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The Tate Intermediate Value Theorem

Paul Balmer, Beren Sanders

TL;DR

The Tate Intermediate Value Theorem provides a precise mechanism for gluing the open and closed pieces of a tensor-triangular spectrum along the support of a Tate ring. By constructing completions along a Thomason subset $Y$ and analyzing the induced spectral map $\varphi:\mathrm{Spc}(\hat{\mathscr{T}}^{\mathrm{d}})\to \mathrm{Spc}(\mathscr{T}^{\mathrm{d}})$, the authors show that specialization relations across $Y$ are controlled by the Tate ring's support $\mathrm{Supp}(\mathbb{t}_Y)$ and the image of completion. The main result states that $x\leadsto y$ with $x\notin Y$, $y\in Y$ holds iff there exists $z$ in $\mathrm{Supp}(\mathbb{t}_Y)$ with $x\leadsto z\leadsto y$, and they prove a strong version giving an intermediate specialization in the completion that maps between the two sides. This Tate IVT yields a pushout description of $\mathrm{Spc}(\mathscr{T}^{\mathrm{d}})$ as the amalgam of the open piece $\mathrm{Spc}(\mathscr{T}|_{U}^{\mathrm{d}})$ and the completion $\mathrm{Spc}(\hat{\mathscr{T}}^{\mathrm{d}})$ glued along the completion of the intersection, aligning with fracture-square perspectives in modern homotopy theory. The paper also develops extensive functoriality, excision, and localization machinery and provides rich examples spanning chromatic, equivariant, and modular representation theory, illustrating the general gluing principle in concrete settings.

Abstract

We explain how the gluing of a closed piece of the tensor-triangular spectrum with its open complement hinges on the support of the Tate ring.

The Tate Intermediate Value Theorem

TL;DR

The Tate Intermediate Value Theorem provides a precise mechanism for gluing the open and closed pieces of a tensor-triangular spectrum along the support of a Tate ring. By constructing completions along a Thomason subset and analyzing the induced spectral map , the authors show that specialization relations across are controlled by the Tate ring's support and the image of completion. The main result states that with , holds iff there exists in with , and they prove a strong version giving an intermediate specialization in the completion that maps between the two sides. This Tate IVT yields a pushout description of as the amalgam of the open piece and the completion glued along the completion of the intersection, aligning with fracture-square perspectives in modern homotopy theory. The paper also develops extensive functoriality, excision, and localization machinery and provides rich examples spanning chromatic, equivariant, and modular representation theory, illustrating the general gluing principle in concrete settings.

Abstract

We explain how the gluing of a closed piece of the tensor-triangular spectrum with its open complement hinges on the support of the Tate ring.

Paper Structure

This paper contains 12 sections, 38 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Support of big objects
  3. Localization, completion and the Tate construction
  4. The spectral map induced by completion
  5. Functoriality
  6. The Tate Intermediate Value Theorem
  7. The gluing of completion and localization
  8. Examples
  9. Chromatic examples
  10. Examples from equivariant homotopy theory
  11. Examples from modular representation theory
  12. Excisive examples

Key Result

Theorem 1.3

Let $Y\subset\mathop{\mathrm{Spc}}\nolimits(\mathscr{T}^{\mathop{\mathrm{d}}\nolimits})$ be a Thomason subset ( e. g. a closed subset with quasi-compact complement) and let $x\in\mathop{\mathrm{Spc}}\nolimits(\mathscr{T}^{\mathop{\mathrm{d}}\nolimits})\smallsetminus Y$ and $y\in Y$ be two points, on

Theorems & Definitions (131)

  • Theorem 1.3: \ref{['Cor:TIV']}
  • Definition 2.8
  • Definition 2.9
  • Remark 2.12
  • Remark 2.13
  • Remark 2.14
  • Remark 2.15
  • Proposition 2.17
  • Remark 2.18
  • Theorem 2.20
  • ...and 121 more