Non-existence of Ulrich modules over Cohen-Macaulay local rings

TL;DR

The paper settles Ulrich's question in the negative by constructing two-dimensional Cohen-Macaulay local rings with no Ulrich modules, including Gorenstein normal domains and complete intersections. The method ties the existence of Ulrich modules to Ulrich sheaves on the exceptional fiber $E$ of the blow-up at the maximal ideal and leverages geometric criteria on $E$ to obstruct extendability. It provides explicit families, via section rings of ample line bundles on curves (and their twists), for which no Ulrich modules can exist, and develops a corollary criterion (co:noU-geometric) that yields all the nonexistence results. This work highlights a deep link between Ulrich theory and blow-up geometry, showing that Ulrich phenomena are not universal even among CM rings, while leaving the question of Ulrich sheaves on these spaces open for further study.

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.

Theorems & Definitions (20)

• Theorem 1.1
• Remark 2.1
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Corollary 3.3
• proof