# Equivariant Homotopy-Invariant Commutative Algebra

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

My project involves generalisation of ideas from commutative algebra to stable homotopy theory. In particular, many duality phenomena throughout algebra and algebraic topology can be expressed in terms of the Gorenstein property of ring spectra over a field in the sense of Dwyer-Greenlees-Iyengar. This has been studied extensively, and an aim of mine is to explore this in an equivariant context. For ring spectra with G-action, a big question is to see when the Gorenstein property descends from the ring to the homotopy-fixed point spectrum. Examples include the complex theory spectrum (and its connective cover) with the action of conjugation, and classically in algebra a theorem of Watanabe for polynomial invariant rings. It is also of interest to consider genuine ring G-spectra, whose Gorenstein shifts have an RO(G)-graded shift. Examples include singular cochains on the double cover of an orientable manifold, equivariant via deck transformations, where Poincaré duality is recovered with an appropriate twist.

Another commutative-algebraic notion that can be considered in stable homotopy theory is the singularity category, defined for ring spectra by Greenlees-Stevenson. A crucial example is the cochains on the classifying space of a compact Lie group - a project of mine has been to adapt a notion of the ``nucleus'' in modular representation theory to a more topological context, where I have shown it to coincide with the support of the singularity category of the cochains, as conjectured in the original context by Benson-Greenlees. I expect understanding this support theory for singularity categories of ring spectra will form a part of my project, especially in using it to classify thick subcategories for hypersurface ring spectra - possibly extending results of Stevenson and Takahashi with discrete rings.

Another commutative-algebraic notion that can be considered in stable homotopy theory is the singularity category, defined for ring spectra by Greenlees-Stevenson. A crucial example is the cochains on the classifying space of a compact Lie group - a project of mine has been to adapt a notion of the ``nucleus'' in modular representation theory to a more topological context, where I have shown it to coincide with the support of the singularity category of the cochains, as conjectured in the original context by Benson-Greenlees. I expect understanding this support theory for singularity categories of ring spectra will form a part of my project, especially in using it to classify thick subcategories for hypersurface ring spectra - possibly extending results of Stevenson and Takahashi with discrete rings.

### Organisations

## People |
## ORCID iD |

John Greenlees (Primary Supervisor) | |

Thomas Peirce (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/W523793/1 | 30/09/2021 | 29/09/2025 | |||

2737776 | Studentship | EP/W523793/1 | 02/10/2022 | 29/09/2026 | Thomas Peirce |