Elkik in Solutions d'equations a coefficients dans un anneu Henselian, Theorem 7 proves that:

Let $A$ be a Noetherian ring that is Henselian with respect to a principal ideal $(a)$.

That is, if $f(x)$ is a monic polynomial with a root $\alpha \in A/(a)$ such that $df/dx (\alpha)$ is a unit in $A/(a)$, then $\alpha$ lifts to a root of $f$ in $A$.

Let $\hat{A}$ be its $a$-adic completion, and let $\hat{B}$ be a formally finitely generated algebra over $\hat{a}$ that is formally smooth over $\hat{A}[a^{-1}]$. Then there exists a finitely generated algebra $B$ over $A$ that is smooth over $A[a^{-1}]$ and such that its $a$-adic completion is isomorphic to $\hat{B}$.

Is this theorem true in the non-Noetherian case? If not, what is a counterexample?

I would ideally prefer counterexamples of relative dimension $0$ such as finite etale covers.

  • $\begingroup$ $B$ is only smooth over $A[a^{-1}]$. $\endgroup$ – Laurent Moret-Bailly Aug 28 '14 at 6:59

If I understand the notation of Gabber and Romero corectly, the theorem is true in the non-Noetherian case for finite etale covers. Finite etale torsors on $X$ of degree $n$ are classified by elements of $H^1(X,S_n)$.

Thus we apply Theorem 5.8.14 of Almost Ring Theory by Gabber and Ramero.

To convert from my notation to theirs, take $R=A$, $t=a$, $I=(1)$, so that $R^ = \hat{A}$. This satisfies the conditions of Proposition 4.21.

Let $G$ be $S_n$. Then $G$ satisfies the conditions of 5.8.4 by explicit construction, or because of Lemma 5.8.5 and the fact that it is defined over the Dedekind domain $\mathbb Z$.

So we may apply the theorem and get an equality

$$H^1( \operatorname{Spec} A[a^{-1}], S_n) = H^1( \operatorname{Spec} \hat{A}[a^{-1}],S_n)$$

giving what I asked for (in the finite etale case).


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