The Stacks project

Lemma 58.20.6. In Situation 58.19.1. Let $V$ be finite étale over $U$. Assume

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

  2. $V_0 = V \times _ U U_0$ is equal to $Y_0 \times _{X_0} U_0$ for some $Y_0 \to X_0$ finite étale.

Then $V = Y \times _ X U$ for some $Y \to X$ finite étale.

Proof. We reduce to the complete case using Lemma 58.20.4. (The assumptions carry over; use Dualizing Complexes, Lemma 47.9.3.)

In the complete case we can lift $Y_0 \to X_0$ to a finite étale morphism $Y \to X$ by More on Algebra, Lemma 15.13.2; observe that $(A, fA)$ is a henselian pair by More on Algebra, Lemma 15.11.4. Then we can use Lemma 58.19.6 to see that $V$ is isomorphic to $Y \times _ X U$ and the proof is complete. $\square$

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