Generalization of bi-canonical degrees

Abstract

We discuss invariants of Cohen-Macaulay local rings that admit a canonical module ω. Attached to each such ring R, when ω is an ideal, there are integers-the type of R, the reduction number of ω-that provide valuable metrics to express the deviation of R from being a Gorenstein ring. In [6] and [7] we enlarged this list with the canonical degree and the bi-canonical degree. In this work we extend the bi-canonical degree to rings where ω is not necessarily an ideal. We also discuss generalizations to rings without canonical modules but admitting modules sharing some of their properties.

Rings with q-torsionfree canonical modules

“…The following proposition shows the case where K A is torsionless without assuming Q(A) is Gorenstein. It is also a generalization of [4,Proposition 3.2].…”

“…Therefore, behaviors of canonical modules, even for non-Cohen-Macaulay rings, are interesting and the q-torsionfree property is well worth studying. The motivation for the present research started with this question that arose while the second and fourth authors were writing the last paper with Vasconcelos concerning (torsionless) canonical modules [4].…”