Pro-affine varieties over a local field - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T02:00:35Z http://mathoverflow.net/feeds/question/95159 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95159/pro-affine-varieties-over-a-local-field Pro-affine varieties over a local field Harry 2012-04-25T13:43:33Z 2012-04-26T17:29:14Z <p>Let $K$ be a (perfect) local field, and let $S = \lim (\mathrm{Spec} A_i)_{i=0}^\infty$ be a pro-affine variety over $K$. This means that each $A_i$ is a finite type $K$-algebra and that the affine varieties $\mathrm{Spec} A_i$ form a projective system. Note that $S$ is an affine $K$-scheme which is not of finite type in general.</p> <p>I have some elementary questions about such schemes.</p> <p>Q1.Is $S$ regular if and only if $\mathrm{Spec} A_i$ is regular for all $i=0,\ldots$?</p> <p>Q1*.Is $S$ normal (resp. irreducible or reduced) if and only if $\mathrm{Spec} A_i$ is normal (resp. irreducible or reduced) for all $i=0,\ldots$?</p> <p>Q2. How can I determine the dimension of $S$ from the dimension of $A_i$. (Assume the schemes to be integral for this question.)</p> <p>Unfortunately, I have little feeling for such pro-affine varieties at the moment.</p> <p>A last (and bit vague) question:</p> <p>Q3. Is there a moduli space of smooth connected pro-affine varieties of given dimension?</p> http://mathoverflow.net/questions/95159/pro-affine-varieties-over-a-local-field/95278#95278 Answer by anon for Pro-affine varieties over a local field anon 2012-04-26T17:29:14Z 2012-04-26T17:29:14Z <p>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 $i$.</p>