Chains of semidualizing modules

TL;DR

This paper investigates chains of semidualizing modules over Artinian local rings, revealing their impact on Poincaré and Bass series, and providing insights into the behavior of Bass numbers.

Contribution

It establishes a connection between the existence of chains of semidualizing modules and specific forms of Poincaré and Bass series in Artinian rings, addressing a question by Huneke.

Findings

Existence of suitable chains implies specific forms of Poincaré and Bass series.

Bass numbers are strictly increasing in this context.

Provides insight into Huneke's question about Bass numbers.

Abstract

Let $(R, \mathfrak{m}, k)$ be a commutative Noetherian local ring. We study the suitable chains of semidualizing $R$-modules. We prove that when $R$ is Artinian, the existence of a suitable chain of semidualizing modules of length $n=\mathrm{max}\,\{\,i\geqslant 0\ |\ \mathfrak{m}^{i}\neq 0\,\}$ implies that the the Poincar$\acute{\mathrm{e}}$ series of $k$ and the Bass series of $R$ have very specific forms. Also, in this case we show that the Bass numbers of $R$ are strictly increasing. This gives an insight into the question of Huneke about the Bass numbers of $R$.