L}/\mathcal{J}\mathcal{I}\mathcal{L} \to \mathcal{I}'\mathcal{L}|_ Y$ whose kernel is supported in closed points. Since $\omega _ Y$ is a Cohen-Macaulay module, the map above factors through an injective map $\mathcal{I}'\mathcal{L}|_ Y \to \omega _ Y$. We see that we get an exact sequence

of coherent sheaves on $Y$ where $\mathcal{Q}$ is supported in dimension $0$ (this uses that $\omega _ Y$ is an invertible module in the generic points of $Y$). We conclude that

by Lemma 53.5.1 and Varieties, Lemma 33.33.3. If $Y$ is connected, then this proves the lemma. If not, then we repeat the last part of the argument for one of the connected components of $Y$. $\square$