In reverse chronological order, so newest first. Early papers are not available here in electronic form.
Consider a finite group G acting on a graded Noetherian k-algebra S, for some field k of characteristic p; for example S might be a polynomial ring. Regard S as a kG-module and consider the multiplicity of a particular indecomposable module as a summand in each degree. We show how this can be described in terms of homological algebra and how it is linked to the geometry of the group action on the spectrum of S.
We extend the results in the paper Rank, Coclass and Cohomology to profinite groups. (In the published version in Section 3 there is an incorrect mention of a Mackey functor for a group G; it should be a global Mackey functor.)
(Transactions London Math. Soc. 8 (2021) 435-439)
We prove that for any prime p the finite p-groups of fixed coclass have only finitely many different mod-p cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of bounded rank.
(International Math. Research Notices, IMRN (2021) 17399-17412)
We construct a well-behaved stable category of modules for a large class of infinite groups. We then consider its Picard group, which is the group of invertible (or endotrivial) modules. We show how this group can be calculated when the group acts on a tree with finite stabilisers.
(Advances in Math. 358 (2019) 106853)
Let M be a finite dimensional modular representation of a finite group G. We consider the generating function for the non-projective part of the tensor powers of M, and consider the reciprocal of the radius of convergence of this power series. We investigate the properties of this invariant, using tools from representation theory and from the theory of commutative Banach algebras.
(Journal London Math. Soc. 101 (2020) 828?856)
We generalize a theorem of Weiss about integral permutation lattices to profinite lattices of infinite rank.
(Advances in Math. 361 (2020) 106925)
Notes for a summer school course.
(in Geometric and Topological Aspects of the Representation Theory of Finite Groups, Springer 2018)
We define the Frobenius limit of a module over a ring of prime characteristic to be the limit of the normalized Frobenius direct images in a certain Grothendieck group. When a finite group acts on a polynomial ring, we calculate this limit for all the modules over the twisted group algebra that are free over the polynomial ring; there are applications to invariant theory.
(Advances in Math. 305 (2017), 144-164)
We give a counterexample to a conjecture of Derksen concerning syzygies of rings of invariants. We also prove a modified version of the conjecture and some general results giving bounds on syzygies.
(Compositio Math. 152 (2016), 2041-2049)
We consider certain group actions on algebraic curves, called Katz-Gabber covers, and ask when the group action can be extended. We also consider the problem of finding explicit groups of automorphisms of k[[t]] and show how this is related to the existence of Katz-Gabber covers.
(Math. Annalen 368 (2017), 811-836)
We generalize much of the machinery of Brauer theory to the setting of profinite groups.
(Jour. Algebra 398 (2014) 496-508)
This paper gives a simpler proof of our previous results on regularity of invariants and of finitely many isomorphism classes of indecomposables for a group action on a polynomial ring. It also applies to a more general class of rings.
(Advances in Math. 240 (2013) 291-301)
We introduce the notion of a pro-fusion system on a pro-p group, which generalizes the notion of a fusion system on a p-group. We also prove a version of Alperin's Fusion Theorem for pro-fusion systems.
(Jour. London Math. Soc. 89 (2014) 461-481)
We prove a recursive formula for the exterior and symmetric powers of modules for a cyclic 2-group. This makes computation straightforward. Previously, a complete description was only known for cyclic groups of prime order.
(Jour. Algebra 410 (2014) 393-420)
We prove Benson's Regularity Conjecture, that the Castelnuovo-Mumford regularity of the mod-p cohomology ring of a finite group is zero. We also prove versions for compact Lie groups and virtual Poincare duality groups. One consequence is a bound on the degrees of the generators and relations of the cohomology ring.
(Jour. American Math. Soc. 23 (2010) 1159-1173)
We show that, when a group acts on a polynomial ring over a finite field, the ring of invariants has Castelnuovo-Mumford regularity at most zero. As a consequence, we prove a well-known conjecture that the invariants are always generated in degrees at most n(|G|-1), where n >1 is the number of polynomial generators and |G|>1 is the order of the group. We also prove some other conjectures in invariant theory.
(Annals of Math. 174 (2011) 499-517)
We produce an explicit element of order four in the Nottingham group.
We consider a finite group acting on a graded module and define an equivariant degree that generalizes the usual non-equivariant degree. The value of this degree is a module for the group, up to a rational multiple. We investigate how this behaves when the module is a ring and apply our results to reprove some results of Kuhn on the cohomology of groups.
(Algebra and Number Theory 3 (2009) 423-443)
We ask when the mod $p$ Galois cohomology of a pro-$p$ group is equal to its cohomology as an abstract group, with particular attention given to finitely generated groups and $H^2$, where an explicit example shows that the cohomologies are different.
