Normal structure of the unitary group over a local ring

A new approach is used to describe the normal subgroups of unitary groups over the ring of integers of an unramified quadratic extension of a local field of characteristic 2. This method is based on the author's results concerning subgroups of unitary groups normalized by diagonal unitary matrices. By means of this method it has been possible to remove restrictions imposed by other authors in a number of papers pertaining to the same question.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 150–163, 1982.The results in this paper were obtained by the author while he was a post-graduate student under the direction of Z. I. Borevich. The author would like to express his sincere gratitude to Professor Borevich.     

The index of a net subgroup of a symplectic group over a Dedekind ring

An explicit formula is derived for the index of a net congruence subgroup of a symplectic group over a Dedekind ring. A classification of symplectic D-nets over a field is obtained along the way.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 75, pp. 74–86, 1978.     

The orthogonal group over a local ring is 4-reflectional

LetV be a free finite-dimensional module over a commutative local ringR where 2 is a unit. Letf:V × V R be a regular symmetric bilinear form. We prove that every element of the orthogonal group O(V) is a product of four involutions in O(V).     

The Brauer group of a curve over a strictly local discrete valuation ring

LetK be the field of fractions of a curve overR whereR is the henselization of a regular local ring on an algebraic curve over a field which is algebraically closed and has characteristic 0. ThenK has the exponent=degree property for division algebras. In fact every central finite dimensionalK-division algebra with exponentn is a cyclic algebra of degreen. In memory of Professor S. A. Amitsur     

Automorphisms of a chain ring

Let A be a chain ring that is a faithful algebra over a commutative chain ring R, such that is a separable, normal, algebraic field extension of and is countably generated over . It has been recently proved by Alkhamees and Singh that A has a coefficient ring R ₀, and there exists a pair (θ, σ) with θ ∈ A, σ an R-automorphism of R ₀ such that J (A) = θ A = Aθ, and θa = σ (a) θ, a ∈ R ₀. The question of the extension of certain R-automorphisms of R ₀ to R-automorphisms of A is investigated. Mathematics Subject Classification (2000) Primary 16H05, 16W20, secondary 13F20, 13J15     

Isomorphisms of the unitary group over a ring

We describe isomorphisms of unitary groups over associative rings with 1/2 that contain a compact system of idempotents. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 55–70, 2006.     

Factorization of transformations over a local ring

Let V be a finitely generated free module over a local ring R, and π an invertible linear transformation for V. Then π is a product of simple mappings. In addition, the determinant of each simple factor but one can be chosen to be any given unit in R. The smallest number of factors needed is not greater than r+1, where r is the codimension of the vector space that is associated with the module of vectors fixed under π.     

On the flatness of local models for the symplectic group

We investigate the bad reduction of certain Shimura varieties (associated to the symplectic group). More precisely, we look at a model of the Shimura variety at a prime p, with parahoric level structure at p. We show that this model is flat, as conjectured by Rapoport and Zink (Ann. of Math. Stud. 141 (1996)), and that its special fibre is reduced.A crucial ingredient is Faltings’ theorem on the normality of Schubert varieties in the affine flag variety.     

Net determinant over a Bezoutian local ring

Let be any D-net of ideals of order n over a commutative local Bezoutian ring R and denote by G() the corresponding net subgroup in the general linear group of degree n over R (RZhMat, 1977, 2A280). We give an explicit computation of the factor group G()/E(), where E() is the subgroup generated by all elementary transvections in G().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 5–13, 1982.