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Dominant Auslander-Gorenstein algebras and mixed cluster tilting

Aaron Chan, Osamu Iyama, Rene Marczinzik

Abstract

We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules. We show that every trivial extension algebra $T(A)$ of a $d$-representation-finite algebra A admits a mixed cluster tilting module and show that this can be seen as a generalisation of the well known result that $d$-representation-finite algebras are fractionally Calabi-Yau. We show that iterated SGC-extensions of a gendo-symmetric dominant Auslander-Gorenstein algebra admit mixed precluster tilting modules.

Dominant Auslander-Gorenstein algebras and mixed cluster tilting

Abstract

We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules. We show that every trivial extension algebra of a -representation-finite algebra A admits a mixed cluster tilting module and show that this can be seen as a generalisation of the well known result that -representation-finite algebras are fractionally Calabi-Yau. We show that iterated SGC-extensions of a gendo-symmetric dominant Auslander-Gorenstein algebra admit mixed precluster tilting modules.
Paper Structure (10 sections, 39 theorems, 81 equations)

This paper contains 10 sections, 39 theorems, 81 equations.

Table of Contents

  1. Preliminaries
  2. Dominant Auslander-Gorenstein algebras and basic properties
  3. The definition and the first properties
  4. Relation with the Gorenstein condition
  5. Mixed precluster tilting modules and dominant Auslander-Gorenstein algebras
  6. Dominant Auslander-Solberg correspondence
  7. Mixed cluster tilting modules and Dominant Auslander correspondence
  8. Mixed cluster tilting for trivial extensions of $d$-representation finite algebras
  9. Mixed precluster tilting for gendo-symmetric algebras
  10. Gluing dominant Auslander-Gorenstein algebras

Key Result

Theorem 1

There exists a bijection between the following objects. The correspondence from (2) to (1) is given by $(B,M)\mapsto A:=\operatorname{End}_B(M)$.

Theorems & Definitions (81)

  • Theorem 1: Theorem \ref{['main correspondence 0']}
  • Theorem 2: Theorem \ref{['main correspondence 0 2']}
  • Theorem 3: Theorem \ref{['ErdmannHolmgeneralisation']}
  • Theorem 4: Theorem \ref{['B+N']}
  • Theorem 5: Corollary \ref{['gendosymtheorem']}
  • Lemma 1.1
  • Proposition 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • ...and 71 more