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

TL;DR

This paper investigates the construction of specific modules over local rings to characterize complete intersection properties, extending known results from derived categories to module-level analysis with an algorithmic approach.

Contribution

It introduces an algorithm to construct modules that are not proxy small in non-complete intersection rings, bridging derived category characterizations with module-level constructions.

Findings

Provides an explicit construction method for non-proxy small modules

Extends the characterization of complete intersections to module-level analysis

Applies the algorithm to equipresented and Stanley-Reisner rings

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 $\mathsf{D}^f(R)$, 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 $\mathsf{D}^f(R)$ 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.