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Periodic trivial extension algebras and fractionally Calabi-Yau algebras

Aaron Chan, Erik Darpö, Osamu Iyama, René Marczinzik

Abstract

We study periodicity and twisted periodicity of the trivial extension algebra $T(A)$ of a finite-dimensional algebra $A$. Our main results show that (twisted) periodicity of $T(A)$ is equivalent to $A$ being (twisted) fractionally Calabi-Yau of finite global dimension. We also extend this result to a large class of self-injective orbit algebras. As a significant consequence, these results give a partial answer to the periodicity conjecture of Erdmann-Skowroński, which expects the classes of periodic and twisted periodic algebras to coincide. On the practical side, it allows us to construct a large number of new examples of periodic algebras and fractionally Calabi-Yau algebras. We also establish a connection between periodicity and cluster tilting theory, by showing that twisted periodicity of $T(A)$ is equivalent the $d$-representation-finiteness of the $r$-fold trivial extension algebra $T_r(A)$ for some $r,d\ge 1$. This answers a question by Darpö and Iyama. As applications of our results, we give answers to some other open questions. We construct periodic symmetric algebras of wild representation type with arbitrary large minimal period, answering a question by Skowroński. We also show that the class of twisted fractionally Calabi-Yau algebras is closed under derived equivalence, answering a question by Herschend and Iyama.

Periodic trivial extension algebras and fractionally Calabi-Yau algebras

Abstract

We study periodicity and twisted periodicity of the trivial extension algebra of a finite-dimensional algebra . Our main results show that (twisted) periodicity of is equivalent to being (twisted) fractionally Calabi-Yau of finite global dimension. We also extend this result to a large class of self-injective orbit algebras. As a significant consequence, these results give a partial answer to the periodicity conjecture of Erdmann-Skowroński, which expects the classes of periodic and twisted periodic algebras to coincide. On the practical side, it allows us to construct a large number of new examples of periodic algebras and fractionally Calabi-Yau algebras. We also establish a connection between periodicity and cluster tilting theory, by showing that twisted periodicity of is equivalent the -representation-finiteness of the -fold trivial extension algebra for some . This answers a question by Darpö and Iyama. As applications of our results, we give answers to some other open questions. We construct periodic symmetric algebras of wild representation type with arbitrary large minimal period, answering a question by Skowroński. We also show that the class of twisted fractionally Calabi-Yau algebras is closed under derived equivalence, answering a question by Herschend and Iyama.

Paper Structure

This paper contains 18 sections, 51 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions and basic facts
  4. Trivial extension algebras
  5. Serre functor, fractionally Calabi--Yau algebras and cluster tilting
  6. Preliminaries on periodicity and twisted periodicity
  7. Characterisations of twisted periodicity
  8. On $\ell$-th roots of the $m$-th suspension functor $[m]$
  9. Reminder on chain complexes
  10. Cofibrant root of $A[m]$ and a certain central element
  11. Fractionally Calabi--Yau algebras have periodic trivial extensions
  12. Application to self-injective algebras
  13. Examples
  14. Examples from representation-finite and $d$-representation-finite algebras
  15. Examples from tensor products and geometry
  16. ...and 3 more sections

Key Result

Theorem 1.3

Let $A$ be a finite-dimensional algebra over a field $k$ such that $A/\operatorname{rad}\nolimits A$ is a separable $k$-algebra (e.g., when $k$ is perfect). Then the following conditions are equivalent. Moreover, let $G$ be an admissible group of automorphisms of the repetitive category $\widehat{A}$ (see Section subsec:trivext) containing $\nu_{\widehat{A}}^\ell$ for some $\ell\ge1$. Then the fo

Figures (1)

  • Figure 1: Relations between various notions of periodicity

Theorems & Definitions (109)

  • Theorem 1.3: Corollaries \ref{['finite out']} and \ref{['application to orbit algebra 2']}
  • Theorem 1.4: Theorem \ref{['main theorem 2 again']}, Corollary \ref{['application to orbit algebra 2']}
  • Corollary 1.5: Corollaries \ref{['finite out']}, \ref{['application to orbit algebra 2']}
  • Corollary 1.7
  • Corollary 1.8: Corollary \ref{['derived closed']}
  • Theorem 1.9: Theorem \ref{['stable Auslander is CY']}
  • Proposition 2.1
  • Proposition 2.2: GG1GG2
  • proof
  • Lemma 2.3
  • ...and 99 more