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Simple-minded systems and coherent rings

Zhen Zhang

Abstract

Let $A$ be a finite dimensional algebra over an algebraically closed field. We present a relationship between simple-minded systems and coherent rings.

Simple-minded systems and coherent rings

Abstract

Let be a finite dimensional algebra over an algebraically closed field. We present a relationship between simple-minded systems and coherent rings.
Paper Structure (9 sections, 26 theorems, 11 equations)

This paper contains 9 sections, 26 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Right triangulated categories
  4. Simple-minded systems
  5. Coherent rings and direct limit
  6. Sms's and coherent rings
  7. Hereditary algebras cases
  8. pure-semisimple ring case
  9. Declarations

Key Result

Theorem 2.11

$($Dugas$)$ Let $\mathcal{T}$ be a Hom-finite Krull-Schmidt right (or left) triangulated category. Suppose $\mathcal{X}\subseteq\mathcal{S}$ for a simple-minded system $\mathcal{S}$ in $\mathcal{T}$. Then $(^{\perp}\mathcal{X},\mathcal{F}(\mathcal{X}))$ and $(\mathcal{F}(\mathcal{X}),\mathcal{X}^{\p

Theorems & Definitions (52)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Remark 2.10
  • ...and 42 more