# Artin/Popescu approximation for (some) big rings

Fix a prime number $p$. Let $A = \overline{\mathbf{Z}_p}$ be the integral closure of the $p$-adic integers $\mathbf{Z}_p$ in some fixed algebraic closure of its fraction field, and let $B$ be the $p$-adic completion of $A$. Is the map $A \to B$ an inductive limit of smooth morphisms? If $A$ was excellent, this would follow from Artin/Popescu approximation theorems, but $A$ is not even noetherian. Of course, one can ask similar questions much more generally, but this is the case I am interested in.

• It may be helpful if you indicate why you pose the question (e.g., idle curiosity or something more substantial). For example, some non-archimedean geometry (most naturally, Berkovich spaces) ensures that for a finite system of polynomial equations over $A$ (or even something more general), any solution in $B$ can be approximated arbitrarily well by a solution in $A$. So if that is your goal then the Artin-Popescu (and "smoothening") viewpoint in such generality is unnecessary. – grp Sep 12 '12 at 4:25
• I would like to know what the cotangent complex of $B$ relative to $\mathbf{Z}_p$ is. If the question I ask has an affirmative answer, then the relative cotangent complex of $B$ over $A$ is concentrated in degree $0$ (and is moreover a $\mathbf{Q}_p$-module). – anon Sep 13 '12 at 12:42

This is not an answer to the question in the main body, but the one raised by the OP in the comments.

The goal is to show that $L_{B/A}$ is concentrated in degree $0$ and $\mathbf{Q}_p$-module for $A = \overline{\mathbf{Z}_p}$ and $B$ the $p$-adic completion of $A$. The map $A \to B$ is flat and an isomorphism after reduction modulo $p$. The flat base change formula for the cotangent complex shows

$$L_{B/A} \otimes_B^L B/p \simeq 0,$$

i.e., that $p$ acts invertibly on $L_{B/A}$. By flat base change again, we get

$$L_{B/A} = L_{B/A}[p^{-1}] \simeq L_{B[p^-1]/A[p^{-1}]}.$$

As $A[p^{-1}] \to B[p^{-1}]$ is an extension of fields of characteristic $0$, it is enough to show:

Claim: If $E/K$ is an extension of fields of characteristic $0$, then $L_{E/K}$ is an $E$-vector space placed in degree $0$.

Proof: By expressing $E$ as a filtered colimit of finitely generated field extensions, as the cotangent complex commtues with such colimits, we may assume $E/K$ is finitely generated, i.e., there exists a finitely generated $K$-algebra $A$ which is a domain such that $E$ is the fraction field of $A$. By generic smoothness (as we are in characteristic $0$), we may assume $A$ is smooth over $K$. Then $L_{A/K} = \Omega^1_{A/K}$ is a finite locally free $A$-module in degree $0$. Localising at the generic point and using $L_{A/K} \otimes_A E \simeq L_{E/K}$ then proves the claim.

Let me answer (?) your question with a question. What is this question really asking? It seems that it asks something like this. Let's call an A-variety a finitely presented, irreducible, scheme X over A. Now, suppose that we have a B-point x ∈ X(B). Assume that the image of x is dense in X. Question: Can we find a morphism Y ---> X of A-varieties and a B-point y of Y which maps to X.

OK, and we can certainly find an alteration Y ---> X such that Y is the base change of a strictly semi-stable scheme over a suitable dvr, say R, contained in A. Since Y ---> X is an alteration and since B is algebraically closed (right?) we can lift x to a y in Y(B). Right?