The Prime Ideal Structure of Precompletions by Prof. Susan Loepp, Mathematics Faculty Seminar, Friday, November 15, 1 – 1:45 pm, Stetson Court Classroom 105
Abstract: Given a ring with exactly one maximal ideal, we can define a metric on the ring with respect to its maximal ideal. We then complete the ring with respect to this metric, and call the resulting ring the completion of the original ring. In this talk, we explore the relationship between the prime ideals of a local ring and the prime ideals of its completion. In particular, we ask the following question: If T is a complete local ring and X a partially ordered set, under what conditions is there a local ring A whose completion is T and whose prime ideal structure, when viewed as a partially ordered set, is X? The results presented in the talk were proved by the SMALL 2019 commutative algebra group (including Erica Barrett ’21 and Emil Graf ’21). Instead of stating general theorems, we will focus mostly on examples that illustrate the power of the theorems.