Document Type
Lecture
Publication Date
9-27-2017
Abstract
Semidualizing modules arise in the study of dualities over commutative rings. Here a nitely generated R-module C is semidualizing provided that HomR(C,C) ≅ R and ExtiR(C,C) = 0 for all i ≥ 1. Examples include the free R-module of rank 1 and Grothendiecks dualizing module for R (if one exists). Applications of these examples include Auslander and Bridgers G-dimension and Grothendiecks local duality. It has been argued that the number s0(R) of (isomorphism classes of) semidualizing modules over a local ring R is a measure of the severity of the singularity of R, namely, how far R is from being Gorenstein. For instance, if R is Gorenstein, then s0(R) = 1; and the converse holds if R is Cohen-Macaulay with a dualizing module. In this talk, however, we will show that s0 does not satisfy a standard condition one usually expects from such a measure: this invariant can increase after localizing. This is joint work with Saeed Nasseh, Ryo Takahashi, and Keller VandeBogert. Much of the talk will be spent on examples and big-picture perspective accessible to graduate students.
Relational Format
presentation
Recommended Citation
Sather-Wagstaff, Sean, "Semidualizing modules give a defective Gorenstein defect" (2017). Algebra/Number Theory Seminar. 32.
https://egrove.olemiss.edu/math_algebra_number_theory/32
