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One version of Hensel's lemma for schemes is simply the definition of formally unramified:

A scheme $X$ is said to be formally etale if:

For a ring $R$ with an ideal $I$ such that $I^2 = 0$, one has that the map $X(R) \to X(R/I)$ is bijective.

One easily sees that this implies that for a complete local ring $R$ with reside field $k$, the map $X(R) \to X(k)$ is bijective. There are similar versions for formally smooth/unramified.

Question: In the last paragraph, I would like to replace complete by henselian. Is it still true and if so, what is a reference?

Asvin
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1 Answers1

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Since posting the question, I have learnt a few thing:

Let $(R,\mathfrak m)$ be a noetherian henselian local ring with residue field $k$ and completion $\hat R$. Then if $X$ is formally unramified over $$, the map $X(R) \to X(k)$ is injective.:

This is simply because by defn of formal unramified, $X(\hat R) \to X(k)$ is injective and since $R$ is noetherian, the map $R\to \hat R$ is injective. Therefore, $X(R) \to X(\hat R)$ is injective and composing with the first injection, we are done.

Now if $X$ is etale, then the map $X(R) \to X(k)$ is bijective:

This is simply expressing the fact that if $R \to S$ is etale, then for any closed point $s \in S$ with residue field $k$, there is a unique section $S\to R$ thats sends $s$ to $\mathfrak m$.

I still don't know what is supposed to replace formally smooth.

Asvin
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