  For any cyclic Nakayama algebra $\Lambda$, we construct \emph{syzygy filtered
algebra} $\bm\varepsilon(\Lambda)$ which corresponds to various syzygy modules
as the name suggests. We prove that the category of modules over the syzygy
filtered algebra $\bm\varepsilon(\Lambda)$ 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
$\varphi$-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 $\varphi$-dimensions are equal to the right finitistic and
$\varphi$-dimensions respectively, as well as the difference between
$\varphi$-dimension and the finitistic dimension is at most one. Furthermore,
we recover various seemingly unrelated results in a uniform way.
