Related papers: The hyperring of ad\`ele classes

In this paper, we study multiplicative structures on the K-theory of the core $A:=C^*(E)^{U(1)}$ of the C*-algebra $C^*(E)$ of a directed graph $E$. In the first part of the paper, we study embeddings $E\to E\times E$ that induce a…

We define abelian extensions of algebras in congruence-modular varieties. The theory is sufficiently general that it includes, in a natural way, extensions of R-modules for a ring R. We also define a cohomology theory, which we call clone…

We consider a generalization $K_0^{\operatorname{gr}}(R)$ of the standard Grothendieck group $K_0(R)$ of a graded ring $R$ with involution. If $\Gamma$ is an abelian group, we show that $K_0^{\operatorname{gr}}$ completely classifies graded…

Let $E$ be a field satisfying the following conditions: (i) the $p$-component of the Brauer group Br$(E)$ is nontrivial whenever $p$ is a prime number for which $E$ is properly included in its maximal $p$-extension; (ii) the relative Brauer…

We prove the conjectures of Hodge and Tate for any four-dimensional hyper-K\"ahler variety of generalized Kummer type. For an arbitrary variety $X$ of generalized Kummer type, we show that all Hodge classes in the subalgebra of the rational…

Hyperfields and systems are two algebraic frameworks which have been developed to provide a unified approach to classical and tropical structures. All hyperfields, and more generally hyperrings, can be represented by systems. Conversely, we…

In this note we characterize the affine semigroup rings K[S] over an arbitrary field K that satisfy condition R_l of Serre. Our characterization is in terms of the face lattice of the positive cone pos(S) of S. We start by reviewing some…

Let $L$ be the function field of a projective space ${\mathbb P}^n_k$ over an algebraically closed field $k$ of characteristic zero, and $H$ be the group of projective transformations. An $H$-sheaf ${\mathcal V}$ on ${\mathbb P}^n_k$ is a…

Let G be a connected real reductive algebraic group, and let K be a maximal compact subgroup of G. We prove that the conjugation orbit space Hom(Z^{2d},K)/K is a strong deformation retract of the space Hom(Z^{2d},G)/G of equivalence classes…

We describe relations between maximal subfields in a division ring and in its rational extensions. More precisely, we prove that properties such as being Galois or purely inseparable over the centre generically carry over from one to…

Let $E$ be an algebraic extension of a global field $E_{0}$ with a nontrivial Brauer group Br$(E)$, and let $P(E)$ be the set of those prime numbers $p$, for which $E$ does not equal its maximal $p$-extension $E(p)$. This paper shows that…

Motivated by the polynomial representation theory of the general linear group and the theory of symplectic singularities, we study a category of perverse sheaves with coefficients in a field $k$ on any affine unimodular hypertoric variety.…

The multiplicative group of a number field acts by multiplication on the full adele ring of the field. Generalising a theorem of Laca and Raeburn, we explicitly describe the primitive ideal space of the crossed product C*-algebra associated…

In this note we give a new existence proof for the universal extension classes for $GL_2$ previously constructed by Friedlander and Suslin via the theory of strict polynomial functors. The key tool in our approach is a calculation of Parker…

For any finite abelian group $G$ and commutative unitary ring $R$, by $R[G]$ we denote the group algebra over $R$. Let $T=(g_1,\ldots,g_{\ell})$ be a sequence over the group $G$. We say $T$ is algebraically zero-sum free over R if…

We show that genus-one fibrations lacking a global section fit naturally into the geometric moduli space of Weierstrass models. Elliptic fibrations with multiple sections (nontrivial Mordell-Weil rank), which give rise in F-theory to…

Our goal is to give a purely algebraic characterization of finite abelian Galois covers of a complete, irreducible, non-singular curve $X$ over an algebraically closed field $\k$. To achieve this, we make use of the Galois theory 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…

Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated…

By a [$K$-]approximate subring of a ring we mean an additively symmetric subset $X$ such that $X \cdot X \cup (X + X)$ is covered by finitely many [resp.\ $K$] additive translates of $X$. We prove a structure theorem for finite approximate…