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William Woods

Publications and source records attributed to William Woods.

12 recordsLinked to original sources
arXiv2026

New directions in the study of prime ideals in rational, nilpotent Iwasawa algebras

Let G be a nilpotent p-valuable (compact p-adic Lie) group. There is an ongoing investigation into the prime ideals of its completed group algebra (Iwasawa algebra), and there remains an open conjecture that they can all be proved to have a canonical standard form. We very this conjecture for several new classes of nilpotent groups, including those corresponding to the positive subalgebra of almost all classical and exceptional types, curiously excluding those of type C.

math.RT↗
arXiv2025

Skew power series rings with automorphisms of finite inner order

We investigate the algebraic properties of the bounded skew power series ring $Q^+[[x;σ,δ]]$ over a (complete, simple) \emph{standard} filtered artinian algebra $Q$ of positive characteristic. Here we are assuming that $(σ,δ)$ is a commuting skew derivation of $Q$, where $δ$ is inner, satisfying the appropriate compatibility conditions. In a previous work of the authors, it was proved that $Q^+[[x;σ,δ]]$ is a simple ring whenever $σ$ has infinite inner order. We now extend this result to the case when $σ$ has finite inner order, proving that this ring is often simple, and always prime in cases of interest. This solves an important special case of an open question of Letzter, and yields important consequences for the classification of prime ideals in Iwasawa algebras of solvable groups.

math.RA↗
arXiv2024

Bounded skew power series rings for inner $σ$-derivations

We define and explore the bounded skew power series ring $R^+[[x;σ,δ]]$ defined over a complete, filtered, Noetherian prime ring $R$ with a commuting skew derivation $(σ,δ)$. We establish precise criteria for when this ring is well-defined, and for an appropriate completion $Q$ of $Q(R)$, we prove that if $Q$ has characteristic $p$, $δ$ is an inner $σ$-derivation and no positive power of $σ$ is inner as an automorphism of $Q$, then $Q^+[[x;σ,δ]]$ is often prime, and even simple under certain mild restrictions on $δ$. It follows from this result that $R^+[[x;σ,δ]]$ is itself prime.

math.RA↗
arXiv2023

The conjugation action in completed group rings

Let $k = \mathbb{F}_p$ or $\mathbb{Z}_p$ (or finite extensions of these). Let $G$ be a $p$-valuable group, and form its completed group algebra $kG$. By analysing the conjugation action of $G$ on itself, we prove two structural results. Firstly, we show that all inner automorphisms of $kG$ that preserve $G$ are induced from inner automorphisms of $G$. Secondly, for a closed subgroup $Γ$ of $G$, we calculate the $Γ$-fixed ring of $kG/I$ under the conjugation action of $Γ$, for certain ideals $I$ induced from the $G$-centraliser of $Γ$.

math.RA↗
arXiv2023

Skew power series rings over a prime base ring

In this paper, we investigate the structure of skew power series rings of the form $S = R[[x;σ,δ]]$, where $R$ is a complete filtered ring and $(σ,δ)$ is a skew derivation respecting the filtration. Our main focus is on the case in which $σδ= δσ$, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if $P$ is an invariant prime ideal of $R$, is $PS$ a prime ideal of $S$? When $R$ has characteristic $p$, our results reduce this to a finite-index problem. We also give preliminary results in the "Iwasawa algebra" case $δ= σ- \mathrm{id}_R$ in arbitrary characteristic. A key step in our argument will be to show that for a large class of Noetherian algebras, the nilradical is "almost" $(σ,δ)$-invariant in a certain sense.

math.AC↗
arXiv2023

Filtered skew derivations on simple artinian rings

Given a complete, positively filtered ring $(R,f)$ and a compatible skew derivation $(σ,δ)$, we may construct its skew power series ring $R[[x;σ,δ]]$. Due to topological obstructions, even if $δ$ is an \emph{inner} $σ$-derivation, in general we cannot ``untwist" it, i.e. reparametrise to find a filtered isomorphism $R[[x; σ, δ]] \cong R[[x'; σ]]$, as might be expected from the theory of skew polynomial rings; similarly when $σ$ is an inner automorphism. We find general conditions under which it is possible to untwist the multiplication data, and use this to analyse the structure of $R[[x;σ,δ]]$ in the simplest case when $R$ is a matrix ring over a (noncommutative) noetherian discrete valuation ring.

