On the theory of generalized Ulrich modules

Abstract

In this paper we further develop the theory of generalized Ulrich modules over Cohen-Macaulay local rings introduced in 2014 by Goto, Ozeki, Takahashi, Watanabe and Yoshida. The term generalized refers to the fact that Ulrich modules are taken with respect to a zero-dimensional ideal which is not necessarily the maximal ideal, the latter situation corresponding to the classical theory from the 80's; despite the apparent naivety of the idea, this passage adds considerable depth to the theory and enlarges its horizon of applications. First, we address the problem of when the Hom functor preserves the Ulrich property, and in particular we study relations with semidualizing modules. Second, we explore horizontal linkage of Ulrich modules, which we use to provide a characterization of Gorensteiness. Finally, we investigate connections between Ulrich modules and modules with minimal multiplicity, including characterizations in terms of relative reduction numbers as well as the Castelnuovo-Mumford regularity of certain blowup modules.