Geometric Points in Tensor Triangular Geometry

TL;DR

This paper explores geometric points in tensor triangular geometry, introduces constructible spectra, and constructs a counter-example to a longstanding conjecture, advancing understanding of prime spectra in tensor categories.

Contribution

It constructs a counter-example to Balmer’s Nerves of Steel conjecture and introduces constructible spectra for tensor triangular categories.

Findings

Counter-example to Balmer's conjecture using higher Zariski geometry

Introduction of constructible spectra in tensor triangular geometry

Geometric realization of primes via maps to pointlike categories

Abstract

In this paper, we study geometric points in tensor triangular geometry. In doing so, we construct a counter-example to Balmer's Nerves of Steel conjecture using free constructions in higher Zariski geometry. We then go on to introduce and discuss constructible spectra in the context of tensor triangular geometry. For tensor triangulated categories satisfying a mild enhancement condition, we use these spectra to construct geometric incarnations of (homological or triangular) primes via maps to "pointlike" tensor triangulated categories.