The Stacks project

Proof. The difference with Lemma 59.83.7 is that here we do not assume $X$ is smooth. Let $\nu : X^\nu \to X$ be the normalization morphism. Then $\nu $ is finite (Varieties, Lemma 33.27.1) and $X^\nu $ is smooth (Varieties, Lemma 33.43.8). Let $j^\nu : U^\nu \to X^\nu $ be the inverse image of $U$. By Lemma 59.83.7 the result holds for $j^\nu _!\underline{\mathbf{Z}/\ell \mathbf{Z}}$. By Lemma 59.83.5 the result holds for $\nu _*j^\nu _!\underline{\mathbf{Z}/\ell \mathbf{Z}}$. In general it won't be true that $\nu _*j^\nu _!\underline{\mathbf{Z}/\ell \mathbf{Z}}$ is equal to $j_!\underline{\mathbf{Z}/\ell \mathbf{Z}}$ but we can work around this as follows. As $X$ is reduced the morphism $\nu : X^\nu \to X$ is an isomorphism over a dense open $j' : X' \to X$ (Varieties, Lemma 33.27.1). Over this open we have agreement

Using Lemma 59.83.6 twice for $j' : X' \to X$ and the sheaves above we conclude. $\square$