Retractions and Gorenstein Homological Properties

Abstract

We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for example, being Gorenstein algebras or CM-free).We apply the results to Nakayama algebras. It turns out that for a connected Nakayama algebra A, there exists a connected self-injective Nakayama algebra A ′ such that there is a sequence of left retractions linking A to A ′ ; in particular, the singularity category of A is triangle equivalent to the stable category of A ′ . We classify connected Nakayama algebras with at most three simple modules according to Gorenstein homological properties.

A note on homological properties of Nakayama algebras

“…Since the vectors {ξ C } C∈Γ∪Ψ are linearly independent, we have ξ = C∈BΓ a C ξ C . This proves (3).…”

The singularity category of a Nakayama algebra

“…Example 5.3. Let A be a connected Nakayama algebra with admissible sequence (7,6,6,5). Assume that {S 1 , S 2 , S 3 , S 4 } is a complete set of pairwise non-isomorphic simple A-modules such that τ S i = S i+1 for 1 ≤ i ≤ 4.…”

Derived invariance of support varieties

“…Furthermore, there exists a concrete algorithm by Ringel [Rin13] for determining whether a Nakayama algebra is Gorenstein. For Nakayama algebras with at most three simples, there is also a method by Chen and Ye [CY14] for determining Gorensteinness. Thus, for Nakayama algebras, the problem of determining (Fg) is completely solved, in the sense that there exists an algorithm for determining whether any given Nakayama algebra satisfies (Fg).…”