Completions of Extremely Noncatenary Local UFDs by Eli Dugan '25

Completions of Extremely Noncatenary Local UFDs by Eli Dugan ’25, Monday April 7, 1:00 – 1:50pm, North Science Building 113, Wachenheim, Mathematics Colloquium

Abstract: In commutative algebra, we study commutative rings by investigating the structure of their prime ideals. We treat the set of prime ideals of a ring as a partially ordered set (poset), ordered by inclusion. With this view, it is natural to ask when a given poset is realized inside the prime spectrum of some commutative ring. This question can be made more challenging by considering increasingly complex posets, and by restricting to relatively well-behaved families of rings. In this talk, we build on work of Bonat and Loepp to fully characterize the completions of local UFDs that are noncatenary at infinitely many places. We also establish a general set of sufficient conditions for a ring to be the completion of a local UFD that is noncatenary at every height 1 prime ideal.