Searcharxiv⌕ Search

arXiv subjects

J. P. C. Greenlees

Publications and source records attributed to J. P. C. Greenlees.

At least 19 recordsLinked to original sources
arXiv2026

Rational G-spectra over blocks with finite Weyl groups

We show that for any clopen collection X of subgroups of G with finite Weyl groups, the category of G-spectra with geometric isotropy in X is equivalent to the category of equivariant sheaves over X. This gives an algebraic model of injective dimension at most the rank of G. [Version 2 fills several gaps. (a) The selection of conjugacy class representatives (b) smallness of generators (c) compatibility of Morita equivalences with diagram (d) definition of complete model. Exposition and notational consistency improved.]

math.AT↗
arXiv2026

The singularity category and duality for complete intersection groups

If G is a finite group, some aspects of the modular representation theory depend on the cochains C^*(BG; k), viewed as a commutative ring spectrum. We consider its singularity category (in the sense of the author and Stevenson arxiv 1702.07957) and show that it is the bounded derived category of the Ω-Tate ring spectrum (k-nullification of the Koszul dual, C_*(ΩBG_p)). We establish a form of Gorenstein duality for C_*(ΩBG_p) and a form of Tate duality for the Ω-Tate homology. If C^*(BG; k) is a homotopical complete intersection in a strong sense there is a stable Koszul complex construction of the Ω-Tate spectrum. [v3: (1) role of ci condition clarified.(2) Novel statements flagged, Ω-Tate named and highlighted.(3) Study of the norm map expanded.]

math.AT↗
arXiv2026

The rational homotopy groups of virtual spheres for rank 1 compact Lie groups

We calculate the rational representation-ring-graded stable stems for rank 1 groups, SU(2), SO(3), Pin (2), O(2), Spin(2) and SO(2), in the same spirit as the calculations for finite groups in arXiv:2205.02382 with J.D.Quigley. This illustrates the effectiveness of the algebraic models for these categories of G-spectra, and the way tom Dieck splitting fails for desuspensions. [v4: typos and tweaks in wording]

math.AT↗
arXiv2025

Spaces of subgroups of toral groups

We study the space of conjugacy classes of subgroups of a compact Lie group G whose identity component is a torus, and consider how various invariants of subgroups behave as sheaves over this space. This feeds in to the author's programme to give algebraic models of rational G-equivariant cohomology theories. The methods are illustrated by making the outcome explicit for all toral subgroups of compact connected rank 2 groups.

math.AT↗
arXiv2025

The spectral space of conjugacy classes of subgroups of a compact Lie group

The space of conjugacy classes of subgroups of a compact Lie groups with its Zariski topology splits as a disjoint union of clopen blocks, each dominated by a subgroup H with finite Weyl group, and the structure of each block is controlled by an integral representation of the component group of H

math.GR↗
arXiv2025

Rational SU(3)-equivariant cohomology theories

We describe the spectral space of conjugacy classes of subgroups of SU(3), together with the additional structure of a sheaf of rings and a component structure. It is a disjoint union of 18 blocks each dominated by a subgroup. For each of these blocks we identify a sheaf of rings and component structure. Taken together, this gives an abelian category A(SU(3)) designed to reflect the structure of rational SU(3)-equivariant cohomology theories, and we assemble the results from elsewhere to show that the category of rational SU(3)-spectra is Quillen equivalent to the category of differential graded objects of A(SU(3)).

math.AT↗
arXiv2025

An algebraic model for rational U(2)-spectra

We construct an explicit and calculable models for rational U(2)-spectra. This is obtained by assembling seven blocks obtained in previous work: the toral part and earlier work on small toral groups. The assembly process requires detailed input on fusion and Weyl groups.

math.AT↗
arXiv2025

Rational G-spectra for rank 2 toral groups of mixed type

We give an explicit and calculable algebraic model for the block of rational G-spectra on full subgroups when G has identity component a 2-torus T, and component group of order 2 acting non-trivially on H_1(T). The example of particular interest is the normalizer of the maximal torus in U(2), which constitutes one of the most complicated blocks in the analysis of SU(3). This builds on the determination of subgroups up to conjugacy in arXiv 2501.06914

math.AT↗
arXiv2025

Algebraic models for 1-dimensional categories of rational G-spectra

In this paper we give algebraic models for rational G-spectra for a compact Lie group G when the geometric isotropy is restricted to lie in a 1-dimensional block of conjugacy classes. This includes all blocks of all groups of dimension 1, semifree spectra, and 1-dimensional blocks for many other groups G. The results were known previously for G=SO(2) or O(2) due to work of Barnes, Shipley and the author.

