Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Krull-Remak-Schmidt-Azumaya Theorem for idempotent complete categories

Simion Breaz

TL;DR

This work extends Krull-Remak-Schmidt-Azumaya theory to idempotent complete additive categories under the presence of enough compact objects, establishing a unique decomposition result for families of objects with local endomorphism rings. The authors connect the exchange property to endomorphism rings, prove an equivalence with exchange rings via Warfield-type arguments, and leverage compactness to handle infinite direct sums, yielding a KS theorem for compactly generated triangulated categories. A key contribution is showing KS holds in such categories while providing a counterexample in $\mathbf{Ab}^{op}$ that finite exchange does not imply full exchange, highlighting the necessity of compactness. These results broaden decomposition theory to triangulated and other non-ordinary contexts, with practical implications for structural analysis in representation theory and homological algebra.

Abstract

We prove that the Krull-Remak-Schmidt-Azumaya unique decomposition theorem holds in idempotent complete additive categories with enough compact objects. In particular, this result applies to compactly generated triangulated categories. In addition, we provide an example of an object with the finite exchange property that does not have the exchange property.

The Krull-Remak-Schmidt-Azumaya Theorem for idempotent complete categories

TL;DR

This work extends Krull-Remak-Schmidt-Azumaya theory to idempotent complete additive categories under the presence of enough compact objects, establishing a unique decomposition result for families of objects with local endomorphism rings. The authors connect the exchange property to endomorphism rings, prove an equivalence with exchange rings via Warfield-type arguments, and leverage compactness to handle infinite direct sums, yielding a KS theorem for compactly generated triangulated categories. A key contribution is showing KS holds in such categories while providing a counterexample in that finite exchange does not imply full exchange, highlighting the necessity of compactness. These results broaden decomposition theory to triangulated and other non-ordinary contexts, with practical implications for structural analysis in representation theory and homological algebra.

Abstract

We prove that the Krull-Remak-Schmidt-Azumaya unique decomposition theorem holds in idempotent complete additive categories with enough compact objects. In particular, this result applies to compactly generated triangulated categories. In addition, we provide an example of an object with the finite exchange property that does not have the exchange property.

Paper Structure

This paper contains 4 sections, 13 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. The exchange property in idempotent complete additive categories
  3. Categories with enough compact objects
  4. The Krull-Schmidt-Azumaya Theorem in Additive Categories

Key Result

Lemma 2.1

An additive category $\mathcal{C}$ is idempotent complete if and only if for every $A\in \mathcal{C}$ and every idempotent endomorphism $e$ of $A$ there exist morphisms $q:B\to A$ and $p:A\to B$ such that $qp=1_B$ and $pq=e$. Moreover, under these conditions, the following are true:

Theorems & Definitions (24)

  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • Corollary 2.7
  • Proposition 2.8
  • proof
  • ...and 14 more