2021/04/12 by Farrah Yhee, Yhee, Farrah C. · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2104.05766
openalex publication_date 2021/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The existence of Ulrich modules for local domains has been a difficult and elusive open question. For over thirty years, it was unknown whether local domains always have Ulrich modules. In this paper, we answer the question of existence for both Ulrich modules and weakly lim Ulrich sequences -- a weaker notion recently introduced by Ma -- in the negative. We construct many local domains in all dimensions d ≥ 2 that do not have any Ulrich modules. Moreover, we show that when d = 2, these local domains do not have weakly lim Ulrich sequences. A key insight in our proofs is the classification of MCM R-modules via the S2-ification of R. For local domains of dimension 2, we show that the existence of weakly lim Ulrich sequences implies the existence of lim Ulrich sequences.