Related papers: Representations of finite groups on modules over K…

Let $G$ be a finite group and let $k$ be a sufficiently large finite field. Let $R(G)$ denote the character ring of $G$ (i.e. the Grothendieck ring of the category of ${\mathbb{C}}G$-modules). We study the structure and the representations…

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…

Our aim is to transfer several foundational results from the modular representation theory of finite groups to the wider context of profinite groups. We are thus interested in profinite modules over the completed group algebra k[[G]] of a…

Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy groups. We show…

We prove an induction theorem for the higher algebraic K-groups of group algebras $kG$ of finite groups $G$ over characteristic $p$ finite fields $k$. For a certain class of finite groups, which we call $p$-isolated, this reduces…

In this paper, we study the K-theory on higher modules in spectral algebraic geometry. We relate the K-theory of an $\infty$-category of finitely generated projective modules on certain $\mathbb{E}_{\infty}$-rings with the K-theory of an…

In this paper, we examine the `derived completion' of the representation ring of a pro-p group G_p^ with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over…

Let K be a complete discretely valued field of mixed characteristics (0, p) with perfect residue field. One of the central objects of study in p-adic Hodge theory is the category of continuous representations of the absolute Galois group of…

We review some recent results on $K$-theory of perfection of commutative $\mF_p$-algebras and provide an alternative proof.

Let $\mathcal{G}$ be a finite group scheme over an algebraically closed field $k$ of characteristic ${\rm char}(k)=p\geq 3$. In generalization of the familiar notion from the modular representation theory of finite groups, we define the…

We study the action of the Galois group $G$ of a finite extension $K/k$ of number fields on the points on an elliptic curve $E$. For an odd prime $p$, we aim to determine the structure of the $p$-adic completion of the Mordell-Weil group…

Let p be prime, k a finite field of characteristic p, and G a virtually pro-p group. We prove an analogue of the Green Correspondence for finitely generated modules over the completed group algebra k[[G]].

We find the model completion of the theory modules over $A$, where $A$ is a finitely generated commutative algebra over a field $K$. This is done in a context where the field $K$ and the module are represented by sorts in the theory, so…

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely…

The conjugation representation of a finite group $G$ is the complex permutation module defined by the action of $G$ on itself by conjugation. Addressing a problem raised by Hain motivated by the study of a Hecke action on iterated Shimura…

A group action on the input ring or category induces an action on the algebraic $K$-theory spectrum. However, a shortcoming of this naive approach to equivariant algebraic $K$-theory is, for example, that the map of spectra with $G$-action…

Let a compact group G act on real or complex C*-algebras A and B, with A separable and B sigma-unital. We express the G-equivariant Kasparov groups KK_n(A,B) by algebraic K-groups of a certain additive category.

We give an explicit algebraic description, based on prismatic cohomology, of the algebraic K-groups of rings of the form $O_K/I$ where $K$ is a p-adic field and $I$ is a non-trivial ideal in the ring of integers $O_K$; this class includes…

This paper investigates the structure of $K_2$-groups for certain quotient rings of the integral group ring $\mathbb{Z}[G]$. Let $G$ be a finite abelian $p$-group with $p$-rank $r$, let $\Gamma$ be the maximal $\mathbb{Z}$-order of…

Let $G$ be a direct product of inner forms of general linear groups over non-archimedean locally compact fields of residue characteristic $p$ and let $K^1$ be the pro-$p$-radical of a maximal compact open subgroup of $G$. In this paper we…