Related papers: The norm map and the capitulation kernel

Given a generic field extension F/k of degree n>3 (i.e. the Galois group of the normal closure of F is isomorphic to the symmetric group $S_n$), we prove that the norm torus, defined as the kernel of the norm map $N:R_{F/k}(G_m)\to\G_m$, is…

Let k be a field of characteristic 0. Let G be a reductive group over the ring of Laurent polynomials R=k[x_1^{\pm 1},...,x_n^{\pm 1}]. We prove that G has isotropic rank >=1 over R iff it has isotropic rank >=1 over the field of fractions…

For an odd prime $p$ and a number field $F$ containing a $p$th root of unity, we study generalised Tate kernels, $D_F^{[i,n]}$, for $i\in \mathbb{Z}$ and $n\geq 1$, having the properties that if $i\geq 2$ and if either $p$ does not divide…

For $R$ a Euclidean number ring, and let $\Gamma_n(p)$ be the level-$p$ principal congruence subgroup of $\text{SL}_n(R)$. Borel--Serre showed that the cohomology of $\Gamma_n(p)$ vanishes above a degree $\nu$ that is quadratic in $n$. Let…

It is a well-known fact that every group $G$ has a presentation of the form $G = F/R$, where $F$ is a free group and $R$ the kernel of the natural epimorphism from $F$ onto $G$. Driven by the desire to obtain a similar presentation of the…

We ask, following Bartholdi, whether it is true that the kernel of the restriction map from the cohomology of a group G to the cohomology of a finite index subgroup H is finitely generated as an ideal. We show that in case the group has…

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general…

Let U be a smooth quasi-projective variety over a field k that is finite, the algebraic closure of a finite field or algebraically closed of characteristic 0. Let X be a suitable projective compactification of U, and D an effective divisor…

For an algebraically closed field K, let G be a finite abelian group of K-linear automorphisms of a finite-dimensional path algebra KQ of a quiver Q. Under certain assumptions on the action of G, we show the existence of a certain kind of…

In analogy to the classical isomorphism between $\mathcal{L}(\mathcal{S}(\mathbb{R}^{n}) ,\mathcal{S}^{\prime}(\mathbb{R}^{m}) ) $ and $\mathcal{S}^{\prime}(\mathbb{R}^{n+m}) $, we show that a large class of moderate linear mappings acting…

Let K be a global function field of positive characteristic p and let M be a (commutative) finite and flat K-group scheme. We show that the kernel of the canonical localization map H^{1}(K,M)\to\prod_{all v}H^{1}(K_{v},M) in flat (fppf)…

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the…

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any…

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers where this homomorphism is prescribed by the action of a semiregular subgroup of $\textrm{Aut}(G)$. We study…

Let R be a semi-local Dedekind domain and let K be the field of fractions of R. Let G be a reductive semisimple simply connected R-group scheme such that every semisimple normal R-subgroup scheme of G contains a split R-torus G_m. We prove…

Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in…

We establish a purely geometric form of the concentration theorem (also called localization theorem) for actions of a linearly reductive group $G$ on an affine scheme $X$ over an affine base scheme $S$. It asserts the existence of a…

We prove the centrality of the congruence kernel for the elementary subgroup of a Chevalley group G of rank >1 over an arbitrary noetherian ring R (under some minor restrictions on R if G is of type C_n or G_2).

For quasi-projective varieties over a higher local field $k_N$, we prove that its $K$-groups, above a suitable degree, are divisible-by-finite. We also prove the finiteness of the prime-to-$p$ torsion subgroup of certain higher Chow groups…

Let S be a locally noetherian scheme and consider two extensions G_1 and G_2 of abelian S-schemes by S-tori. In this note we prove that the fppf-sheaf Corr _S(G_1,G_2) of divisorial correspondences between G_1 and G_2 is representable.…