Quasi-Galois theory in symmetric monoidal categories

Pauwels, Bregje

doi:10.2140/ant.2017.11.1891

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Quasi-Galois theory in symmetric monoidal categories

Bregje Pauwels

Vol. 11 (2017), No. 8, 1891–1920

DOI: 10.2140/ant.2017.11.1891

Abstract

Given a ring object $A$ in a symmetric monoidal category, we investigate what it means for the extension $⊮ \to A$ to be (quasi-)Galois. In particular, we define splitting ring extensions and examine how they occur. Specializing to tensor-triangulated categories, we study how extension-of-scalars along a quasi-Galois ring object affects the Balmer spectrum. We define what it means for a separable ring to have constant degree, which is a necessary and sufficient condition for the existence of a quasi-Galois closure. Finally, we illustrate the above for separable rings occurring in modular representation theory.

Keywords

tensor triangulated category, separable, etale, Galois, ring-object, stable category

Mathematical Subject Classification 2010

Primary: 18BXX

Secondary: 16GXX, 18GXX

Milestones

Received: 1 September 2016

Revised: 29 May 2017

Accepted: 9 July 2017

Published: 15 October 2017

Authors

Bregje Pauwels
Mathematical Sciences Institute The Australian National University Acton ACT 2601 Australia

Vol. 11, No. 8, 2017