[Paper Review] Periodic resolutions and self-injective algebras of finite type

Alex Dugas|arXiv (Cornell University)|Aug 9, 2008

Algebraic structures and combinatorial models12 references4 citations

TL;DR

This paper proves that all self-injective algebras of finite representation type over an algebraically closed field are periodic, meaning their Hochschild cohomology has a periodic resolution. Using Galois coverings, smash products, and stable Auslander algebras, the authors establish that periodicity is preserved under these constructions and compute explicit period bounds, particularly for algebras of type $(\mathbb{D}_{3m},1/3,1)$, showing the period divides $4(2m-1)$ and is at least $2m-1$. This resolves a long-standing question and corrects errors in prior work on Calabi-Yau dimensions.

ABSTRACT

We 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.

Motivation & Objective

• To resolve the open question of whether all self-injective algebras of finite representation type are periodic.
• To extend periodicity results from standard to nonstandard algebras of type $(\mathbb{D}_{3m},1/3,1)$.
• To establish that periodicity is preserved under finite Galois coverings and via the stable Auslander algebra construction.
• To compute explicit period bounds for self-injective algebras of finite type, particularly for $\mathbb{D}_{3m}$-type algebras.
• To correct errors in prior work on stable Calabi-Yau dimensions and clarify which finite-type self-injective algebras are Calabi-Yau.

Proposed method

• Use of smash products and Galois coverings to transfer periodicity between an algebra $A$ and its finite Galois cover $B$, showing $A$ is periodic iff $B$ is.
• Application of Buchweitz’s results to relate periodicity of $A$ to that of its stable Auslander algebra.
• Construction of a $\mathbb{Z}/\langle 2\rangle$-grading on nonstandard algebras of type $(\mathbb{D}_{3m},1/3,1)$ to analyze syzygy functors and bimodule resolutions.
• Leveraging derived equivalence classifications via Asashiba’s work to bound periods in terms of algebra types.
• Use of graded projective resolutions in the category $\mathrm{mod}_G\mbox{-}A^e$ to ensure degree-preserving automorphisms in syzygy functors.
• Application of Lemma 3.6 to show that if $\Omega^r(A) \cong {}_1A_\sigma$ with $\sigma$ degree-preserving and $\sigma^2$ inner, then the period divides $2r$.

Experimental results

Research questions

• RQ1 Are all self-injective algebras of finite representation type periodic, particularly in the nonstandard case?
• RQ2 Does periodicity of a self-injective algebra of finite type descend or ascend through finite Galois coverings?
• RQ3 What are the precise period bounds for self-injective algebras of type $(\mathbb{D}_{3m},1/3,1)$?
• RQ4 How does the periodicity of the stable Auslander algebra relate to that of the original algebra?
• RQ5 Can the results correct inaccuracies in prior claims about Calabi-Yau dimensions of stable module categories?

Key findings

• All self-injective algebras of finite representation type are periodic, resolving a key open problem in Hochschild cohomology.
• For nonstandard algebras of type $(\mathbb{D}_{3m},1/3,1)$, the period $p_A$ satisfies $(2m-1) \mid p_A \mid 4(2m-1)$.
• The period of the standard algebra of type $(\mathbb{D}_{3m},1/3,1)$ divides $2(2m-1)$, and this bound lifts to the nonstandard case.
• The stable Auslander algebra of such algebras is also periodic, as periodicity is preserved under this construction.
• For $m=2$, the nonstandard algebra of type $(\mathbb{D}_6,1/3,1)$ has period 6, while the standard one has period 3 in characteristic 2.
• The results correct an error in [13] regarding Calabi-Yau dimensions and clarify the conditions under which finite-type self-injective algebras are Calabi-Yau.