Constructing non-proxy small test modules for the complete intersection property

Authors

Benjamin Briggs, University of UtahFollow
Eloísa Grifo, University of California, RiversideFollow
Josh Pollitz, University of UtahFollow

Document Type

Article

Date of this Version

5-2021

Citation

arXiv:2009.11800v3 [math.AC] 13 May 2021

Comments

To appear in the Nagoya Mathematical Journal

Copyright © 2021 by the authors.

Abstract

A local ring R is regular if and only if every finitely generated R-module has finite projective dimension. Moreover, the residue field k is a test module: R is regular if and only if k has finite projective dimension. This characterization can be extended to the bounded derived category Df(R), which contains only small objects if and only if R is regular. Recent results of Pollitz, completing work initiated by Dwyer-Greenlees-Iyengar, yield an analogous characterization for complete intersections: R is a complete intersection if and only if every object in Df(R) is proxy small. In this paper, we study a return to the world of R-modules, and search for finitely generated R-modules that are not proxy small whenever R is not a complete intersection. We give an algorithm to construct such modules in certain settings, including over equipresented rings and Stanley-Reisner rings.