Open Access
Access granted
Subscription Access
Vol 232, No 5 (2018)
- Year: 2018
- Articles: 12
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14935
Article
Local-Global Principle for the General Quadratic and General Hermitian Groups and the Nilpotency of KH1
Abstract
In this article, an analog of the Quillen–Suslin’s local-global principle was established for the elementary subgroup of the general quadratic group and the general Hermitian group. It is shown that unstable K1-groups of the general Hermitian groups over module finite rings are nilpotent-by-Abelian. This generalizes earlier results by A. Bak, R. Hazrat, and N. Vavilov.
591-609
Normalizers of Elementary Overgroups of Ep(2, A)
Abstract
Let A be an involution ring, e1 , . . . , en be a full system of Hermitian idempotents in A, let every ei generate A as a two-sided ideal, and 2 ∈ A∗. In this paper, the normalizers of the groups Ep(2,A) · E(2,A, I) are calculated under natural assumptions on A, where Ep(2,A) denotes the elementary symplectic group, E(2,A, I) stands for the elementary subgroup of level I.
610-621
Hochschild Cohomology for Algebras of Semidihedral Type. VII. Algebras with a Small Parameter
Abstract
The Hochschild cohomology groups are computed for algebras of semidihedral type, which are contained in the family SD(2ℬ)2(k, t, c) (from the famous K. Erdmann’s classification) in the case where k = 1. In the calculation, the beforehand construction of the minimal bimodule resolution for algebras from the subfamily under discussion is used.
622-634
On Derived Equivalence of Algebras of Semidihedral Type with Two Simple Modules
Abstract
The Hochschild cohomology groups of degrees not exceeding 3 are computed for algebras of semidihedral type that form the family SD(2B)1 (from the famous K. Erdmann’s classification). In the calculation, the beforehand construction of the initial part of the minimal projective bimodule resolution is used for algebras from the family under discussion. The obtained results imply that the algebras from the families SD(2B)1 and SD(2B)2 with the same parameters in defining relations are not derived equivalent.
635-646
Double Cosets of Stabilizers of Totally Isotropic Subspaces in a Special Unitary Group. I
Abstract
Let D be a division algebra with a fixed involution, and let V be the corresponding unitary space over D with T -condition (see N. Bourbaki, Algèbre). For a pair of totally isotropic subspaces u, v ≤ V, the double cosets PuγPv of their stabilizers Pu, Pv in Γ = SU(V ) are considered. A description of the cosets PuγPv in terms of the intersection distance din(u, γ(v)) and the Witt index of u + γ(v) is given.
647-661
On the Ultrasolvability of p-Extensions of an Abelian Group by a Cyclic Kernel
Abstract
The paper contains a solution of A. V. Yakovlev’s problem in the embedding theory for p-extensions of odd order with a cyclic normal subgroup and an Abelian quotient group: for such nonsplit extensions there exists a realization for the quotient group as a Galois group over number fields such that the corresponding embedding problem is ultrasolvable (i.e., this embedding problem is solvable and has only fields as solutions). A solution for embedding problems of p-extensions of odd order with kernel of order p and with a quotient group that is represented by a direct product of its proper subgroups is also given – this is a generalization for p > 2 of an analogous result for p = 2 due to A. Ledet.
662-676
On the Ultrasolvability of Some Classes of Minimal Nonsplit p-Extensions with Cyclic Kernel for p > 2
Abstract
For any nonsplit p > 2-extension of finite groups with a cyclic kernel and a quotient group with two generators all the accompanying extensions of which split, there exists a realization of the quotient group as a Galois group of number fields such that the corresponding embedding problem is ultrasolvable (i.e., this embedding problem is solvable and has only fields as solutions).
677-692
693-703
Formal Modules for Relative Formal Lubin–Tate Groups
Abstract
Relative formal Lubin–Tate groups are studied, namely, their structure, the ring of endomorphisms, and the group of points. The primary elements are considered, and an explicit formula for the generalized Hilbert symbol is derived.
704-716
On the Normalizer of a Unipotent Root Subgroup in a Chevalley Group
Abstract
In the present paper, the normalizer of a unipotent short and long root subgroup in a Chevalley group over an arbitrary field is calculated in detail. Surely this result is known to specialists. However the author could not find a reference to it.
717-720
Vector Bundles on P1ℤ with Simple Jumps
Abstract
Vector bundles of rank 2 on the projective line over ℤ are considered. It is assumed that such a bundle E is trivial on a generic fiber, and its restriction to any special fiber is isomorphic either to O2 or to O(−1)⊕O(1). Under these assumptions it is proved that an exact sequence of the form 0→O(−2) → E →O(2) → 0 exists.
721-731
Vector Bundles on P1ℤ with Generic Fiber O ⊕ O(1) and Simple Jumps
Abstract
Vector bundles on the projective line over a Dedekind domain A are studied. In the case where A is a principal ideal domain, a complete classification is obtained for rank 2 vector bundles with generic fiber O ⊕ O(1) and special fibers isomorphic either to O ⊕ O(1) or to O(−1) ⊕ O(2).
732-745