Lemma 10.136.4. Let $R \to S$ be a ring map. Suppose we have $g_1, \ldots g_ m \in S$ which generate the unit ideal such that each $R \to S_{g_ i}$ is syntomic. Then $R \to S$ is syntomic.

**Proof.**
This is true for being flat and for being of finite presentation by Lemmas 10.39.18 and 10.23.3. The property of having fibre rings which are local complete intersections is local on $S$ by its very definition, see Definition 10.135.1.
$\square$

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