If $X$ is a variety and $x \in X$, there are several ways to look locally around the point $x$:
- Localisation: taking the direct limit over open immersions around $x$.
- Henselisation: taking the direct limit over étale maps around $x$.
- Completion: taking an inverse limit.
I would like to know if completing can be seen as taking a direct limit over a certain class of maps. Let me state the problem more formally.
Let $A$ be a local ring with maximal ideal $\frak{m}$ and let $\hat{A}$ be its completion with respect to $\mathfrak{m}$.
Suppose that the map $A \rightarrow \hat{A}$ is regular (apparently this is the case if $A$ is an excellent local ring). By Popescu's theorem $\hat{A}$ is a direct limit of smooth (finitely presented) $A$-algebras.
I would like to know if $\hat{A}$ is actually a direct limit of étale $A$-algebras or if there is another characterisation of such maps.
To put the result into context, the henselisation $A^h$ of $A$ is obtained as a direct limit of étale $A$-algebras with a distinguished point and prescribed residue field.
