Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Notes on Tate cohomology

Arpon Raksit

TL;DR

Notes on Tate cohomology develops a universal, functorial framework for Tate cohomology within the abstract setting of three functor formalisms. It defines norm maps via weak adjoints in (\infty,2)-categories and establishes a universal property akin to Nikolaus-Scholze, yielding functoriality, base change, and descent results that apply to generalized spaces and stacks. The approach connects classical Tate theory for finite groups and BG to ambidexterity phenomena, and it extends to stable, presentable settings such as locally valued systems and quasicoherent sheaves, enabling applications to topological cyclic homology and related invariants. Overall, the work provides a robust toolkit for computing and functorially relating Tate cohomology in diverse geometric and categorical contexts.

Abstract

We formulate a definition of Tate cohomology in the context of three functor formalisms, and we establish basic monoidality and functoriality properties of it in this context. Our approach to these properties is based on the treatment of Nikolaus-Scholze in the setting of local systems of spectra on spaces. We discuss a couple of other specific settings of interest that are accommodated by our generalization.

Notes on Tate cohomology

TL;DR

Notes on Tate cohomology develops a universal, functorial framework for Tate cohomology within the abstract setting of three functor formalisms. It defines norm maps via weak adjoints in (\infty,2)-categories and establishes a universal property akin to Nikolaus-Scholze, yielding functoriality, base change, and descent results that apply to generalized spaces and stacks. The approach connects classical Tate theory for finite groups and BG to ambidexterity phenomena, and it extends to stable, presentable settings such as locally valued systems and quasicoherent sheaves, enabling applications to topological cyclic homology and related invariants. Overall, the work provides a robust toolkit for computing and functorially relating Tate cohomology in diverse geometric and categorical contexts.

Abstract

We formulate a definition of Tate cohomology in the context of three functor formalisms, and we establish basic monoidality and functoriality properties of it in this context. Our approach to these properties is based on the treatment of Nikolaus-Scholze in the setting of local systems of spectra on spaces. We discuss a couple of other specific settings of interest that are accommodated by our generalization.
Paper Structure (18 sections, 38 theorems, 71 equations)

This paper contains 18 sections, 38 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Overview
  4. Acknowledgements
  5. Conventions
  6. Generalities
  7. Three functor formalisms
  8. Weak adjoints
  9. Norm maps and Tate cohomology
  10. Poincaré duality
  11. Poincaré duality for H-spaces
  12. Norm and base change
  13. Norm and composition
  14. Universal property of Tate cohomology
  15. Functoriality of Tate cohomology
  16. ...and 3 more sections

Key Result

Proposition 1.2.10

In the situation of t--c--categorical-comp-right, suppose that $\epsilon'$ and $\epsilon"$ exhibit $H'$ and $H"$ as weak right adjoints of $G'$ and $G"$, and that one of the following further conditions is satisfied: Then the map $\mu : H'H" \to H$ is an isomorphism.

Theorems & Definitions (131)

  • Example 2.1
  • Example 2.2
  • Definition 1.2.1
  • Remark 1.2.2
  • Example 1.2.3
  • Remark 1.2.5
  • Example 1.2.6
  • Remark 1.2.7
  • Remark 1.2.8
  • Proposition 1.2.10
  • ...and 121 more