The $\phi$-Dimension of Cyclic Nakayama Algebras

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