Some notes on tensor triangular geometry

TL;DR

This paper introduces tensor triangular geometry using lattice theory, covering small and large categories, and provides a tensor triangular proof of Thomason's theorem, with some new ideas for the large case.

Contribution

It offers an accessible introduction to tensor triangular geometry and presents a novel tensor triangular proof of Thomason's theorem for perfect complexes.

Findings

Tensor triangular geometry can be effectively studied through lattice theory.

A tensor triangular proof of Thomason's theorem is provided.

Exploratory ideas are introduced for the compactly generated case.

Abstract

These are notes from the lectures I gave at the Oberwolfach seminar `Tensor Triangular Geometry and Interactions' which was held in October 2025. The aim of these notes is to give an introduction to tensor triangular geometry, for both small and large categories, through the lens of lattice theory. We do not try to be exhaustive and this is reflected in both the content and the bibliography. For instance we are quite light on triangulated preliminaries, especially for compactly generated categories. The first three sections treat the essentially small case and conclude with a tensor triangular proof of Thomason's theorem computing the spectrum of the perfect complexes on a quasi-compact and quasi-separated scheme. The last section treats the compactly generated case. This final section is somewhat experimental and contains some new thoughts.