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Let's say $\mathcal{O}$ is a complete DVR with fraction field $K$ and algebraically closed residue field $k$. (The case I had in mind here was with $\mathcal{O}$ of equicharacteristic $p$, so assume this if you like.)

Let $A$ be a complete local $\mathcal{O}$-algebra of finite Krull dimension. Say we are given a presentation of $A$ as $$ A=\mathcal{O}[[X_1,\dots,X_n]]/(f_1,\dots,f_m)$$

Let's also assume that $A$ is a domain, and that $A$ is $\mathcal{O}$-flat.

My question is:

Let $B$ be the integral closure of $A$ in $A\otimes K$. When is $B\otimes k$ a regular local ring (i.e. when is it a power series ring over $k$)?

This is the same as asking whether $B$ is formally smooth over $\mathcal{O}$.

Let's say $t$ lies in the maximal ideal of $\mathcal{O}$. The ring $\mathcal{O}[[X,Y]]/(Y^2-t^2X)$ has this property, but $\mathcal{O}[[X,Y]]/(XY=t)$ does not.

Is there an "algorithm" to determine whether $A$ has this property, based on the presentation above? I put algorithm in quotes because the problem probably involves knowing $f_1,\dots,f_m$ to infinite accuracy. To get around this point, let's assume that all the $f_i$ lie in a polynomial ring $D[X_1,\dots,X_n]$, where $D\subset\mathcal{O}$ is finitely generated, as in the above examples.

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