K_0 and the dimension filtration for p-torsion Iwasawa modules

Let G be a compact p-adic analytic group. We study K-theoretic questions related to the representation theory of the completed group algebra kG of G with coefficients in a finite field k of characteristic p. We show that if M is a finitely generated kG-module whose dimension is smaller than the dimension of the centralizer of any p-regular element of G, then the Euler characteristic of M is trivial. Writing F_i for the abelian category consisting of all finitely generated kG-modules of dimension at most i, we provide an upper bound for the rank of the natural map from the Grothendieck group of F_i to that of F_d, where d denotes the dimension of G. We show that this upper bound is attained in some special cases, but is not attained in general

Verma modules for Iwasawa algebras are faithful

We establish the faithfulness of Verma modules for rational Iwasawa algebras of split semisimple compact $L$-analytic groups. We also prove the algebraic independence of Arens-Michael envelopes over Iwasawa algebras and compute the centre of affinoid enveloping algebras of semisimple $p$-adic Lie algebras.Comment: 19 page

On the Cartan map for crossed products and Hopf-Galois extensions

We study certain aspects of the algebraic K-theory of Hopf-Galois extensions. We show that the Cartan map from K-theory to G-theory of such an extension is a rational isomorphism, provided the ring of coinvariants is regular, the Hopf algebra is finite dimensional and its Cartan map is injective in degree zero. This covers the case of a crossed product of a regular ring with a finite group and has an application to the study of Iwasawa modules

D-modules on rigid analytic spaces I

We introduce a sheaf of infinite order differential operators D-cap on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle. We show that the sections of this sheaf over sufficiently small affinoid varieties are Fr\'echet-Stein algebras, and use this to define co-admissible sheaves of D-cap-modules. We prove analogues of Cartan's Theorems A and B for co-admissible D-cap-modules.Comment: 48 pages. Comments welcom

Gamma-invariant ideals in Iwasawa algebras

Let kG be the completed group algebra of a uniform pro-p group G with coefficients in a field k of characteristic p. We study right ideals I in kG that are invariant under the action of another uniform pro-p group Gamma. We prove that if I is non-zero then an irreducible component of the characteristic support of kG/I must be contained in a certain finite union of rational linear subspaces of Spec gr kG. The minimal codimension of these subspaces gives a lower bound on the homological height of I in terms of the action of a certain Lie algebra on G/G^p. If we take Gamma to be G acting on itself by conjugation, then Gamma-invariant right ideals of kG are precisely the two-sided ideals of kG, and we obtain a non-trivial lower bound on the homological height of a possible non-zero two-sided ideal. For example, when G is open in SL_n(\Zp) this lower bound equals 2n - 2. This gives a significant improvement of the results of Ardakov, Wei and Zhang on reflexive ideals in Iwasawa algebras

Bars and spirals in tidal interactions with an ensemble of galaxy mass models

We present simulations of the gaseous and stellar material in several different galaxy mass models under the influence of different tidal fly-bys to assess the changes in their bar and spiral morphology. Five different mass models are chosen to represent the variety of rotation curves seen in nature. We find a multitude of different spiral and bar structures can be created, with their properties dependent on the strength of the interaction. We calculate pattern speeds, spiral wind-up rates, bar lengths, and angular momentum exchange to quantify the changes in disc morphology in each scenario. The wind-up rates of the tidal spirals follow the 2:1 resonance very closely for the flat and dark matter dominated rotation curves, whereas the more baryon dominated curves tend to wind-up faster, influenced by their inner bars. Clear spurs are seen in most of the tidal spirals, most noticeable in the flat rotation curve models. Bars formed both in isolation and interactions agree well with those seen in real galaxies, with a mixture of "fast" and "slow" rotators. We find no strong correlation between bar length or pattern speed and the interaction strength. Bar formation is, however, accelerated/induced in four out of five of our models. We close by briefly comparing the morphology of our models to real galaxies, easily finding analogues for nearly all simulations presenter here, showing passages of small companions can easily reproduce an ensemble of observed morphologies.Comment: 30 pages, 29 colour figures, accepted for publication in MNRAS. Videos of simulations can be found at http://www.youtube.com/playlist?list=PLQKy--XcWrIVBc1sS2RNc-ekyfeBsGtD