in the middle of working on some deformation related questions, I bumped into a question regarding finite type subalgebras of some completed algebras. I hope someone may know the answer, or may possibly be able to offer some ideas on it.
To make the question simple enough, let me take the following simplest case:
Let $k$ be a field and $k[[t]]$ be the ring of formal power series in 1 variable with the coefficients in $k$.
Suppose we have a finite type $k$-subalgebra $R \subset k[[t]]$, i.e. there exist some finitely many formal power series $f_1, \cdots, f_r$ in $t$ such that $R= k[f_1, \cdots, f_r]$.
Then, can one have a good way to compute the Krull dimension of $R$? Of course we know it is bounded by $r$ by Noether normalization, but I am asking whether we have some ways to "reduce" those $f_i$ in a sense.
Apparently, the case $r=1$ is trivial, so the question is interesting only when $r \geq 2$.
If all of $f_i$ are given by polynomials, then the question is very simple: the ring $R$ is contained in $k[x]$ and it is integral over $R$, so $R$ must be of dimension $1$.
However, when they are given by formal power series in general, I am not sure whether I can use some sorts of Weierstrass theorems here. Can one see any possible ways to handle these?
