Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cluster tilting modules for local algebras

Rene Marczinzik, Daniel Owens

TL;DR

The paper addresses the existence of non-trivial cluster tilting modules in local finite-dimensional algebras by constructing the first explicit example. It identifies a local algebra $A=KQ/I$ with a generator-cogenerator $M=DA\oplus\tau_2 DA\oplus\tau_2^2 DA\oplus\tau_2^3 DA\oplus\tau_2^4 DA$ (with $\tau_2=\tau\Omega^1$) that forms a $2$-cluster tilting module, and proves this via computation of the endomorphism ring $B=\operatorname{End}_A(M)$ using the GAP/QPA package. The calculation shows $\operatorname{gldim} B=\operatorname{domdim} B=3$ (over $\mathbb{Q}$) and leverages field-extension arguments to extend the result to all characteristic-0 fields, invoking Iyama's higher Auslander correspondence to conclude the $2$-cluster tilting property. The work also provides QPA code to reproduce the endomorphism-ring computations and outlines open problems and conjectures, including potential Cartan-determinant criteria and the search for additional local algebras with higher cluster-tilting structures.

Abstract

We give the first example of a non-trivial cluster tilting module in a local finite dimensional algebra. To do this, we give an explicit calculation of the corresponding higher Auslander algebra by quiver and relations using the GAP-package QPA. We discuss related problems and conjectures for local finite-dimensional algebras.

Cluster tilting modules for local algebras

TL;DR

The paper addresses the existence of non-trivial cluster tilting modules in local finite-dimensional algebras by constructing the first explicit example. It identifies a local algebra with a generator-cogenerator (with ) that forms a -cluster tilting module, and proves this via computation of the endomorphism ring using the GAP/QPA package. The calculation shows (over ) and leverages field-extension arguments to extend the result to all characteristic-0 fields, invoking Iyama's higher Auslander correspondence to conclude the -cluster tilting property. The work also provides QPA code to reproduce the endomorphism-ring computations and outlines open problems and conjectures, including potential Cartan-determinant criteria and the search for additional local algebras with higher cluster-tilting structures.

Abstract

We give the first example of a non-trivial cluster tilting module in a local finite dimensional algebra. To do this, we give an explicit calculation of the corresponding higher Auslander algebra by quiver and relations using the GAP-package QPA. We discuss related problems and conjectures for local finite-dimensional algebras.
Paper Structure (4 sections, 3 theorems, 4 equations)

This paper contains 4 sections, 3 theorems, 4 equations.

Table of Contents

  1. Calculation of the endomorphism ring in QPA
  2. Remarks, Open problems and questions
  3. Appendix: QPA code
  4. Acknowledgment

Key Result

Theorem 1

(Iyama) Let $n \geq 1$. There is a bijective correspondence between algebras $A$ with an $n$-cluster tilting module $M$ and higher Auslander algebras $B$ with $\operatorname{gldim} B \leq n+1 \leq \operatorname{domdim} B$ up to Morita equivalence. In particular, a generator-cogenerator $M$ over a no

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 1.1
  • proof
  • Conjecture 2.1