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John Greenlees

Publications and source records attributed to John Greenlees.

6 recordsLinked to original sources
arXiv2026

Rational Sp(2)-equivariant cohomology theories I: dominant subgroups

We give a general description of the spectral space of conjugacy classes of subgroups of Sp(2): it is a disjoint union of finitely many blocks, each dominated by a subgroup: of these blocks, 26 are of dimension 1, 6 are of dimension 2 and the remainder are isolated points. On each of these blocks there is a sheaf of polynomial rings and a component structure. These are the ingredients for constructing an abelian category A(Sp(2)) designed to reflect the structure of rational Sp(2)-equivariant cohomology theories. We assemble the results from earlier papers in the series to show that the category of rational Sp(2)-spectra is Quillen equivalent to the category of differential graded objects of A(Sp(2)). In the sequel we will make the fine structure of A(Sp(2)) explicit, and make calculations based upon it.

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arXiv2023

Modules with finitely generated cohomology, and singularities of $C^*BG$

Let $G$ be a finite group and $k$ a field of characteristic $p$. We conjecture that if $M$ is a $kG$-module with $H^*(G,M)$ finitely generated as a module over $H^*(G,k)$ then as an element of the stable module category $\mathsf{StMod}(kG)$, $M$ is contained in the thick subcategory generated by the finitely generated $kG$-modules and the modules $M'$ with $H^*(G,M')=0$. We show that this is equivalent to a conjecture of the second author about generation of the bounded derived category of cochains $C^*(BG;k)$, and we prove the conjecture in the case where the centraliser of every element of $G$ of order $p$ is $p$-nilpotent. In this case some stronger statements are true, that probably fail for more general finite groups.

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arXiv2022

Formality of cochains on BG

Let $G$ be a compact Lie group with maximal torus $T$. If $|N_G(T)/T|$ is invertible in the field $k$ then the algebra of cochains $C^*(BG;k)$ is formal as an $A_\infty$ algebra, or equivalently as a DG algebra.

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arXiv2021

The singularity and cosingularity categories of $C^*BG$ for groups with cyclic Sylow $p$-subgroups

We construct a differential graded algebra (DGA) modelling certain $A_\infty$ algebras associated with a finite group $G$ with cyclic Sylow subgroups, namely $H^*BG$ and $H_*ΩBG^{^\wedge}_p$. We use our construction to investigate the singularity and cosingularity categories of these algebras. We give a complete classification of the indecomposables in these categories, and describe the Auslander--Reiten quiver. The theory applies to Brauer tree algebras in arbitrary characteristic, and we end with an example in characteristic zero coming from the Hecke algebras of symmetric groups.

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arXiv2021

The local cohomology spectral sequence for topological modular forms

We discuss proofs of local cohomology theorems for topological modular forms, based on Mahowald-Rezk duality and on Gorenstein duality, and then make the associated local cohomology spectral sequences explicit, including their differential patterns and hidden extensions.

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