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Geometric Points in Tensor Triangular Geometry

Tobias Barthel, Logan Hyslop, Maxime Ramzi

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.

Geometric Points in Tensor Triangular Geometry

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.

Paper Structure

This paper contains 36 sections, 104 theorems, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Context and main results
  3. Methodology
  4. Structure of the document
  5. Conventions and notation
  6. A Counterexample to the Nerves of Steel Conjecture
  7. Preliminaries on Tensor Triangular Geometry
  8. Basic notions of tensor-triangular geometry
  9. The homological spectrum
  10. Weak rings
  11. Nilpotence detection
  12. The nerves of steel conjecture
  13. The exact-nilpotence condition
  14. Tensor triangular fields
  15. Free Constructions
  16. ...and 21 more sections

Key Result

Theorem A

The Nerves of Steel Conjecture is false. More precisely, the comparison map $\phi$ fails to be injective for the free rigid commutative 2-ring $\mathbb{A}^{1,+}$ on a pointed object.

Figures (1)

  • Figure 1: On the points, Wassily Kandinsky, 1928

Theorems & Definitions (228)

  • Conjecture : Balmer's Nerves of Steel Conjecture
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Theorem F
  • Definition 2.1: Balmer2005
  • Proposition 2.1: BalmerSpSpSp
  • Definition 2.2
  • ...and 218 more