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

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.