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Let $K$ be a (perfect) local field, and let $S = \lim (\operatorname{Spec} A_i)_{i=0}^\infty$ be a pro-affine variety over $K$. This means that each $A_i$ is a finite type $K$-algebra and that the affine varieties $\operatorname{Spec} A_i$ form a projective system. Note that $S$ is an affine $K$-scheme which is not of finite type in general.

I have some elementary questions about such schemes.

Q1.Is $S$ regular if and only if $\operatorname{Spec} A_i$ is regular for all $i=0,\ldots$?

Q1*.Is $S$ normal (resp. irreducible or reduced) if and only if $\operatorname{Spec} A_i$ is normal (resp. irreducible or reduced) for all $i=0,\ldots$?

Q2. How can I determine the dimension of $S$ from the dimension of $A_i$. (Assume the schemes to be integral for this question.)

Unfortunately, I have little feeling for such pro-affine varieties at the moment.

A last (and bit vague) question:

Q3. Is there a moduli space of smooth connected pro-affine varieties of given dimension?

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    $\begingroup$ It looks like any affine scheme of countable type over $K$ is a pro-affine $K$-variety in your sense. Am I mistaken? $\endgroup$
    – S. Carnahan ♦
    Commented Apr 26, 2012 at 3:59
  • $\begingroup$ I think you're right. You just write $A$ as a direct limit of its finitely generated sub-$K$-algebras. $\endgroup$
    – Harry
    Commented Apr 26, 2012 at 7:16

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The nature of the ground field $K$ doesn't seem to be very relevant. In general $S$ won't even be noetherian. The only positive result I know is the following: if $S_i$ is regular for all sufficiently large $i$, and the map $S_j\to S_i$ is etale for all sufficiently large $i,j$, then $S$ is noetherian and regular (and $\dim S=\dim S_i$ for all sufficiently large $i$).

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