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On syzygy categories over Iwanaga-Gorenstein algebras: Reduction, minimality and finiteness

Ralf Schiffler, Khrystyna Serhiyenko

TL;DR

The paper addresses the structure of syzygy categories for Iwanaga-Gorenstein and 2-Calabi-Yau tilted algebras by proving that the stable CMP category is generated by the radical in the 1-Gorenstein setting and by developing an explicit reduction framework for vertex-removed quotients. It establishes equivalences between CMP categories under reduction via $J=Ae_iA$ and $B=A/J$, providing explicit functors and multiple equivalent conditions for when CMP categories are invariant under reduction. These theoretical results yield practical criteria for preserving finite Cohen-Macaulay type in CM-finite families, notably dimer tree algebras and their skew group algebras. The applications show that CM-minimality is preserved between a dimer tree algebra and its skew group partner, and provide combinatorial reductions for CM-type stability in dimer-tree contexts.

Abstract

We study 2-Calabi-Yau tilted algebras which are non-commutative Iwanaga-Gorenstein algebras of Gorenstein dimension 1. In particular, we are interested in their syzygy categories or equivalently the stable categories of Cohen-Macauley modules $\underline{\text{CMP}}$. First we show that if an algebra $A$ is Iwanaga-Gorenstein of Gorenstein dimension 1 then its stable category is generated under extensions by its radical $\text{rad}\,A$. Next, for a 2-Calabi-Yau tilted algebra $A$ we provide an explicit relationship between the $\underline{\text{CMP}}$ category of $A$ and its quotient $A/Ae_iA$ by an ideal generated by an idempotent $e_i$. Consequently, we obtain various equivalent characterizations of when the $\underline{\text{CMP}}$ category remains the same after passing to the quotient. We also obtain applications to two classes of algebras that are CM finite, the dimer tree algebras and their skew group algebras.

On syzygy categories over Iwanaga-Gorenstein algebras: Reduction, minimality and finiteness

TL;DR

The paper addresses the structure of syzygy categories for Iwanaga-Gorenstein and 2-Calabi-Yau tilted algebras by proving that the stable CMP category is generated by the radical in the 1-Gorenstein setting and by developing an explicit reduction framework for vertex-removed quotients. It establishes equivalences between CMP categories under reduction via and , providing explicit functors and multiple equivalent conditions for when CMP categories are invariant under reduction. These theoretical results yield practical criteria for preserving finite Cohen-Macaulay type in CM-finite families, notably dimer tree algebras and their skew group algebras. The applications show that CM-minimality is preserved between a dimer tree algebra and its skew group partner, and provide combinatorial reductions for CM-type stability in dimer-tree contexts.

Abstract

We study 2-Calabi-Yau tilted algebras which are non-commutative Iwanaga-Gorenstein algebras of Gorenstein dimension 1. In particular, we are interested in their syzygy categories or equivalently the stable categories of Cohen-Macauley modules . First we show that if an algebra is Iwanaga-Gorenstein of Gorenstein dimension 1 then its stable category is generated under extensions by its radical . Next, for a 2-Calabi-Yau tilted algebra we provide an explicit relationship between the category of and its quotient by an ideal generated by an idempotent . Consequently, we obtain various equivalent characterizations of when the category remains the same after passing to the quotient. We also obtain applications to two classes of algebras that are CM finite, the dimer tree algebras and their skew group algebras.

Paper Structure

This paper contains 7 sections, 18 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Finite CM-type
  3. Syzygies of 1-Iwanaga Gorenstein algebras are generated by radicals
  4. Reduction of 2-Calabi-Yau tilted algebra
  5. Applications in finite CM-type
  6. Reduction of dimer tree algebras
  7. Minimal dimer tree algebras have minimal skew group algebras

Key Result

Theorem 1.1

(Theorem prop:genrad) Let $A$ be an Iwanaga-Gorenstein algebra of Gorenstein dimension 1. Then $\underline{\textup{CMP}} A$ is generated under extensions by the $A$-module $\textup{rad}\, A$.

Figures (1)

  • Figure 1: A dimer algebra $A$ given by the quiver and its four reductions at the idempotents $e_1, e_2, e_6, e_1+e_2$ respectively. All of these algebras are of CM type $\mathbb{A}_2$, and the algebras in the right column are CM minimal.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • ...and 29 more