Lemma 48.24.6. The property $\mathcal{P}(S) =$“$S$ is Gorenstein” is local in the syntomic topology.

**Proof.**
Let $\{ S_ i \to S\} $ be a syntomic covering. The scheme $S$ is locally Noetherian if and only if each $S_ i$ is Noetherian, see Descent, Lemma 35.16.1. Thus we may now assume $S$ and $S_ i$ are locally Noetherian. If $S$ is Gorenstein, then each $S_ i$ is Gorenstein by Lemma 48.24.5. Conversely, if each $S_ i$ is Gorenstein, then for each point $s \in S$ we can pick $i$ and $t \in S_ i$ mapping to $s$. Then $\mathcal{O}_{S, s} \to \mathcal{O}_{S_ i, t}$ is a flat local ring homomorphism with $\mathcal{O}_{S_ i, t}$ Gorenstein. Hence $\mathcal{O}_{S, s}$ is Gorenstein by Dualizing Complexes, Lemma 47.21.8.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: