Syzygy Filtrations of Cyclic Nakayama Algebras

Abstract

We introduce a method "syzygy filtration" to give building blocks of syzygies appearing in projective resolutions of indecomposable Λ-modules where Λ is a cyclic Nakayama algebra. We interpret homological invariants of Λ including left and right finitistic dimension, left and right ϕ-dimension, Gorenstein dimension, dominant dimension and their upper bounds in terms of this new filtration. For all of them, we obtain a unified upper bound 2r where r is the number of relations defining the algebra Λ. We give a simpler proof of theorem relating Gorenstein dimension to ϕ-dimension. Moreover we show that ϕ dim Λ − fin.dim Λ ≤ 1.

Delooping Level of Nakayama algebras

“…Hence we consider infinite global dimensional case. Moreover, we can exclude the equalities of the left and right finitistic dimensions, since we proved them in [Sen19]. Now we can use induction to prove the remaining cases of the main result:…”

“…Here, we recall some basic definitions and construction of syzygy filtered algebra ε(Λ) which will be used in the section 3. For details we refer to papers [Sen18] and [Sen19].…”

“…Syzygy Filtered Algebra. [Sen19] Let S(Λ) be the complete set of representatives of socles of projective modules over Λ:…”

“…Remark 2.4. We collect and summarize some useful results about ϕ-dimension and syzygy filtration from [Sen18] and [Sen19].…”

Nakayama Algebras which are Higher Auslander Algebras

“…Here we briefly recall some basics about Nakayama algebras. For details we refer to [Sen18] and [Sen19]. Algebra is called Nakayama algebra if all indecomposable modules are uniserial.…”