Non-existence of Ulrich modules over Cohen-Macaulay local rings
Commutative Algebra
2025-03-13 v2 Algebraic Geometry
Abstract
Over a Cohen-Macaulay local ring, the minimal number of generators of a maximal Cohen-Macaulay module is bounded above by its multiplicity. In 1984 Ulrich asked whether there always exist modules for which equality holds; such modules are known nowadays as Ulrich modules. We answer this question in the negative by constructing families of two dimensional Cohen-Macaulay local rings that have no Ulrich modules. Some of these examples are Gorenstein normal domains; others are even complete intersection domains, though not normal.
Keywords
Cite
@article{arxiv.2403.15566,
title = {Non-existence of Ulrich modules over Cohen-Macaulay local rings},
author = {Srikanth B. Iyengar and Linquan Ma and Mark E. Walker and Ziquan Zhuang},
journal= {arXiv preprint arXiv:2403.15566},
year = {2025}
}
Comments
13 pages. The Introduction has been expanded, and a few minor corrections have been made in the text. This is slated to appear in the Commun. Am. Math. Soc