arXiv subjects

Neil Strickland

Publications and source records attributed to Neil Strickland.

At least 19 recordsLinked to original sources

arXiv2025

The spectrum of global representations for families of bounded rank and VI-modules

A global representation is a compatible collection of representations of the outer automorphism groups of the finite groups belonging to a family $\mathscr{U}$. These arise in classical representation theory, in the study of representation stability, as well as in global homotopy theory. In this paper we begin a systematic study of the derived category $\mathsf{D}(\mathscr{U};k)$ of global representations over fields $k$ of characteristic zero, from the point-of-view of tensor-triangular geometry. We calculate its Balmer spectrum for various infinite families of finite groups including elementary abelian $p$-groups, cyclic groups, and finite abelian $p$-groups of bounded rank. We then deduce that the Balmer spectrum associated to the family of finite abelian $p$-groups has infinite Krull dimension and infinite Cantor--Bendixson rank, illustrating the complex phenomena we encounter. As a concrete application, we provide a complete tt-theoretic classification of finitely generated derived VI-modules. Our proofs rely on subtle information about the growth behaviour of global representations studied in a companion paper, as well as novel methods from non-rigid tt-geometry.

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arXiv2025

Global representation theory: Homological foundations

A global representation is a compatible collection of representations of the outer automorphism groups of the groups belonging to some collection of finite groups $\mathscr{U}$. Global representations assemble into an abelian category $\mathsf{A}(\mathscr{U})$, simultaneously generalising classical representation theory and the category of VI-modules appearing in the representation theory of the general linear groups. In this paper we establish homological foundations of its derived category $\mathsf{D}(\mathscr{U})$. We prove that any complex of projective global representations is DG-projective, and hence conclude that the derived category admits an explicit model as the homotopy category of projective global representations. We show that from a tensor-triangular perspective it exhibits some unusual features: for example, there are very few dualizable objects and in general many more compact objects. Under more restrictive conditions on the family $\mathscr{U}$, we then construct torsion-free classes for global representations which encode certain growth properties in $\mathscr{U}$. This lays the foundations for a detailed study of the tensor-triangular geometry of derived global representations which we pursue in forthcoming work.

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arXiv2024

Arithmetic localisation and completion of spectra

This is an exposition of facts about p-local spectra, p-complete spectra and modules over the p-complete sphere spectrum, including homological criteria for finiteness. Most things are well-known to the experts, with a couple of potential exceptions: every dualisable p-complete spectrum is the p-completion of a finite spectrum, and the category of modules over the p-complete sphere has homological Brown representability.

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arXiv2020

An introduction to the category of spectra

These notes give a brief introduction to the category of spectra as defined in stable homotopy theory. In particular, Section 5 discusses an extensive list of examples of spectra whose properties have been found to be interesting.

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arXiv2020

Algebraic theory of abelian groups

This document aims to give a self-contained account of the parts of abelian group theory that are most relevant for algebraic topology. It is almost purely expository, although there are some slightly unusual features in the treatment of tensor products, torsion products and $\text{Ext}$ groups.

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arXiv2020

The model structure for chain complexes

Let $\text{Ch}$ be the category of (possibly unbounded) chain complexes of abelian groups. In this note we construct the standard Quillen model structure on $\text{Ch}$, by a method that is somewhat different from the standard one. Essentially, we use a functorial two-stage projective resolution for abelian groups, and build everything directly from that. This has the advantage of being very concrete, explicit and functorial. It does not rely on the small object argument, or make any explicit use of transfinite induction. On the other hand, it is not so conceptual, and it does use the fact that subgroups of free abelian groups are free, so it does not generalise to many rings other than $\mathbb{Z}$. We do not claim any great technical benefit for this approach, but it seems like an interesting alternative, and may be pedagogically useful.

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arXiv2020

Is $D$ symmetric monoidal?

We verify that a certain functor $D\colon\text{Sp}^Σ(\text{Ch}^+)\to\text{Ch}$ is symmetric monoidal. This functor is used elsewhere in developing the model category theory of symmetric spectra and of chain complexes graded over $\mathbb{N}$ or $\mathbb{Z}$.

