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The Stacks project

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


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