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CONSTRUCTING NONPROXY SMALL TEST MODULES FOR THE COMPLETE INTERSECTION PROPERTY

Part of: Local rings and semilocal rings Commutative algebra: Homological methods

Published online by Cambridge University Press:  21 June 2021

BENJAMIN BRIGGS Open the ORCID record for BENJAMIN BRIGGS [Opens in a new window] ,
ELOÍSA GRIFO Open the ORCID record for ELOÍSA GRIFO [Opens in a new window]  and
JOSH POLLITZ Open the ORCID record for JOSH POLLITZ [Opens in a new window]
BENJAMIN BRIGGS
Affiliation:
Department of Mathematics University of Utah Salt Lake City, UT 84112 USA briggs@math.utah.edu
ELOÍSA GRIFO
Affiliation:
Department of Mathematics University of Nebraska—Lincoln NE 68588 USA grifo@unl.edu
JOSH POLLITZ
Affiliation:
Department of Mathematics University of Utah Salt Lake City, UT 84112 USA pollitz@math.utah.edu
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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 $\mathsf {D}^{\mathsf f}(R)$, 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 $\mathsf {D}^{\mathsf f}(R)$ 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.

MSC classification

Primary: 13D09: Derived categories
Secondary: 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

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Type
Article
Information
Nagoya Mathematical Journal , Volume 246 , June 2022 , pp. 412 - 429
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Copyright
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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Footnotes

The second author was supported by the National Science Foundation grant DMS-2001445. The third author was supported by the National Science Foundation grant DMS-2002173 and the National Science Foundation Research Training Group grant DMS-1840190.

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