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A characterization of finite étale morphisms in tensor triangular geometry

Beren Sanders

TL;DR

This work characterizes finite étale morphisms in tensor triangular geometry by showing they are exactly the geometric functors that satisfy Grothendieck–Neeman duality, have conservative right adjoints, and admit a trivial relative dualizing object via a canonical map 1 → ω_f. The main result rests on an intrinsic monadic framework built from strongly separable algebras and a strengthened separable monadicity theory, yielding an equivalence between finite étale morphisms and extension-of-scalars by compact separable commutative algebras. The paper also furnishes corollaries in locally monogenic settings and provides a suite of examples from equivariant homotopy theory, derived categories of schemes, and motivic contexts to illustrate the scope and limitations of the theory. Overall, it clarifies how finite étale extensions in tensor triangulated geometry interface with classical étale theory and when the relative dualizing object must be trivial.

Abstract

We provide a characterization of finite étale 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).

A characterization of finite étale morphisms in tensor triangular geometry

TL;DR

This work characterizes finite étale morphisms in tensor triangular geometry by showing they are exactly the geometric functors that satisfy Grothendieck–Neeman duality, have conservative right adjoints, and admit a trivial relative dualizing object via a canonical map 1 → ω_f. The main result rests on an intrinsic monadic framework built from strongly separable algebras and a strengthened separable monadicity theory, yielding an equivalence between finite étale morphisms and extension-of-scalars by compact separable commutative algebras. The paper also furnishes corollaries in locally monogenic settings and provides a suite of examples from equivariant homotopy theory, derived categories of schemes, and motivic contexts to illustrate the scope and limitations of the theory. Overall, it clarifies how finite étale extensions in tensor triangulated geometry interface with classical étale theory and when the relative dualizing object must be trivial.

Abstract

We provide a characterization of finite étale 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).

Paper Structure

This paper contains 5 sections, 19 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Strongly separable algebras
  3. Separable algebras and triangulated categories
  4. Finite étale morphisms
  5. Examples

Key Result

Theorem 1.1

Smashing localizations of a rigidly-compactly generated tensor-triangulated category $\mathscr{T}$ are precisely those geometric functors $f^*\colon\mathscr{T} \to \mathscr{S}$ between rigidly-compactly generated tensor-triangulated categories whose right adjoint $f_*$ is fully faithful.

Theorems & Definitions (101)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.9
  • proof
  • Corollary 2.12
  • ...and 91 more