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arXiv2019

Iterated chromatic localisation

We study a certain monoid of endofunctors of the stable homotopy category that includes localizations with respect to finite unions of Morava $K$-theories. We work in an axiomatic framework that can also be applied to analogous questions in equivariant stable homotopy theory. Our results should be helpful for the study of transchromatic phenomena, including the Chromatic Splitting Conjecture. The combinatorial parts of this work have been formalised in the Lean proof assistant.

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arXiv2019

Chromatic (co)homology of finite general linear groups

We study the Morava $E$-theory (at a prime $p$) of $BGL_d(F)$, where $F$ is a finite field with $|F|=1\pmod{p}$. Taking all $d$ together, we obtain a structure with two products $\times$ and $\bullet$. We prove that it is a polynomial ring under $\times$, and that the module of $\times$-indecomposables inherits a $\bullet$-product, and we describe the structure of the resulting ring. In the process, we prove many auxiliary structural results.

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arXiv2018

A combinatorial model for the known Bousfield classes

We give a combinatorial construction of an ordered semiring A, and show that it can be identified with a certain subquotient of the semiring of p-local Bousfield classes, containing almost all of the classes that have previously been named and studied. This is a convenient way to encapsulate most of the known results about Bousfield classes.

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arXiv2018

Level three structures

We give a detailed discussion of the universal example of an elliptic curve equipped with a level three structure over a base on which three is invertible. This is intended as a convenient reference for applications in elliptic cohomology and stable homotopy theory.

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arXiv2016

Uniformization of embedded surfaces

Let X be a closed surface of genus two embedded in the 3-sphere. Then X inherits a metric and an orientation, which give an almost complex structure, which automatically integrates to a genuine complex structure, making X a Riemann surface. It follows that X is conformally isomorphic to a branched cover of the Riemann sphere, or to the quotient of the unit disc by the action of a Fuchsian group. The theorems behind these statements are important, well-known, and a century old. Nonetheless, we believe that the literature contains no examples where a significant fraction of the structure can be made explicit. This monograph is a partially successful attempt to provide such an example, starting with a particular surface X that has interesting geometry. The required theory is surprisingly rich, and is supported by a large body of Maple code, which is used for semi-formal verification of many proofs, as well as for numerical calculation.

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arXiv2013

Large self-injective rings and the generating hypothesis

We construct a number of different examples of non-Noetherian graded rings that are injective as modules over themselves (or have some related but weaker properties). We discuss how these are related to the theory of triangulated categories, and to Freyd's Generating Hypothesis in stable homotopy theory.

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arXiv2013

An abelian embedding for Moore spectra

We embed the category of Moore spectra as a full subcategory of an abelian category, and make some remarks about abelian embeddings of various other categories of spectra.

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arXiv2012

Tambara functors

We survey and extend the theory of Tambara functors. These are algebraic structures similar to Mackey functors, but with multiplicative norm maps as well as additive transfer maps, and a rule governing their interaction that is most easily formulated in an abstract categorical framework. Examples include Burnside rings, representation rings, and homotopy groups of equivariant E-infinity ring spectra in stable homotopy theory. Some other examples are related to Witt rings in the sense of Dress and Siebeneicher.

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arXiv2012

Geometry and cohomology of Khovanov-Springer varieties

The Khovanov-Springer variety X(n) is a certain subvariety of the variety of flags of length 2n, which has been studied from various different points of view. We give a new proof of the ring structure of the cohomology of X(n) and relate it to some interesting geometric, combinatorial and algebraic phenomena.

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arXiv2008

Chains on suspension spectra

We define and study a homological version of Sullivan's rational de Rham complex for simplicial sets. This new functor can be generalised to simplicial symmetric spectra and in that context it has excellent categorical properties which promise to make a number of interesting applications much more straightforward.

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arXiv2007

Triangulated categories without models

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories.

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