Syzygy Filtrations of Cyclic Nakayama Algebras

Abstract

For any cyclic Nakayama algebra $\Lambda$, we construct \emph{syzygy filtered algebra} $\mathbf{\epsilon }(\Lambda )$ which corresponds to various syzygy modules as the name suggests. We prove that the category of modules over the syzygy filtered algebra $\mathbf{\epsilon }(\Lambda )$ is equivalent to the wide subcategory cogenerated by projective-injective modules of the original algebra $\Lambda$ along with other categorical equivalences. In terms of this new algebra, we interpret the following homological invariants of $\Lambda$: left and right finitistic dimension, left and right $\phi$-dimension, Gorenstein dimension, dominant dimension and their upper bounds. For all of them, we obtain a unified upper bound $2r$ where $r$ is the number of relations defining the algebra $\Lambda$. We show that the left finitistic and $\phi$-dimensions are equal to the right finitistic and $\phi$-dimensions respectively, as well as the difference between $\phi$-dimension and the finitistic dimension is at most one. Furthermore, we recover various seemingly unrelated results in a uniform way.

v.3 Major update following referee report. Changed abstract. Exposition is made more clear, proofs are elaborated. We add new result: syzygy filtered algebra is equivalent to the wide subcategory of projective-injective cogenerator. Relying on this, we add a subsection for applications of the method besides Nakayama algebras. Comments are welcome!