(St. Petersburg Math. Jour. 19 (2008) 961-973)
We show how Mislin's theorem on group homomorphisms that induce an isomorphism in cohomology can be proved based on ideas of Alperin and the use of Lannes's T-functor. This enables us to extend the class of groups to include groups of finite virtual cohomological dimension and profinite groups.
(Jour. Algebra 313 (2007) 802-810)
We consider a module over a polynomial ring with a group action and show that the existence of structure theorem, as in our work with Karagueuzian, is equivalent to the presence of only finitely many indecomposable modules and also equivalent to various other homological formulations.
(Advances in Mathematics 208 (2007), 408-421)
We prove a relative version of the theorem of Webb that the augmented chain complex of the Quillen complex of a finite group is homotopy euivalent to a complex of projectives. This allows us to take into account the group of automorphisms of the group.
(Jour. Pure and Applied Algebra 212 (2008) 1984-1986)
We show that in certain circumstances there is a sort of double coset formula for induction followed by restriction for representations of profinite groups.
(Communications in Algebra 36 (2008) 1059-1066)
Consider a cyclic group of order $p^n$ acting on a module in characteristic $p$. We show how to reduce the calculation of the symmetric algebra to that of the exterior algebra.
(Bulletin London Math. Soc. 39 (2007) 181-188)
Goerss, Henn, Mahowald and Rezk have constructed certain complexes of permutation modules for a Morava stabilizer group. We describe a more conceptual approach.
(Geometry & Topology Monographs 11 (2007) 369–378)
We show how to extend work of Rickard to associate a complex of projective coefficient systems to a group action on a variety. Then Smith Theory becomes applicable.
(Algebraic and Geometric Topology 4 (2004), 121-131)
We calculate the Tate-Farrell cohomology of the Morava stabilizer group $S_{p-1}$ with coefficients in the moduli space $E_{p-1}$ for an odd prime p.
(Contemp. Math. 346 (2004), 485-492)
Mislin proved that a subgroup H of G controls p-fusion in G if and only if the restriction map is an isomorphism in mod-p cohomology. His proof used deep results from algebraic topology, notably Carlsson's proof of the Segal Conjecture. We show how this problem can be approached algebraically using Mackey functors. As a corollary we obtain a result about the cohomology of p-permutation modules.
(Bull. London Math. Soc. 36 (2004) 623-632)
An important tool in the analysis of discrete groups of finite virtual cohomological dimension is the existence of a contractible CW-complex on which the group acts with finite stabilizers. We develop an analogue for profinite groups.
(Commentarii Mathematici Helvetici 82 (2007), 1-37)
A commentary on the previous paper.
(CRM Proceedings and Lecture Notes 35 (2004) 139-158)
We consider a group acting on a polynomial ring over a finite field, and study the polynomial ring as a module for the group. We prove a structure theorem with several striking corollaries. For example, any indecomposable module that occurs as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite. There are also applications to invariant theory, giving a priori bounds on the degrees of the generators.
(Jour. American Math. Soc. 20 (2007), 931-967)
We show how various sequences associated to a class of subgroups of a given group and a coefficient system can be analysed by methods from homological algebra. We are particularly interested in when these sequences are exact, or, if not, when their homology is equal to the higher limits of the coefficient system.
(Jour. Pure and Applied Algebra 199 (2005) 261-298)
We show that the mod-p cohomology ring of a pro-p group is finitely generated modulo nilpotents if and only if the number of conjugacy classes of elementary abelian subgroups is finite, and that the ring is finitely generated if and only if the number of conjugacy classes of finite subgroups is finite.
(Proc. American Math. Soc. 132 (2004) 1581-1588)
We calculate the cohomology of a pro-p group with an extendable and almost powerfully embedded subgroup.
(Jour. Pure and Applied Algebra 189 (2004) 221-246)
We have shown previously how a complex representation of a finite group can be split into a virtual sum of representations induced from 1-dimensional representations in a natural way (sometimes known as explicit Brauer induction). Here we consider the modular case.
(Bull. London Math. Soc. 34 (2002) 551-560)
(In New Horizons in Pro-p Groups, du Sautoy, Segal and Shalev eds., Birkhauser (2000) 349-416)
(Pacific Jour. Math. 195 (2000) 225-230)
(Math. Proc. Cam. Phil. Soc. 127 (1999) 495-496)
(Jour. Algebra 218 (1999) 672-692)
(Comm. Math. Helv. 73 (1998) 400-405)