math.AT↗
arXiv2025

Torsion models for tensor-triangulated categories

Given a rigidly-compactly generated tensor-triangulated category whose Balmer spectrum is finite dimensional and Noetherian, we construct a torsion model for it, which is equivalent to the original tensor-triangulated category. The torsion model is determined in an adelic fashion by objects with singleton supports. This categorifies the Cousin complex from algebra, and the process of reconstructing a spectrum from its monochromatic layers in chromatic stable homotopy theory. This model is inspired by work of the second author in rational equivariant stable homotopy theory, and extends previous work of the authors from the one-dimensional setting.

math.AT↗
arXiv2023

Prismatic decompositions and rational $G$-spectra

We study the tensor-triangular geometry of the category of rational $G$-spectra for a compact Lie group $G$. In particular, we prove that this category can be naturally decomposed into local factors supported on individual subgroups, each of which admits an algebraic model. This is an important step and strong evidence towards the third author's conjecture that the category of rational $G$-spectra admits an algebraic model for all compact Lie groups. To facilitate these results, we relate topological properties of the associated Balmer spectrum to structural features of the group $G$ and the category of rational $G$-spectra. A key ingredient is our presentation of the spectrum as a Priestley space, separating the Hausdorff topology on conjugacy classes of closed subgroups of $G$ from the cotoral ordering. We use this to prove that the telescope conjecture holds in general for rational $G$-spectra, and we determine exactly when the Balmer spectrum is Noetherian. In order to construct the desired decomposition of the category, we develop a general theory of `prismatic decompositions' of rigidly-compactly generated tensor-triangulated categories, which in favourable cases gives a series of recollements for reconstructing the category from local factors over individual points of the spectrum.

math.AT↗
arXiv2022

Torsion models for tensor-triangulated categories: the one-step case

Given a suitable stable monoidal model category $\mathscr{C}$ and a specialization closed subset $V$ of its Balmer spectrum one can produce a Tate square for decomposing objects into the part supported over $V$ and the part supported over $V^c$ spliced with the Tate object. Using this one can show that $\mathscr{C}$ is Quillen equivalent to a model built from the data of local torsion objects, and the splicing data lies in a rather rich category. As an application, we promote the torsion model for the homotopy category of rational circle-equivariant spectra from [18] to a Quillen equivalence. In addition, a close analysis of the one step case highlights important features needed for general torsion models which we will return to in future work.

math.AT↗
arXiv2022

Rational torus-equivariant stable homotopy V: the torsion Adams spectral sequence

We provide a calculational method for rational stable equivariant homotopy theory for a torus G based on the homology of the Borel construction on fixed points. More precisely we define an abelian torsion model, A_t(G) of finite injective dimension, a homology theory \piAt_* taking values in A_t(G) based on the homology of the Borel construction, and a finite Adams spectral sequence Ext_{A_t(G)}^{*,*}(\piAt_*(X), \piAt_*(Y)) ==> [X,Y]^G_* for rational G-spectra X and Y.

math.AT↗
arXiv2022

Gorenstein Duality and Universal Coefficient Theorems

The paper describes a duality phenomenon for cohomology theories with the character of Gorenstein rings. For a connective cohomology theory with the p-local integers in degree 0, and coefficient ring R_* Gorenstein of shift 0, this states that for X with R_*(X) torsion, we have R^*(X)=Σ^a Hom( R_*(X), Z/p^{\infty}). A corresponding statement for modules over a commutative Gorenstein ring spectrum is also proved. [Minor typographical and bibliographic changes to the last version.]

math.AT↗
arXiv2022

Separated and complete adelic models for one-dimensional Noetherian tensor-triangulated categories

We prove the existence of various adelic-style models for rigidly small-generated tensor-triangulated categories whose Balmer spectrum is a one-dimensional Noetherian topological space. This special case of our general programme of giving adelic models is particularly concrete and accessible, and we illustrate it with examples from algebra, geometry, topology and representation theory. This version: minor improvements, additional references and examples.

math.AT↗
arXiv2020

Adelic models of tensor-triangulated categories

We show that a well behaved Noetherian, finite dimensional, stable, monoidal model category is equivalent to a model built from categories of modules over completed rings in an adelic fashion. For abelian groups this is based on the Hasse square, for chromatic homotopy theory this is based on the chromatic fracture square, and for rational torus-equivariant homotopy theory this is the model of Greenlees-Shipley arXiv:1101.2511.

math.AT↗
arXiv2020

Morita theory and singularity categories

We propose an analogue of the bounded derived category for an augmented ring spectrum, defined in terms of a notion of Noether normalization. In many cases we show this category is independent of the chosen normalization. Based on this, we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are Koszul dual in the style of the BGG correspondence. Examples of interest include Koszul algebras and Ginzburg DG-algebras, $C^*(BG)$ for finite groups (or for compact Lie groups with orientable adjoint representation), cochains in rational homotopy theory and various examples from chromatic homotopy theory.

math.AT↗