On reduction numbers and Castelnuovo–Mumford regularity of blowup rings and modules

We prove new results on the interplay between reduction numbers and the Castelnuovo–Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of 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 largely 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 a characterization of (polynomial) ideals of linear type, progress on the theory of generalized Ulrich ideals, and improvements of results by other authors.