Ulrich modules over local rings of dimension two

TL;DR

The paper proves the existence of Ulrich modules for a broad class of two-dimensional local rings by relating them to Ulrich sheaves on the exceptional fiber $E$ of the blow-up. It shows that, when $R$ is complete and equidimensional with $E$ geometrically reduced, there exists an Ulrich $R$-module locally free on the punctured spectrum, and in the algebraically closed case this module can be taken of rank one. This lifting from Ulrich sheaves on $E$ to modules on $R$ yields the Length Conjecture for this class of rings and extends to non-equidimensional cases via reduction to equidimensional quotients using $j(R)$ and Dutta multiplicity. The results connect geometric Ulrich data on $E$ with algebraic Ulrich modules on $R$, providing concrete existence results and quantitative length bounds for modules of finite projective dimension.

Abstract

It is proved that Ulrich modules exist for a large class of local rings of dimension two. This complements earlier work of the authors and Ziquan Zhuang that described complete intersection domains of dimension two that admit no Ulrich modules. As an application, it is proved that, for this class of rings, the length of a nonzero module of finite projective dimension is at least the multiplicity of the local ring.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2
• Proposition 2.5
• proof
• Corollary 2.6
• Theorem 3.1
• Example 3.2