Relative reduction numbers, regularity of blowup algebras, and Ulrich modules

Abstract

We prove new results concerning the connection between (relative) reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules. A key basic tool is the operation of (relative) Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension and allowing for the setting relative to a Cohen-Macaulay module. We derive a number of applications, including progress on the theory of generalized Ulrich ideals and modules and improvements of results by other authors.