The Stacks project

[Corollary 1.11, Bhatt-local]

Lemma 58.19.6. In Situation 58.19.1. Assume

  1. $H^1_\mathfrak m(A)$ and $H^2_\mathfrak m(A)$ are annihilated by a power of $f$, and

  2. $A$ is henselian or more generally $(A, (f))$ is a henselian pair.

Then the restriction functor $\textit{FÉt}_ U \longrightarrow \textit{FÉt}_{U_0}$ is fully faithful.

Proof. By Lemma 58.19.4 we may assume that $A$ is a Noetherian complete local ring. (The assumptions carry over; use Dualizing Complexes, Lemma 47.9.3.) By Lemma 58.17.1 the result follows from Algebraic and Formal Geometry, Lemma 52.15.5. $\square$

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