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A Characteristic free approach to skew-gentle algebras

Yiping Chen

Abstract

To each skew-gentle algebra, one can assign a gentle algebra in terms of combinatorial data. In order to relate the structures of the two algebras, we establish a homological epimorphism and a recollement of derived module categories. This approach is characteristic free and works in particular also in characteristic two, which is the difficult case for skew-gentle algebras. This allows to solve open problems and to uniformly reprove and strengthen known results, for instance, (1) a complete classification of selfinjective skew-gentle algebras; (2) the finitistic dimension conjecture, Auslander and Reiten's conjecture, and Keller's conjecture hold for all skew-gentle algebras; (3) a precise connection of K-theory between skew-gentle algebras and gentle algebras; (4) all skew-gentle algebras are Gorenstein, and skew-gentle algebras and their gentle algebras share the same singularity categories.

A Characteristic free approach to skew-gentle algebras

Abstract

To each skew-gentle algebra, one can assign a gentle algebra in terms of combinatorial data. In order to relate the structures of the two algebras, we establish a homological epimorphism and a recollement of derived module categories. This approach is characteristic free and works in particular also in characteristic two, which is the difficult case for skew-gentle algebras. This allows to solve open problems and to uniformly reprove and strengthen known results, for instance, (1) a complete classification of selfinjective skew-gentle algebras; (2) the finitistic dimension conjecture, Auslander and Reiten's conjecture, and Keller's conjecture hold for all skew-gentle algebras; (3) a precise connection of K-theory between skew-gentle algebras and gentle algebras; (4) all skew-gentle algebras are Gorenstein, and skew-gentle algebras and their gentle algebras share the same singularity categories.
Paper Structure (8 sections, 13 theorems, 37 equations)

This paper contains 8 sections, 13 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Recollements of triangulated categories
  4. Singular equivalences of Morita type (with level)
  5. Homological conjectures
  6. Keller's conjecture
  7. Skew-gentle algebras
  8. Main results

Key Result

Theorem 1.1

Let $A=A(Q, I, Sp)$ be a skew-gentle algebra with $Sp\neq\emptyset$. Then,

Theorems & Definitions (20)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • ...and 10 more