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Let $K[[T_1,...,T_n]]$ be a finitely many variables formal power series ring over a field $K$.

Dorin Popescu proved that there are smooth algebras $A_{\lambda}$'s which are of finite type over $K$ and index set $\Lambda$ such that we have an isomorphism $K[[T_1,...,T_n]] \cong \underset{\lambda \in \Lambda}{\varinjlim}\,A_{\lambda}$. Now, we define the infinitely many variables formal power series ring $K[[T_1,...,T_{\infty}]]$ by the following.

$K[[T_1,...,T_{\infty}]] \colon= \underset{n \geq 1}{\varprojlim}\,K[[T_1,...,T_n]]$.

Q. Is it possible to write $K[[T_1,...,T_{\infty}]] \cong \underset{\omega \in \Omega}{\varinjlim}\,B_{\omega}$ by some smooth algebras $B_{\omega}$'s which are of finite type over a field $K$ and index set $\Omega$ ?

$K[[T_1,...,T_{\infty}]]$ is a non-noetherian local ring with the unique maximal ideal $(T_1,T_2,...)$. I guess that $K[[T_1,...,T_{\infty}]]$ should be 'regular' in a certain sense, although it is not noetherian.

I feel inclined to think that the above question would hold.

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    $\begingroup$ With your definition, $K[[T_1,\dots,T_\infty]]$ has an element deserving the name $\sum_{n=1}^{\infty}T_n$. It is in the maximal ideal, but not in the ideal $(T_1,T_2,\dots)$. By the way, the notation $K[[T_1,\dots,T_\infty]]$ is misleading since there is no $T_\infty$. $\endgroup$
    – Laurent Moret-Bailly
    Jun 18, 2016 at 7:24
  • $\begingroup$ Dear Prof. Moret-Bailly, thanks for your comment. Yes I should say the unique closed maximal ideal, but $(T_1,...,T_{\infty})$ is open. I am thinking that the similar question, but each ring is polynomial ring will suffice. That is, $\underset{n \geq 1}{\varprojlim}\,K[T_1,...,T_n]$ will be written by the inductive limit of smooth algebras over $K$, which I cannot show. $\endgroup$
    – Pierre MATSUMI
    Jun 27, 2016 at 5:19

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