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On endomorphism algebras of silting complexes over hereditary abelian categories

Wei Dai, Changjian Fu, Liangang Peng

Abstract

Let $\mathcal{E}$ be the class of finite-dimensional algebras isomorphic to endomorphism algebras of silting complexes over hereditary abelian categories. It is proved that the class $\mathcal{E}$ is closed under taking idempotent quotients, idempotent subalgebras and $τ$-reduction. We also show that the proper class consisting of shod algebras is also closed under these operations. In addition, several classic classes of algebras -- including laura, glued, weakly shod algebras -- are proved to be closed under idempotent quotients, thereby generalizing a known result originally established for specific idempotents.

On endomorphism algebras of silting complexes over hereditary abelian categories

Abstract

Let be the class of finite-dimensional algebras isomorphic to endomorphism algebras of silting complexes over hereditary abelian categories. It is proved that the class is closed under taking idempotent quotients, idempotent subalgebras and -reduction. We also show that the proper class consisting of shod algebras is also closed under these operations. In addition, several classic classes of algebras -- including laura, glued, weakly shod algebras -- are proved to be closed under idempotent quotients, thereby generalizing a known result originally established for specific idempotents.
Paper Structure (13 sections, 37 theorems, 40 equations)

This paper contains 13 sections, 37 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Perpendicular subcategories in triangulated categories
  4. Silting theory
  5. Perpendicular subcategories in hereditary categories
  6. Rigid objects and presilting complexes over hereditary categories
  7. Completion
  8. Reduction of endomorphism algebras
  9. Idempotent quotients
  10. Idempotent subalgebras
  11. $\tau$-tilting reduction
  12. Idempotent quotients for several classes of algebras
  13. Examples

Key Result

Theorem 1.1

Let $A\in \mathcal{E}$ and $e\in A$ be an idempotent element. Then

Theorems & Definitions (69)

  • Theorem 1.1: Theorem \ref{['p: idempotent-factor-closed']}, Theorem \ref{['p: idempotent-subalg-closed']}
  • Theorem 1.2: Theorem \ref{['t: quasisilted is tau-tilting reduction closed']}
  • Theorem 1.3: Theorem \ref{['t: quotient-closed']}
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • ...and 59 more