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Constructing nonproxy small test modules for the complete intersection property

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Title: Constructing nonproxy small test modules for the complete intersection property
Authors: BRIGGS, B
GRIFO, E
POLLITZ, J
Item 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.
Issue Date: Jun-2022
Date of Acceptance: 1-Jun-2021
URI: http://hdl.handle.net/10044/1/114004
DOI: 10.1017/nmj.2021.7
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/).
Publication Status: Published
Online Publication Date: 2021-06-21
Appears in Collections:Pure Mathematics
Mathematics



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