Some notes on tensor triangular geometry
Greg Stevenson
TL;DR
The notes develop tensor triangular geometry from a lattice-theoretic viewpoint, building a bridge between tt-categories and Stone duality to define the spectrum Spc(K). Starting in the essentially small setting, the author provides a hands-on computation for D^perf(k[x]) and then proves Thomason’s theorem in tt-language for quasi-compact, quasi-separated schemes, illustrating how the spectrum recovers the underlying geometry. The framework is then extended to big tt-categories, introducing localizing and smashing ideals, categorified lattices, and refined supports, with criteria (e.g., Cantor–Bendixson rank) ensuring local-to-global principles. The resulting picture unifies algebraic geometry, homotopy theory, and modular representation theory under a common lattice-theoretic and spectral paradigm, while highlighting both the power and limitations of rigidity and smashing in non-noetherian and non-rigid contexts. The practical upshot is a robust, universal language for classifying localizing structures via supports and spectra, enabling concrete computations and conceptual insights across domains.
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.