Quasi-Galois theory in symmetric-monoidal categories

TL;DR

This paper develops a generalized Galois theory framework within symmetric monoidal categories, exploring conditions for Galois extensions, their spectral effects, and applications in modular representation theory.

Contribution

It introduces the concept of quasi-Galois extensions in symmetric monoidal categories and characterizes when such extensions have a Galois closure, extending classical Galois theory.

Findings

Defined splitting ring extensions and their occurrence.

Analyzed the impact of quasi-Galois extensions on the Balmer spectrum.

Provided conditions for the existence of Galois closures in tensor-triangulated categories.

Abstract

Given a ring object $A$ in a symmetric monoidal category, we investigate what it means for the extension $\mathbb{1}\rightarrow A$ 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.