arXiv subjects
Publications and source records attributed to Konstantin Ardakov.
Let $Φ$ be a root system and let $Φ(\Zp)$ be the standard Chevalley $\Zp$-Lie algebra associated to $Φ$. For any integer $t\geq 1$, let $G$ be the uniform pro-$p$ group corresponding to the powerful Lie algebra $p^t Φ(\Zp)$ and suppose that $p\geq 5$. Then the Iwasawa algebra $Ω_G$ has no nontrivial reflexive two-sided ideals. This was previously proved by the authors for the root system $A_1$.
Let H be a torsionfree compact p-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra Λ_H of H has trivial centre and use this result to classify the prime c-ideals in the Iwasawa algebra Λ_G of G := H \times \Zp. We also show that a finitely generated torsion Λ_G-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.
Let $G$ be a torsionfree compact $p$-adic analytic group. We give sufficient conditions on $p$ and $G$ which ensure that the Iwasawa algebra $Ω_G$ of $G$ has no non-trivial two-sided reflexive ideals. Consequently, these conditions imply that every nonzero normal element in $Ω_G$ is a unit. We show that these conditions hold in the case when $G$ is an open subgroup of $\SL_2(\Zp)$ and $p$ is arbitrary. Using a previous result of the first author, we show that there are only two prime ideals in $Ω_G$ when $G$ is a congruence subgroup of $\SL_2(\Zp)$: the zero ideal and the unique maximal ideal. These statements partially answer some questions asked by the first author and Brown.
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.
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.
Necessary and sufficient conditions are given for the completed group algebras of a compact p-adic analytic group with coefficient ring the p-adic integers or the field of p elements to be prime, semiprime and a domain. Necessary and sufficient conditions for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups are found. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coates et al.