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The invariance of the Auslander-Reiten Formula for hereditary algebras

Andrew Hubery

Abstract

We show that the Auslander-Reiten Formula for a finite dimensional hereditary algebra is invariant under the Auslander-Reiten translate.

The invariance of the Auslander-Reiten Formula for hereditary algebras

Abstract

We show that the Auslander-Reiten Formula for a finite dimensional hereditary algebra is invariant under the Auslander-Reiten translate.
Paper Structure (8 sections, 17 theorems, 75 equations)

This paper contains 8 sections, 17 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. Finite dimensional hereditary algebras
  3. Duality for semisimple algebras
  4. The Auslander--Reiten Formula
  5. Rewriting the Auslander--Reiten translate
  6. The evaluation map
  7. Rewriting the Auslander--Reiten Formula
  8. Invariance under the translate

Key Result

Proposition 2.2

Every right $\Lambda$-module $X$ admits a standard projective resolution \begin{tikzcd} \mathbb P_X \ \colon &[-20pt] 0 \arrow[r] & X\otimes\Omega \arrow[r,"i_X"] & X\otimes_A\Lambda \arrow[r,"p_X"] & X \arrow[r] & 0 \end{tikzcd}where $i_X(x\otimes d\lambda)=x\otimes_A\lambda-x\lambda\otimes_A1$ and

Theorems & Definitions (41)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 31 more