Comparing complexities of pairs of modules

TL;DR

This paper investigates the relationships and inequalities involving the complexities of pairs of modules over local rings, focusing on the polynomial growth of Ext modules.

Contribution

It establishes fundamental equalities and inequalities relating the complexities of different module pairs over local rings.

Findings

Derived key inequalities between complexities of module pairs.

Identified conditions for equalities involving complexities.

Enhanced understanding of polynomial growth patterns in Ext modules.

Abstract

Let $R$ be a local ring and $M,N$ be finitely generated $R$-modules. The complexity of $(M,N)$, denoted by $\cxx RMN$, measures the polynomial growth rate of the number of generators of the modules $\Ext nRMN$. In this paper we study several basic equalities and inequalities involving complexities of different pairs of modules.