Periodic resolutions and self-injective algebras of finite type

Abstract

a b s t r a c tWe say that an algebra A is periodic if it has a periodic projective resolution as an (A, A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B → A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba's classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.

Radical cube zero selfinjective algebras of finite complexity

“…This implies that t (3,1) (2, 1), (1, 2), (1, 1) 4 (2, 1, 1), (1, 2, 1), (1, 1, 2) (1, 1, 1) t λ max = 2 λ max < 2 For t ≥ 5 and λ max ≤ 2, then the sequence (a 1 b 1 , a 2 b 2 , . .…”

“…If Λ is a finite-dimensional selfinjective algebra over an algebraically closed field of finite representation type, then it follows from [3] that Λ satisfies (Fg). Hence to further limit the classes of algebras we need to analyze in characterizing when a selfinjective algebra Λ = kQ /I with radical cube zero satisfies (Fg), we recall in this section how the representation type of Λ is determined by the spectral radius of the adjacency matrix of Λ.…”

“…If Λ has finite representation type, then it satisfies (Fg) by [3]. If Λ has infinite representation type, then it is a Koszul algebra by [7,8].…”

Partial silting objects and smashing subcategories

“…Take, for example, T to be the stable category of not necessarily finitely generated modules over a representation-finite self-injective and finite dimensional algebra. By [16], such an algebra is periodic and, thus, every object in T is isomorphic to infinitely many of its positive (respectively, negative) shifts. As a consequence, there are no nondegenerate t-structures in T .…”

The symmetry, period and Calabi–Yau dimension of finite dimensional mesh algebras

“…When ϕ is the identity (or an inner automorphism), the algebra Λ is called periodic and the problem of determining the self-injective algebras which are periodic is widely open. However, there is a lot of work in the literature were several classes of periodic algebras have been identified (see, e.g., [9], [16], [12]). Even when an algebra Λ is known to be periodic, it is usually hard to calculate explicitly its period, that is, the smallest of the integers r > 0 such that Ω r Λ e (Λ) is isomorphic to Λ as a bimodule.…”