Complexity and curvature of pairs of Burch modules and ideals

TL;DR

This paper investigates the complexity and curvature of pairs of Burch modules and ideals, establishing their extremal properties and characterizations of complete intersection rings through these invariants.

Contribution

It extends the study of complexity and curvature to pairs of Burch modules and ideals, providing new extremal results and characterizations of complete intersection rings.

Findings

Complexity and curvature of Burch ideal pairs equal that of the residue field.

Finite complexity or curvature at most 1 characterizes complete intersection rings.

Results are derived from general properties of Burch modules.

Abstract

The complexity and curvature of a module were first introduced by Avramov to distinguish modules of infinite homological dimension. Later, Avramov-Buchweitz extended the notion of complexity from a single module to that of pairs of modules, which measures the polynomial growth rate of the minimal number of generators of their Ext modules. Dao studied a similar notion of Tor-complexity. Recently, Dey-Ghosh-Saha initiated the study of Ext and Tor curvature of a pair of modules, which measure the exponential growth rates of the corresponding Ext and Tor, respectively. On the other hand, the concept of Burch ideals was introduced by Dao-Kobayashi-Takahashi, motivated by the classical work of Burch, and subsequently extended to modules by Dey-Kobayashi. This class includes several large and well-studied families of modules and ideals over a Noetherian local ring $(R,\mathfrak{m},k)$. For example, these include the residue field $k$ as an $R$-module, every nonzero module of the form $\mathfrak{m} M$ (e.g., $\mathfrak{m}^n$ for $n\ge 1$), and under mild conditions every integrally closed ideal $I$ with $\rm{depth}(R/I)=0$. Suppose $I$ and $J$ are Burch ideals such that $I$ is $\mathfrak{m}$-primary. Motivated by Avramov's result that Burch modules exhibit extremal complexity and curvature, we establish in this article that $\rm{cx}_R(I,J)=\rm{tcx}_R(I,J)=\rm{cx}_R(k)$. Moreover, we show that $R$ is complete intersection if and only if $\rm{cx}_R(I,J)$ or $\rm{tcx}_R(I,J)$ is finite if and only if $\rm{curv}_R(I,J)$ or $\mathrm{tcurv}_R(I,J)$ is at most $1$. We deduce these results from the corresponding more general results on Burch modules.