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Galois Correspondence for Partial Groupoid Actions

Wesley G. Lautenschlaeger, Thaísa Tamusiunas

TL;DR

This work extends Galois theory for groupoid actions on commutative rings from global to partial actions, addressing orthogonality restrictions via an orthogonalization technique and globalization. It first establishes a fundamental correspondence for unital partial actions under an $\alpha$-Galois framework, then transfers the theory to the nonorthogonal setting by passing to an equivalent orthogonal action and introducing strongly $\alpha$-Galois extensions. A global-extension result completes the picture by providing a Galois correspondence for global groupoid actions without restrictive hypotheses. Together, these results broaden the applicability of Galois correspondences in partial symmetry contexts and offer methods to relate partial actions to their globalizations.

Abstract

We prove a Galois correspondence theorem for groupoids acting orthogonally and partially on commutative rings. We also consider partial actions that are not orthogonal, presenting two correspondences in this case: one for strongly Galois partial groupoid actions and one for global groupoid actions (without restriction). Some examples are presented.

Galois Correspondence for Partial Groupoid Actions

TL;DR

This work extends Galois theory for groupoid actions on commutative rings from global to partial actions, addressing orthogonality restrictions via an orthogonalization technique and globalization. It first establishes a fundamental correspondence for unital partial actions under an -Galois framework, then transfers the theory to the nonorthogonal setting by passing to an equivalent orthogonal action and introducing strongly -Galois extensions. A global-extension result completes the picture by providing a Galois correspondence for global groupoid actions without restrictive hypotheses. Together, these results broaden the applicability of Galois correspondences in partial symmetry contexts and offer methods to relate partial actions to their globalizations.

Abstract

We prove a Galois correspondence theorem for groupoids acting orthogonally and partially on commutative rings. We also consider partial actions that are not orthogonal, presenting two correspondences in this case: one for strongly Galois partial groupoid actions and one for global groupoid actions (without restriction). Some examples are presented.
Paper Structure (9 sections, 16 theorems, 39 equations)

This paper contains 9 sections, 16 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial Groupoid Actions
  4. Galois Theory
  5. Correspondence Theorem
  6. General Actions
  7. Orthogonalization
  8. Strongly $\alpha$-Galois Extension
  9. Weakening the hypothesis

Key Result

Proposition 2.2

bagio2012partial Let $\alpha = (A_g,\alpha_g)_{g \in \mathcal{G}}$ be a partial action of the groupoid $\mathcal{G}$ on the ring $A$. Then it hold:

Theorems & Definitions (46)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Lemma 3.1
  • ...and 36 more