Title: Intersections of ultrapowers
Speaker: Alan Loper (Ohio State University)
Abstract: An ultrafilter can be used to build an ultraproduct of an infinite collection of rings. The ultraproduct is a quotient of the direct product. It shares with the direct product the feature that it is very large. In this work we consider each component of the product to be the same ring and refer to the ultraproduct as an ultrapower. Let $D$ be a commutative integral domain and fix a nonprincipal ultrafilter to use building ultrapowers. If $D$ is local then an ultrapower of $D$ is also local. If $D$ has an infinite number of maximal ideals then an ultrapower of $D$ has a lot of maximal ideals - perhaps more than can reasonably be studied. One way to ameliorate this situation a little is to go from $D$ to the localizations $D_M$ for each maximal ideal $M$, construct an ultrapower of each localization, and then intersect the resulting rings. We consider this construction, and then move to a more general setting. Choose a collection of overrings of $D$ (rings that lie between $D$ and its quotient field) that intersect to $D$, construct the ultrapower of each overring, and then intersect the resulting rings. Of particular interest is the case where each overring is a valuation domain. We demonstrate some surprising properties of these intersections.
