Vol 222, No 4 (2017)

The zeroth stable \( \mathbb{A} \)¹-homotopy group of a smooth projective variety is computed. This group is identified with the group of oriented 0-cycles on the variety. The proof heavily exploits properties of strictly homotopy invariant sheaves. Bibliography: 7 titles.

An example of embedding problem with kernel of odd order is constructed so that the compatibility condition cannot be reduced to accompanying Abelian problem. Some remarks concerning examples for groups of smaller order are given. Bibliography: 7 titles

The paper presents an explicit formula for the Hilbert pairing between the Milnor K-group of a higher-dimensional local field and a polynomial formal module. This formula generalizes similar results for the one-dimensional case and the higher-dimensional case of multiplicative group. The case where the field and its first residue field have different characteristics, is considered. Bibliography: 13 titles.

The authors investigate cyclic Galois extensions for quintic equations and construct the resolvent for real fields and fields containing the square root of −1. Also they prove a theorem which characterizes all the Galois extensions for quintics. Bibliography: 9 titles.

In this paper, a structure theorem for the symmetries of a graded flat cosymbol algebra of differential operators is proved. Together with a lemma on equivariant polynomials also proved in the paper, this theorem gives an upper bound on the dimension of the graded Lie algebra associated with the symmetries of geodesic flow on a smooth variety. Bibliography: 14 titles

Let R be a semilocal 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 \( {\mathbb{G}}_{m,R} \). It is proved that the kernel of the map

\( {H}_{\overset{\prime }{e}t}^1\left(R,\kern0.5em G\right)\to {H}_{\overset{\prime }{e}t}^1\left(K,\kern0.5em G\right) \)

In his seminal paper, half a century ago, Hyman Bass established commutator formulas for a (stable) general linear group, which were the key step in defining the group K₁. Namely, he proved that for an associative ring A with identity,

\( E(A)=\left[E(A),E(A)\right]=\left[\mathrm{GL}(A),\mathrm{GL}(A)\right], \)

where GL(A) is the stable general linear group and E(A) is its elementary subgroup. Since then, various commutator formulas have been studied in stable and non-stable settings for classical groups, algebraic groups, and their analogs, and mostly in relation to subnormal subgroups of these groups. The basic classical theorems and methods developed for their proofs are associated with the names of the heroes of classical algebraic K-theory: Bak, Quillen, Milnor, Suslin, Swan, Vaserstein, and others.

One of the dominant techniques in establishing commutator type results is localization. In the present paper, some recent applications of localization methods to the study (higher/relative) commutators in the groups of points of algebraic and algebraic-like groups, such as general linear groups GL(n,A), unitary groups GU(2n,A, Λ), and Chevalley groups G(Φ,A), are described. Some auxiliary results and corollaries of the main results are also stated.

The paper provides a general overview of the subject and covers the current activities. It contains complete proofs borrowed from our previous papers and expositions of several main results to give the reader a self-contained source.