Counterexamples in the Theory of Ulrich Modules

Abstract

The theory of Ulrich modules has many powerful and broad applications ranging from the original purpose of giving a criterion for when a local Cohen-Macaulay ring is Gorenstein to new methods of finding Chow forms of a variety to longstanding open conjectures in multiplicity theory. For example, the existence of Ulrich modules and Ulrich-like objects has been the main approach to Lech's conjecture, which has been open for over 60 years. However, existence results have been very difficult to establish and for over thirty years, it was unknown whether (complete) local domains always have Ulrich modules. Recently, Ma introduced the weaker notion of (weakly) lim Ulrich sequences and showed that their existence for (complete) local domains implies Lech's conjecture. Ma then asks if (weakly) lim Ulrich sequences always exist for complete local domains.

In this thesis, we answer the question of existence for both Ulrich modules and weakly lim Ulrich sequences in the negative by constructing (complete) local domains that do not have any Ulrich modules or weakly lim Ulrich sequences. A key insight in our proofs is the classification of MCM modules over a ring $R$ via the $S_2$-ification of $R$. Moreover, for local domains of dimension 2, we show that the existence of weakly lim Ulrich sequences implies the existence of lim Ulrich sequences. Finally, our counterexamples are not standard-graded or Cohen--Macaulay. As such, we construct candidate counterexample rings that are standard-graded and/or Cohen--Macaulay from our original counterexamples.