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Classification of Auslander-Gorenstein monomial algebras: The acyclic case

Viktória Klász, Markus Kleinau, René Marczinzik

Abstract

We give a linear algebraic classification of Auslander regular acyclic monomial algebras via the Bruhat factorisation of the Coxeter matrix. Namely, we show under mild assumptions that a monomial acyclic quiver algebra is Auslander regular if and only if its Coxeter matrix $C$ has a Bruhat factorisation $U_1 P U_2$ with $U_1$ the identity matrix. In particular, this holds without restrictions for linear Nakayama algebras and we use the Bruhat decomposition to answer a question raised by Ringel by showing that his homological permutation coincides with the permutation coming from the Bruhat factorisation of the Coxeter matrix. We also use our methods to show that general Auslander regular acyclic quiver algebras are echelon-independent, proving a conjecture of Defant-Jiang-Marczinzik-Segovia-Speyer-Thomas-Williams, and we answer another question by Ringel on the delooping level of simple modules over Nakayama algebras.

Classification of Auslander-Gorenstein monomial algebras: The acyclic case

Abstract

We give a linear algebraic classification of Auslander regular acyclic monomial algebras via the Bruhat factorisation of the Coxeter matrix. Namely, we show under mild assumptions that a monomial acyclic quiver algebra is Auslander regular if and only if its Coxeter matrix has a Bruhat factorisation with the identity matrix. In particular, this holds without restrictions for linear Nakayama algebras and we use the Bruhat decomposition to answer a question raised by Ringel by showing that his homological permutation coincides with the permutation coming from the Bruhat factorisation of the Coxeter matrix. We also use our methods to show that general Auslander regular acyclic quiver algebras are echelon-independent, proving a conjecture of Defant-Jiang-Marczinzik-Segovia-Speyer-Thomas-Williams, and we answer another question by Ringel on the delooping level of simple modules over Nakayama algebras.

Paper Structure

This paper contains 8 sections, 35 theorems, 69 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutations on simple modules
  4. 2-Gorenstein monomial algebras
  5. Nakayama algebras
  6. Ringel's bijection and the Coxeter permutation
  7. Auslander regularity and the Coxeter matrix
  8. Ringel's bijection for Auslander-Gorenstein algebras

Key Result

Theorem 1.3

(Theorem thm::C=PU_iff_ARbijnak and thm::AG_iff_C=PU_acyclic_2Gorenstein_monomial) Let $A$ be a linear Nakayama algebra or a 2-Gorenstein, acyclic monomial algebra. Then $A$ is Auslander regular if and only if there exists a Bruhat decomposition $C_A=U_1 P U_2$ of the Coxeter matrix of $A$ where $U_

Figures (3)

  • Figure 1: A picture of a minimal projective resolution of $S_i$ (top) and $I(i)$ (bottom). The modules are depicted as intervals in $\mathbb{Z}$. In the resolution of $S_i$ (resp. $I(i)$), the even terms are shown in purple (resp. blue), while the odd terms are orange. Note that $\textcolor{orange}{P_j=R_j}$ for all odd $j.$ This picture corresponds to the case when $e(S_i)$ is odd. The green intervals represent the terms of a minimal injective coresolution of $M$, the module defined in the proof of Lemma \ref{['lem::existanceOfM']}. The diagram indicates how these intervals align with the terms of the two resolutions.
  • Figure :
  • Figure :

Theorems & Definitions (74)

  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Lemma 2.4
  • ...and 64 more