scholarly journals Crossed Products and Maximal Orders

1965 ◽  
Vol 25 ◽  
pp. 165-174 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Sufficient Conditions ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Crossed Product ◽  
Quotient Field ◽  
Maximal Orders ◽  
Rank One ◽  
Central Simple ◽  
Necessary And Sufficient ◽  
Algebra Over A Field

Let I’ be a maximal order over a complete discrete rank one valuation ring R in a central simple algebra over the quotient field of R. The purpose of this paper is to determine necessary and sufficient conditions for I’ to be equivalent to a crossed product over a tamely ramified extension of R.It is a classical result that every central simple algebra over a field k is equivalent to a crossed product over a Galois extension of k. Furthermore, it has been proved by Auslander and Goldman in [2] that every central separable algebra over a local ring is equivalent to a crossed product over an unramified extension.

scholarly journals SOLVABLE CROSSED PRODUCT ALGEBRAS REVISITED

2019 ◽  
Vol 62 (S1) ◽  
pp. S165-S185 ◽  
Author(s):  
CHRISTIAN BROWN ◽  
SUSANNE PUMPLÜN
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Crossed Product ◽  
Finite Chain ◽  
Central Simple ◽  
Crossed Product Algebra ◽  
Product Algebra ◽  
Maximal Subfield ◽  
Algebra Over A Field ◽  
Crossed Product Algebras

AbstractFor any central simple algebra over a field F which contains a maximal subfield M with non-trivial automorphism group G = AutF(M), G is solvable if and only if the algebra contains a finite chain of subalgebras which are generalized cyclic algebras over their centers (field extensions of F) satisfying certain conditions. These subalgebras are related to a normal subseries of G. A crossed product algebra F is hence solvable if and only if it can be constructed out of such a finite chain of subalgebras. This result was stated for division crossed product algebras by Petit and overlaps with a similar result by Albert which, however, was not explicitly stated in these terms. In particular, every solvable crossed product division algebra is a generalized cyclic algebra over F.

scholarly journals Local selectivity of orders in central simple algebras

2017 ◽  
Vol 13 (04) ◽  
pp. 853-884 ◽  
Author(s):  
Benjamin Linowitz ◽  
Thomas R. Shemanske
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Ring Of Integers ◽  
Isomorphism Classes ◽  
Maximal Orders ◽  
Local Variant ◽  
Simple Algebras ◽  
Central Simple ◽  
Global Principle

Let [Formula: see text] be a central simple algebra of degree [Formula: see text] over a number field [Formula: see text], and [Formula: see text] be a strictly maximal subfield. We say that the ring of integers [Formula: see text] is selective if there exists an isomorphism class of maximal orders in [Formula: see text] no element of which contains [Formula: see text]. In the present work, we consider a local variant of the selectivity problem and applications. We first prove a theorem characterizing which maximal orders in a local central simple algebra contain the global ring of integers [Formula: see text] by leveraging the theory of affine buildings for [Formula: see text] where [Formula: see text] is a local central division algebra. Then as an application, we use the local result and a local–global principle to show how to compute a set of representatives of the isomorphism classes of maximal orders in [Formula: see text], and distinguish those which are guaranteed to contain [Formula: see text]. Having such a set of representatives allows both algebraic and geometric applications. As an algebraic application, we recover a global selectivity result, and give examples which clarify the interesting role of partial ramification in the algebra.

On Projective Modules and Automorphisms of Central Separable Algebras

1969 ◽  
Vol 21 ◽  
pp. 44-53 ◽  
Author(s):  
L. N. Childs
Keyword(s):  
Division Algebra ◽  
Quaternion Algebra ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Algebraic Number Field ◽  
Projective Modules ◽  
Rank One ◽  
Central Simple ◽  
Separable Algebras ◽  
Reduced Norm

This paper developed from, and complements, the paper by F. R. DeMeyer (see 6).In the first section of this paper we note a correspondence between projective modules of a central separable R-algebra A and the two-sided ideals of central separable algebras in the same class as A in the Brauer group of R. When R has the property that rank one projective A-modules are free, this correspondence yields a bijection between isomorphism types of indecomposable projective A-modules and the isomorphism types of algebras in the Brauer class of A which are the analogue of division algebra components in the field case. This bijection was remarked on without proof by DeMeyer in (6).Pursuing the ideas behind this correspondence, we consider the situation for a separable order A in a central simple algebra A over an algebraic number field, and obtain, by means of results involving the reduced norm, a generalization of DeMeyer's remark except when the division algebra component of A is a totally definite quaternion algebra (Theorem 3.3).

