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Chain Conditions for Etale Groupoid Algebras

Sunil Philip

Abstract

Let $R$ be a unital commutative ring with unit and $\mathscr{G}$ an ample groupoid. Using the topology of the groupoid $\mathscr{G}$, Steinberg defined an etale groupoid algebra $R\mathscr{G}$. These etale groupoid algebras generalize various algebras including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for etale groupoid algebras. We characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple etale groupoid algebras, generalizing existing results for Leavitt path algebras.

Chain Conditions for Etale Groupoid Algebras

Abstract

Let be a unital commutative ring with unit and an ample groupoid. Using the topology of the groupoid , Steinberg defined an etale groupoid algebra . These etale groupoid algebras generalize various algebras including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical chain conditions for etale groupoid algebras. We characterize categorically noetherian and artinian, locally noetherian and artinian, and semisimple etale groupoid algebras, generalizing existing results for Leavitt path algebras.

Paper Structure

This paper contains 13 sections, 38 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Groupoid Algebras
  3. Topological, etale, and ample groupoids
  4. Etale Groupoid Algebras
  5. Preliminaries on Chain Conditions
  6. Classical Chain Conditions
  7. Generalized Chain Conditions
  8. Results on Generalized Chain Conditions
  9. Semisimplicity and Local Semisimplicity Results
  10. Main Result
  11. Applications
  12. Leavitt Path Algebras
  13. Future work

Key Result

Proposition 2.4

Suppose $\mathcal{G}$ is an ample groupoid and $R$ a commutative ring with unit. Let $\mathcal{B}$ denote the generalized boolean algebra of compact open subsets of $\mathcal{G}^{(0)}$.

Theorems & Definitions (87)

  • Remark 2.1
  • Definition 2.2: Isotropy Group and Orbit
  • Remark 2.3
  • Proposition 2.4: Proposition 2.1 of groupoidbundles
  • Lemma 2.5: Lemma 3 of gpdchain
  • proof
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 77 more