A characterization of finite \'etale morphisms in tensor triangular geometry

TL;DR

This paper characterizes finite étale morphisms within tensor triangular geometry by identifying key functorial properties such as having a conservative right adjoint, satisfying Grothendieck--Neeman duality, and possessing a trivial relative dualizing object.

Contribution

It provides a precise criterion for finite étale morphisms in tensor triangular geometry based on functorial and duality properties.

Findings

Finite étale morphisms are characterized by specific functorial conditions.

They have a conservative right adjoint and satisfy Grothendieck--Neeman duality.

The relative dualizing object is trivial for these morphisms.

Abstract

We provide a characterization of finite \'etale morphisms in tensor triangular geometry. They are precisely those functors which have a conservative right adjoint, satisfy Grothendieck--Neeman duality, and for which the relative dualizing object is trivial (via a canonically-defined map).