scholarly journals Algebras of Finite Cohomological Dimension

1971 ◽  
Vol 43 ◽  
pp. 127-135
Author(s):  
Joseph A. Wehlen
Keyword(s):  
Commutative Ring ◽  
Matrix Algebra ◽  
Triangular Matrix ◽  
Cohomological Dimension ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Hereditary Algebra ◽  
Triangular Matrix Algebra ◽  
Central Simple ◽  
Algebra Over A Field

The cohomology theory of an associative algebra has been shown to be valuable in the study of the structure of algebras of finite cohomological dimension, especially those of dimension less than or equal to one over a field. M. Harada [9] has shown that every semi-primary hereditary algebra A (for example, A is finitely generated over a field R and has dimension < 1) is isomorphic to a generalized triangular matrix algebra. The concept of a central separable algebra over a commutative ring has been shown to be a useful generalization of the concept of a central simple algebra over a field, where a separable algebra is defined to be an algebra having cohomological dimension zero.

scholarly journals Motivic decomposition of compactifications of certain group varieties

2018 ◽  
Vol 2018 (745) ◽  
pp. 41-58
Author(s):  
Nikita A. Karpenko ◽  
Alexander S. Merkurjev
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Chow Ring ◽  
Prime Degree ◽  
Central Simple

Abstract Let D be a central simple algebra of prime degree over a field and let E be an {\operatorname{\mathbf{SL}}_{1}(D)} -torsor. We determine the complete motivic decomposition of certain compactifications of E. We also compute the Chow ring of E.

Isotropy of orthogonal involutions in characteristic two

2018 ◽  
Vol 17 (12) ◽  
pp. 1850240 ◽  
Author(s):  
A.-H. Nokhodkar
Keyword(s):  
Quadratic Form ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Characteristic Two ◽  
Central Simple ◽  
The Given

A totally singular quadratic form is associated to any central simple algebra with orthogonal involution in characteristic two. It is shown that the given involution is isotropic if and only if its corresponding quadratic form is isotropic.

scholarly journals Splitting of Algebras by Function Fields of One Variable

1966 ◽  
Vol 27 (2) ◽  
pp. 625-642 ◽  
Author(s):  
Peter Roquette
Keyword(s):  
Tensor Product ◽  
Function Fields ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Field Extension ◽  
Brauer Group ◽  
Central Simple Algebras ◽  
Natural Mapping ◽  
Simple Algebras ◽  
Central Simple

Let K be a field and (K) the Brauer group of K. It consists of the similarity classes of finite central simple algebras over K. For any field extension F/K there is a natural mapping (K) → (F) which is obtained by assigning to each central simple algebra A/K the tensor product which is a central simple algebra over F. The kernel of this map is the relative Brauer group (F/K), consisting of those A ∈(K) which are split by F.

Almost valuation property in bi-amalgamations and pairs of rings

2019 ◽  
Vol 18 (06) ◽  
pp. 1950104 ◽  
Author(s):  
Najib Ouled Azaiez ◽  
Moutu Abdou Salam Moutui
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quotient Field ◽  
Integral Domains ◽  
Valuation Rings ◽  
Ring Extensions ◽  
Necessary And Sufficient

This paper examines the transfer of the almost valuation property to various constructions of ring extensions such as bi-amalgamations and pairs of rings. Namely, Sec. 2 studies the transfer of this property to bi-amalgamation rings. Our results cover previous known results on duplications and amalgamations, and provide the construction of various new and original examples satisfying this property. Section 3 investigates pairs of integral domains where all intermediate rings are almost valuation rings. As a consequence of our results, we provide necessary and sufficient conditions for a pair (R, T), where R arises from a (T, M, D) construction, to be an almost valuation pair. Furthermore, we study the class of maximal non-almost valuation subrings of their quotient field.

Cleft comodules over Hopf quasigroups

2015 ◽  
Vol 17 (06) ◽  
pp. 1550007 ◽  
Author(s):  
J. N. Alonso Álvarez ◽  
J. M. Fernández Vilaboa ◽  
R. González Rodríguez ◽  
C. Soneira Calvo
Keyword(s):  
Hopf Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Crossed Product ◽  
Comodule Algebra ◽  
Hopf Quasigroup ◽  
Necessary And Sufficient

In this paper, we provide necessary and sufficient conditions for a cleft right H-comodule algebra (A, ϱA) over a Hopf quasigroup H to be isomorphic as an algebra to the crossed product AH♯σAHH, where AH is the coinvariants subalgebra of A and σAH is a morphism between H ⊗ H and AH. As a consequence, we obtain the corresponding version in the nonassociative setting of the result given by Blattner, Cohen and Montgomery for projections of Hopf algebras with coalgebra splitting. Concrete examples satisfying the obtained conditions are provided.

Sign in / Sign up

Export Citation Format

Share Document