Degrees: Vasconcelos contributions - Ghezzi, Laura; Hong, Jooyoun (De Gruyter Proc. Math., 2025)
A degree of a module M is a numerical measure of information carried by M. We highlight some of Vasconcelos’ outstanding contributions to the theory of degrees, bridging commutative algebra and computational algebra. We present several degrees he introduced and developed, including arithmetic degree, jdeg, homological degree, cohomological degrees, canonical degree, and bicanonical degree. For the canonical and bicanonical degrees, we discuss recent developments motivated by our joint works [25, 19, 9]. ©2025 Walter de Gruyter GmbH,Berlin/Boston.
Keywords: Multiplicity; arithmetic degree; bicanonical degree; canonical degree; canonical ideal; cohomological degree; dept.mat; homological degree; jdeg

Rings with q-torsionfree canonical modules - Endo, Naoki; Ghezzi, Laura; Goto, Shiro; Hong, Jooyoun; Iai, Shin-Ichiro; Kobayashi, Toshinori; Matsuoka, Naoyuki; Takahashi, Ryo (Commutative Algebra, 2025)
Let A be a Noetherian local ring with canonical module KA. We characterize A when KA is a torsionless, reflexive, or q-torsionfree module for an integer q ≥ 3. If A is a Cohen–Macaulay ring, H.-B. Foxby proved in 1974 that the A-module KA is q-torsionfree if and only if the ring A is q-Gorenstein. With mild assumptions, we provide a generalization of Foxby’s result to arbitrary Noetherian local rings admitting the canonical module. In particular, since the reflexivity of the canonical module is closely related to the ring being Gorenstein in low codimension, we also explore quasinormal rings, introduced by W. V. Vasconcelos. We provide several examples as well. ©2025 Walter de Gruyter GmbH,Berlin/Boston.
Keywords: Algebra over a field; Canonical module; Class (philosophy); Cohen–Macaulay ring; Gorenstein ring; Representation (politics); Set (abstract data type); dept.mat; q-Gorenstein ring; q-torsionfree module; quasinormal ring