Gluing Positive Height Prime Ideals in Complete Local Rings by Jackson Ehrenworth '23

Gluing Positive Height Prime Ideals in Complete Local Rings by Jackson Ehrenworth ’23, Mathematics Senior Thesis Defense, Monday, May 15, 1:45 – 2:30 pm, North Science Building 113, Wachenheim.

Abstract:  If we wish to study the prime spectrum of a ring (we algebraists get excited about the strangest things), we might take a prime ideal and either localize at or quotient out by it.  Similarly, although perhaps less familiarly, we can take the completion of a ring as another investigative tool.  While we understand very well what happens to the prime spectrum under localization and quotienting, we know much less about what is preserved under completion.  Another way to look at it is to take a complete local ring and try to determine how “weird” the prime spectrum of a precompletion can get.  As it turns out, that ends up being quite difficult.  In this talk, I will present one tool I developed as a means to investigate this question: gluing positive height prime ideals (modulo some fine print) while preserving completion.