Characterizing certain semidualizing complexes via their Betti and Bass numbers

TL;DR

This paper establishes criteria to identify certain semidualizing complexes over local rings using their Betti and Bass numbers, linking these invariants to shifts of rings and dualizing complexes.

Contribution

It provides new characterizations of semidualizing complexes via Betti and Bass numbers, specifically identifying when they are shifts of rings or dualizing complexes.

Findings

A semidualizing complex is a shift of the ring if its nth Betti number is one.

A semidualizing complex is dualizing if its dth Bass number is one.

The criteria connect numerical invariants to complex types in commutative algebra.

Abstract

It is known that the numerical invariants Betti numbers and Bass numbers are worthwhile tools for decoding a large amount of information about modules over commutative rings. We highlight this fact, further, by establishing some criteria for certain semidualizing complexes via their Betti and Bass numbers. Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let $C$ be a semidualizing complex for an analytically irreducible local ring $R$ and set $n:=\sup C$ and $d:=\dim_RC$. We show that $C$ is quasi-isomorphic to a shift of $R$ if and only if the $n$th Betti number of $C$ is one. Also, we show that $C$ is a dualizing complex for $R$ if and only if the $d$th Bass number of $C$ is one.