Preprints


Home Page

In reverse chronological order, so newest first. Early papers are not available here in electronic form.


The Asymptotic Behavior of Frobenius Direct Images of Rings of Invariants.

(with M. Hashimoto)

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.

(to appear in Advances in Math.)

pdf

Degree Bounds on Homology and a Conjecture of Derksen.

(with M. Chardin)

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.

(to appear in Compositio Math.)

pdf

Automorphisms of Katz-Gabber covers.

(with F. Bleher, T. Chinburg and B. Poonen)

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.

(to appear in Math. Annalen)

pdf

Brauer Theory for Profinite Groups.

(with J. MacQuarrie)

We generalize much of the machinery of Brauer theory to the setting of profinite groups.

(Jour. Algebra 398 (2014) 496-508)

pdf

Group Actions on Rings and the Cech Complex.

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)

dvi
pdf

Fusion Systems for Profinite Groups.

(with R. Stancu)

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)

dvi
pdf

Exterior and Symmetric Powers of Modules for Cyclic 2-groups .

(with F. Himstedt)

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)

dvi
pdf

On the Castelnuovo-Mumford Regularity of the Cohomology Ring of a Group.

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)

dvi
pdf

On the Castelnuovo-Mumford Regularity of Rings of Polynomial Invariants.

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)

dvi
pdf

An Element of Order 4 in the Nottingham Group at the Prime 2.

(with T. Chinburg)

We produce an explicit element of order four in the Nottingham group.

dvi
pdf

Equivariant Hilbert Series.

(with F. Himstedt)

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)

dvi
pdf

Pro-$p$ groups with non-isomorphic discrete and continuous cohomology groups.

(with G. Fern\a'andez-Alcober, I. Kazatchkov and V. Remeslennikov)

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)

dvi
pdf

On Cohomology Isomorphisms of Groups

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)

dvi
pdf

Structure Theorems over Polynomial Rings

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)

dvi
pdf

A Relative Version of a Theorem of Webb

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)

dvi
pdf

Double Coset Formulas for Profinite Groups

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)

dvi
pdf

Cyclic Group Actions on Polynomial Rings

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)

dvi
pdf

On the Construction of Permutation Complexes for Profinite Groups

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) 369378)

dvi
pdf

Smith Theory for Algebraic Varieties

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)

dvi
pdf

The Tate-Farrell Cohomology of the Morava Stabilizer Group $S_{p-1}$ with coefficients in $E_{p-1}$

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)

dvi
pdf

Mackey Functors and Control of Fusion

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)

dvi
pdf
correction

Permutation Complexes for Profinite Groups

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)

dvi
pdf

The Module Structure of a Group Action on a Polynomial Ring: Examples, Generalizations and Applications

With Dikran Karagueuzian

A commentary on the previous paper.

(CRM Proceedings and Lecture Notes 35 (2004) 139-158)

dvi
pdf

The Module Structure of a Group Action on a Polynomial Ring: A Finiteness Theorem

With Dikran Karagueuzian

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)

dvi
pdf

The Bredon Cohomology of Subgroup Complexes

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)

dvi
pdf

Cohomology and Finite Subgroups of Profinite Groups

With Pham Anh Minh

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)

dvi
pdf

The Cohomology of a Pro-p Group with a Powerfully Embedded Subgroup

With Pham Anh Minh

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)

dvi
pdf

A Splitting Principle for Modular Group Representations

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)

dvi
ps

The Cohomology of Analytic Pro-p Groups

With Thomas Weigel

(In New Horizons in Pro-p Groups, du Sautoy, Segal and Shalev eds., Birkhauser (2000) 349-416)

ps

Group Actions on Polynomial and Power Series Rings

(Pacific Jour. Math. 195 (2000) 225-230)

dvi

The Action of the Automorphism Group of a p-group on Cohomology

(Math. Proc. Cam. Phil. Soc. 127 (1999) 495-496)

dvi

The Module Structure of a Group Action on a Polynomial Ring

With Dikran Karagueuzian

(Jour. Algebra 218 (1999) 672-692)

dvi

The Quotient Space of the p-subgroup Complex is Contractible

(Comm. Math. Helv. 73 (1998) 400-405)

dvi