Quasi-Galois theory in symmetric monoidal categories

Abstract

Given a ring object A in a symmetric monoidal category, we investigate what it means for the extension 1 → 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.

Smashing localizations in equivariant stable homotopy

“…If we required additionally that the morphism R → A hG be an equivalence, this would be the usual definition of a Galois extension, due to Rognes [38]. This terminology is used in [33] where quasi-Galois extensions are studied in a tt-geometry context.…”