Constructing nonproxy small test modules for the complete intersection property
File(s)
Accepted version
Author(s)
BRIGGS, BENJAMIN
GRIFO, ELOÍSA
POLLITZ, JOSH
Type
Journal Article
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.
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.
Date Issued
2022-06
Date Acceptance
2021-06-01
Citation
Nagoya Mathematical Journal, 2022, 246, pp.412-429
ISSN
0027-7630
Publisher
Cambridge University Press
Start Page
412
End Page
429
Journal / Book Title
Nagoya Mathematical Journal
Volume
246
Copyright Statement
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license. This is the author’s accepted manuscript made available under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (https://creativecommons.org/licenses/by-nc-nd/4.0/).
Identifier
http://dx.doi.org/10.1017/nmj.2021.7
Publication Status
Published
Date Publish Online
2021-06-21