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.

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 Avariety a finitely presented, irreducible, scheme X over A. Now, suppose that we have a Bpoint x ∈ X(B). Assume that the image of x is dense in X. Question: Can we find a morphism Y > X of Avarieties and a Bpoint 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 semistable 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? This already proves something about your cotangent complex because Y is lci over A. Right? 


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. 

