卷 222, 编号 4 (2017)
- 年: 2017
- 文章: 14
- URL: https://journals.rcsi.science/1072-3374/issue/view/14819
Article
On the Zeroth Stable \( \mathbb{A} \)1-Homotopy Group of a Smooth Projective Variety
摘要
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.
Cubic Forms on Adjoint Representations of Exceptional Groups
摘要
On the adjoint representation of the Chevalley group of type E7, cubic forms are constructed so that their partial derivatives are linear combinations of equations on the orbit of the highest weight vector. These forms are described in terms of new combinatorial notions related to maximal squares in root systems of exceptional types. Bibliography 17 titles.
Compatibility Condition. A Possibility of Reduction to Commutative Situation
摘要
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
Normality of the Elementary Subgroup in Sp(2, A)
摘要
Let A be an associative ring with identity and involution, and let e1, . . . , en be a full system of Hermitian idempotents in A such that every ei generates A as a two-sided ideal. It is proved that the elementary subgroup is normal in Sp(2, A) if n ≥ 3 and A satisfies an analog of the local stable rank condition. Bibliography: 16 titles.
Explicit Form of the Hilbert Symbol on Polynomial Formal Module for Multidimensional Local Field. II
摘要
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.
Hochschild Cohomology for Algebras of Semidihedral Type. VI. The Family \( SD{\left(2\mathrm{\mathcal{B}}\right)}_2 \) in Characteristic Different from 2
摘要
The Hochschild cohomology groups for algebras of semidihedral type that lie in the family \( SD{\left(2\mathrm{\mathcal{B}}\right)}_2 \) (in the famous K.Erdmann’s classification) over an algebraically closed field with characteristic different from two are computed. The calculation, relies upon the minimal projective bimodule resolution for algebras from the above family that was constructed in the previous author’s paper.
Cyclic Galois Extensions for Quintic Equation
摘要
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.
Proof of the Congruence Hypothesis for Generalized Rings
摘要
In 2007, A. L. Smirnov formulated an interesting conjecture on the generalized rings introduced and studied by N. V. Durov. In this paper, the conjecture is proved.
Symmetries of a Flat Cosymbol Algebra of Differential Operators
摘要
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
Reduction Theorems for Triples of Short Root Subgroups in Chevalley Groups
摘要
Reduction theorems for triples of short root unipotent subgroups in Chevalley groups of type Bℓ and Cℓ are proved. The main result asserts that except for one case, any subgroup generated by such a triple is conjugate to a subgroup of G(B4, K)U(B5, K) or G(C4, K)U(C5, K), respectively.
On the Grothendieck–Serre Conjecture Concerning Principal G-Bundles Over Semilocal Dedekind Domains
摘要
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
Chow Ring of Generic Maximal Orthogonal Grassmannians
摘要
The Chow ring of the maximal orthogonal Grassmannian corresponding to a versal torsor is computed. In particular, this shows that the ring has no torsion as an Abelian group. Bibliography: 5 titles.
The Commutators of Classical Groups
摘要
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,
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.
Overgroups of Elementary Block Diagonal Subgroups in Even Unitary Groups over Quasi-Finite Rings: Main Results
摘要
Let H be a subgroup of the hyperbolic unitary group U(2n,R, Λ) that contains an elementary block diagonal subgroup EU(ν, R, Λ) of type ν. Assume that all self-conjugate blocks of EU(ν, R, Λ) are of size at least 6 (at least 4 if the form parameter Λ satisfies the condition RΛ+ΛR = R) and that all non-self-conjugate blocks are of size at least 5. Then there exists a unique major exact form net of ideals (σ, Γ) such that EU(σ, Γ) ≤ H ≤ NU(2n,R,Λ)(U(σ, Γ)), where NU(2n,R,Λ)(U(σ, Γ)) stands for the normalizer in U(2n,R, Λ) of the form net subgroup U(σ, Γ) of level (σ, Γ) and EU(σ, Γ) denotes the corresponding elementary form net subgroup. The normalizer NU(2n,R,Λ)(U(σ, Γ)) is described in terms of congruences.