The Stacks project

Lemma 53.19.16. Let $k$ be a field. Let $X$ be a locally algebraic $k$-scheme equidimensional of dimension $1$ whose singularities are at-worst-nodal. Then $X$ is Gorenstein and geometrically reduced.

Proof. The Gorenstein assertion follows from Lemma 53.19.15 and Duality for Schemes, Lemma 48.24.5. Or you can use that it suffices to check after passing to the algebraic closure (Duality for Schemes, Lemma 48.25.1), then use that a Noetherian local ring is Gorenstein if and only if its completion is so (by Dualizing Complexes, Lemma 47.21.8), and then prove that the local rings $k[[t]]$ and $k[[x, y]]/(xy)$ are Gorenstein by hand.

To see that $X$ is geometrically reduced, it suffices to show that $X_{\overline{k}}$ is reduced (Varieties, Lemmas 33.6.3 and 33.6.4). But $X_{\overline{k}}$ is a nodal curve over an algebraically closed field. Thus the complete local rings of $X_{\overline{k}}$ are isomorphic to either $\overline{k}[[t]]$ or $\overline{k}[[x, y]]/(xy)$ which are reduced as desired. $\square$

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