# Adelic models, rigidity and equivariant cohomology

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

Algebraic topology aims to capture the essence of geometric problems in rigid algebraic invariants. In fact the invariants themselves are a legitimate subject of study: if one seeks to find good invariants, it is useful to be able to look at all invariants and choose those that are particularly well structured, or which capture the phenomena you care about. An example of particular interest in this project is when we take symmetries into account (for example all of the geometric objects might have a specified rotational symmetry that is part of the structure we want to capture).

This collection of invariants has the structure of a so-called tensor-triangulated category (tt-category).

The aim of the project is to give a standard way of building a model for a tt-category. This is a two step process, first identifying an object like an algebraic variety and its ring of functions, but then also identifying additional structure. Once the standard model is constructed we give criteria for showing that the resulting model is not just a good approximation but actually recovers the whole tt-category (rigidity). When it comes to rational equivariant cohomology theories this result is known for tori and rank 1 groups thanks to previous work of the PI and collaborators. A test of the effectiveness of the methods of this project is that it should apply to give an analysis of cohomology groups for a wide range of other compact Lie groups.

Along the way, the project will study new invariants of a tt-category (adelic cohomology groups), calculate them in familiar cases and understand their role in rigidity. Another benefit of the uniform approach to tt-categories is that examples from one arena can sometimes be transported to another, and we aim to consider several examples of this type.

This collection of invariants has the structure of a so-called tensor-triangulated category (tt-category).

The aim of the project is to give a standard way of building a model for a tt-category. This is a two step process, first identifying an object like an algebraic variety and its ring of functions, but then also identifying additional structure. Once the standard model is constructed we give criteria for showing that the resulting model is not just a good approximation but actually recovers the whole tt-category (rigidity). When it comes to rational equivariant cohomology theories this result is known for tori and rank 1 groups thanks to previous work of the PI and collaborators. A test of the effectiveness of the methods of this project is that it should apply to give an analysis of cohomology groups for a wide range of other compact Lie groups.

Along the way, the project will study new invariants of a tt-category (adelic cohomology groups), calculate them in familiar cases and understand their role in rigidity. Another benefit of the uniform approach to tt-categories is that examples from one arena can sometimes be transported to another, and we aim to consider several examples of this type.

### Planned Impact

Who benefits from this research and how:

(1) academic beneficiaries described in the previous section (from the useful results and ways of thinking).

(2) RA who benefits from the training element

(3) PhD students in Sheffield, especially those of the PI. They learn about this work and the techniques that are discovered, the way mathematics is done, the way mathematicians collaborate and the way they disseminate extremely technical and specialized material to a wide audience.

(4) UK industry who either emply the RA or PhD students in (2) and (3) above (if they go on to work in industry) or those trained by the RA or PhD students (if the RA or PhD students continue to work in academia). Either way, the rigour of absolute intellectual honesty and transparency, and the precision of mathematical reasoning are fundamental transferrable skills that are in particular demand in contemporary Britain.

These Impacts are all generic to core mathematical research and not specific to this project, but they are no less valuable for that.

(1) academic beneficiaries described in the previous section (from the useful results and ways of thinking).

(2) RA who benefits from the training element

(3) PhD students in Sheffield, especially those of the PI. They learn about this work and the techniques that are discovered, the way mathematics is done, the way mathematicians collaborate and the way they disseminate extremely technical and specialized material to a wide audience.

(4) UK industry who either emply the RA or PhD students in (2) and (3) above (if they go on to work in industry) or those trained by the RA or PhD students (if the RA or PhD students continue to work in academia). Either way, the rigour of absolute intellectual honesty and transparency, and the precision of mathematical reasoning are fundamental transferrable skills that are in particular demand in contemporary Britain.

These Impacts are all generic to core mathematical research and not specific to this project, but they are no less valuable for that.

## People |
## ORCID iD |

John Greenlees (Principal Investigator) |

### Publications

Balchin S
(2020)

*Adelic models of tensor-triangulated categories*in Advances in Mathematics
Barthel T
(2019)

*On the Balmer spectrum for compact Lie groups*in Compositio Mathematica
Benson D
(2018)

*Corrigendum to "Stratifying the derived category of cochains on BG for G a compact Lie group" [J. Pure Appl. Algebra 218 (4) (2014) 642-650]*in Journal of Pure and Applied Algebra
Benson D
(2021)

*Massey products in the homology of the loop space of a p -completed classifying space: finite groups with cyclic Sylow p -subgroups*in Proceedings of the Edinburgh Mathematical Society
Greenlees J
(2018)

*Correction to the article Gorenstein duality for real spectra*in Algebraic & Geometric Topology
Greenlees J
(2018)

*An Alpine Bouquet of Algebraic Topology*
Greenlees J
(2018)

*An algebraic model for rational torus-equivariant spectra*in Journal of Topology
Greenlees J
(2019)

*The Balmer spectrum of rational equivariant cohomology theories*in Journal of Pure and Applied Algebra### Related Projects

Project Reference | Relationship | Related To | Start | End | Award Value |
---|---|---|---|---|---|

EP/P031080/1 | 22/10/2017 | 30/03/2018 | £318,758 | ||

EP/P031080/2 | Transfer | EP/P031080/1 | 30/04/2018 | 21/07/2022 | £278,916 |

Description | A number of algebraic models have been developed, especially the adelic, separated and complete models. Numerous examples have been studied. Work towards the non-Noetherian and torsion cases has begun. |

Exploitation Route | Further algebraic models and rigidity theorems. Homotopical versions of algebraic models. |

Sectors | Education |