Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.