math.RA↗
arXiv2018

Dimension theory in iterated local skew power series rings

By analysing the structure of the associated graded ring with respect to certain filtrations, we deduce a number of good properties of iterated local skew power series rings over appropriate base rings. In particular, we calculate the Krull dimension, prime length and global dimension in well-behaved cases, obtaining lower bounds to complement the upper bounds obtained by Venjakob and Wang.

math.RA↗
arXiv2018

Maximal prime homomorphic images of mod-$p$ Iwasawa algebras

Let $k$ be a finite field of characteristic $p$, and $G$ a compact $p$-adic analytic group. Write $kG$ for the completed group ring of $G$ over $k$. In this paper, we describe the structure of the ring $kG/P$, where $P$ is a minimal prime ideal of $kG$. We give an isomorphism between $kG/P$ and a matrix ring with coefficients in the ring $(k'G')_α$, where $k'/k$ is a finite field extension, $G'$ is a large subquotient of $G$ with no finite normal subgroups, and $(-)_α$ is a "twisting" operation that preserves several desirable properties of the ring structure. We demonstrate an application of this isomorphism by setting up correspondences between certain ideals and subrings of $kG$ and those of $(k'G')_α$, and showing that these correspondences often preserve some useful properties, such as almost-faithfulness of an ideal, or control of an ideal by a closed normal subgroup.

math.RT↗
arXiv2017

Extensions of almost faithful prime ideals in virtually nilpotent mod-$p$ Iwasawa algebras

Let $G$ be a nilpotent-by-finite compact $p$-adic analytic group for some $p>2$, and $H = \mathbf{FN}_p(G)$ its finite-by-(nilpotent $p$-valuable) radical. Fix a finite field $k$ of characteristic $p$, and write $kG$ for the completed group ring of $G$ over $k$. We show that almost faithful $G$-stable prime ideals $P$ of $kH$ extend to prime ideals $PkG$ of $kG$.

math.RA↗
arXiv2016

On the structure of virtually nilpotent compact $p$-adic analytic groups

Let $G$ be a compact $p$-adic analytic group. We recall the well-understood finite radical $Δ^+$ and FC-centre $Δ$, and introduce a $p$-adic analogue of Roseblade's subgroup $\mathrm{nio}(G)$, the unique largest orbitally sound open normal subgroup of $G$. Further, when $G$ is nilpotent-by-finite, we introduce the finite-by-(nilpotent $p$-valuable) radical $\mathbf{FN}_p(G)$, an open characteristic subgroup of $G$ contained in $\mathrm{nio}(G)$. By relating the already well-known theory of isolators with Lazard's notion of $p$-saturations, we introduce the isolated lower central (resp. isolated derived) series of a nilpotent (resp. soluble) $p$-valuable group of finite rank, and use this to study the conjugation action of $\mathrm{nio}(G)$ on $\mathbf{FN}_p(G)$. We emerge with a structure theorem for $G$, $$1 \leq Δ^+ \leq Δ\leq \mathbf{FN}_p(G) \leq \mathrm{nio}(G) \leq G,$$ in which the various quotients of this series of groups are well understood. This sheds light on the ideal structure of the Iwasawa algebras (i.e. the completed group rings $kG$) of such groups, and will be used in future work to study the prime ideals of these rings.

math.GR↗
arXiv2012

An equivariant covering map from the upper half plane to the complex plane minus a lattice

This paper studies a covering map phi from the upper half plane to the complex plane with a triangular lattice excised. This map is interesting as it factorises Klein's J invariant. Its derivative has properties which are a slight generalisation of modular functions, and (phi')^6 is a modular function of weight 12. There is a homomorphism from the modular group Gamma to the affine transformations of the complex plane which preserve the excised lattice. With respect to this action phi is a map of Gamma-sets. Identification of the excised lattice with the root lattice of sl_3(C) allows functions familiar from the study of modular functions to be expressed in terms of standard constructions on representations of sl_3(C).

math.CA↗