The $\varphi$-Dimension of Cyclic Nakayama Algebras

TL;DR

This paper investigates the $$-dimension of cyclic Nakayama algebras with infinite global dimension, establishing that its supremum is always even, and providing bounds based on algebra relations.

Contribution

It characterizes the $$-dimension for cyclic Nakayama algebras, proving it is always even and giving explicit bounds related to the algebra's relations.

Findings

The supremum of $$-values is always an even number.

The $$-dimension equals 2 under certain symmetry conditions.

A sharp upper bound for $$-dimension is given in terms of monomial relations.

Abstract

K. Igusa and G. Todorov introduced the $\varphi$ function which generalizes the notion of projective dimension. We study the behavior of the $\varphi$ function for cyclic Nakayama algebras of infinite global dimension. We prove that the supremum of values of $\varphi$ is always an even number. In particular we show that the $\varphi$-dimension is $2$ if and only if the algebra satisfies certain symmetry conditions. Also we give a sharp upper bound for $\varphi$-dimension in terms of the number of monomial relations which describes the algebra.