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The formal spectrum of a tensor-triangulated category

Drew Heard, Marius Nielsen

Abstract

To any essentially small tensor-triangulated category $\mathcal{K}$ and Thomason subset $Y \subseteq \mathrm{Spc}(\mathcal{K})$ we associate a ringed space $(\mathrm{Spf}(\mathcal{K},Y), \mathcal{O}_{\mathrm{Spf}(\mathcal{K},Y)}),$ called the formal spectrum of $(\mathcal{K},Y)$. We establish basic properties of this construction and compute it in several examples from algebraic geometry, chromatic homotopy theory, equivariant homotopy theory, and modular representation theory.

The formal spectrum of a tensor-triangulated category

Abstract

To any essentially small tensor-triangulated category and Thomason subset we associate a ringed space called the formal spectrum of . We establish basic properties of this construction and compute it in several examples from algebraic geometry, chromatic homotopy theory, equivariant homotopy theory, and modular representation theory.
Paper Structure (10 sections, 21 theorems, 140 equations)

This paper contains 10 sections, 21 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Formal schemes and formal completions
  3. The Balmer spectrum of a tt-category
  4. Completions of tt-categories
  5. The formal spectrum of a tt-category
  6. The formal Hopkins--Neeman theorem
  7. Globalizing to schemes
  8. A comparison map
  9. Chromatic homotopy
  10. Further examples

Key Result

Theorem A

Let $\mathscr{K}$ be a $2$-ring and let $Y\subseteq \mathop{\mathrm{Spc}}\nolimits(\mathscr{K})$ be a Thomason subset. There is a natural isomorphism of ringed spaces where $\varphi=\mathop{\mathrm{Spc}}\nolimits(\widehat{(-)}_Y)$ is the map on Balmer spectra induced by completion.

Theorems & Definitions (98)

  • Theorem A: \ref{['thm:completion-theorem']}
  • Theorem B: \ref{['thm:formal-hn']}
  • Theorem C: \ref{['Thm:spf-open-restriction']}
  • Theorem D: \ref{['thm:global-formal-hn']}
  • Theorem E: \ref{['thm:formal-chromatic']}
  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 88 more