The $\varphi$-Dimension of Cyclic Nakayama Algebras

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.