Diagonal quartic surfaces and transcendental elements of the Brauer group

ABSTRACTWe exhibit central simple algebras over the function field of a diagonal quartic surface over the complex numbers that represent the 2-torsion part of its Brauer group. We investigate whether the 2-primary part of the Brauer group of a diagonal quartic surface over a number field is algebraic and give sufficient conditions for this to be the case. In the last section we give an obstruction to weak approximation due to a transcendental class on a specific diagonal quartic surface, an obstruction which cannot be explained by the algebraic Brauer group which in this case is just the constant algebras.

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Splitting of Algebras by Function Fields of One Variable

Nagoya Mathematical Journal ◽

10.1017/s0027763000026441 ◽

1966 ◽

Vol 27 (2) ◽

pp. 625-642 ◽

Cited By ~ 14

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.

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On the Brauer Group of Algebras Having a Grading and an Action

Canadian Journal of Mathematics ◽

10.4153/cjm-1976-053-6 ◽

1976 ◽

Vol 28 (3) ◽

pp. 533-552 ◽

Cited By ~ 10

Author(s):

Morris Orzech

Keyword(s):

Abelian Group ◽

Commutative Ring ◽

Bilinear Form ◽

Brauer Group ◽

Central Simple Algebras ◽

Simple Algebras ◽

Central Simple

Beginning with Wall's introduction [19] of Z2-graded central simple algebras over a field K, a number of related generalizations of the Brauer group have been proposed. In [16] the field K was replaced by a commutative ring R, building upon the theory developed in [1]. The concept of a G-graded central simple K-algebra (G an abelian group) was first defined in [12]; this work and that of [16] was subsequently unified in [6] and [7] via the construction and computation of the graded Brauer group Bφ﹛R, G) (φ a bilinear form from G X G to U(R), the units of R).

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Derivations in central separable algebras

Glasgow Mathematical Journal ◽

10.1017/s0017089500003402 ◽

1978 ◽

Vol 19 (1) ◽

pp. 75-77 ◽

Cited By ~ 4

Author(s):

George T. Georgantas

Keyword(s):

Galois Group ◽

Brauer Group ◽

Normal Extension ◽

Noether Theorem ◽

Central Simple Algebras ◽

Group Extensions ◽

Simple Algebras ◽

Central Simple ◽

Inner Automorphisms ◽

Separable Algebras

Given N a finite separable normal extension of a field F, it is well known that the Brauer group Br(N/F) of classes of central simple F-algebras split by N is isomorphic with Ext(N*, G), the classes of group extensions of N* by the Galois group G of N over F. In the construction of this isomorphism, a key role is played by the Skolem-Noether Theorem which extends automorphisms to inner automorphisms in central simple algebras.

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More on the Schur group of a commutative ring

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171285000308 ◽

1985 ◽

Vol 8 (2) ◽

pp. 275-282 ◽

Cited By ~ 2

Author(s):

R. A. Mollin

Keyword(s):

Finite Group ◽

Commutative Ring ◽

Homomorphic Image ◽

Group Ring ◽

Local Field ◽

Natural Generalization ◽

Brauer Group ◽

Central Simple Algebras ◽

Simple Algebras ◽

Central Simple

The Schur group of a commutative ring,R, with identity consists of all classes in the Brauer group ofRwhich contain a homomorphic image of a group ringRGfor some finite groupG. It is the purpose of this article to continue an investigation of this group which was introduced in earlier work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well-known Schur index in the global or local field case.

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More on the Schur group of a commutative ring

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171285000552 ◽

1985 ◽

Vol 8 (3) ◽

pp. 513-520

Author(s):

R. A. Mollin

Keyword(s):

Finite Group ◽

Commutative Ring ◽

Homomorphic Image ◽

Group Ring ◽

Local Field ◽

Natural Generalization ◽

Brauer Group ◽

Central Simple Algebras ◽

Simple Algebras ◽

Central Simple

The Schur group of a commutative ring,R, with identity consists of all classes in the Brauer group ofRwhich contain a homomorphic image of a group ringRGfor some finite groupG. It is the purpose of this article to continue an investigation of this group which was introduced in earler work as a natural generalization of the Schur group of a field. We generalize certain facts pertaining to the latter, among which are results on extensions of automorphisms and decomposition of central simple algebras into a product of cyclics. Finally we introduce the Schur exponent of a ring which equals the well-known Schur index in the global or local field case.

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Central simple algebras and the Brauer group

Graduate Studies in Mathematics - Graduate Algebra: Noncommutative View ◽

10.1090/gsm/091/17 ◽

2008 ◽

pp. 447-484

Author(s):

Louis Rowen

Keyword(s):

Brauer Group ◽

Central Simple Algebras ◽

Simple Algebras ◽

Central Simple

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Patching over Berkovich curves and quadratic forms

Compositio Mathematica ◽

10.1112/s0010437x19007632 ◽

2019 ◽

Vol 155 (12) ◽

pp. 2399-2438

Author(s):

Vlerë Mehmeti

Keyword(s):

Quadratic Forms ◽

Sufficient Conditions ◽

Analytic Geometry ◽

Central Simple Algebras ◽

Local Isotropy ◽

Simple Algebras ◽

Central Simple ◽

Global Principle ◽

Analytic Curves ◽

Invent Math

We extend field patching to the setting of Berkovich analytic geometry and use it to prove a local–global principle over function fields of analytic curves with respect to completions. In the context of quadratic forms, we combine it with sufficient conditions for local isotropy over a Berkovich curve to obtain applications on the $u$-invariant. The patching method we adapt was introduced by Harbater and Hartmann [Patching over fields, Israel J. Math. 176 (2010), 61–107] and further developed by these two authors and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263]. The results presented in this paper generalize those of Harbater, Hartmann, and Krashen [Applications of patching to quadratic forms and central simple algebras, Invent. Math. 178 (2009), 231–263] on the local–global principle and quadratic forms.

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Invariants of degree 3 and torsion in the Chow group of a versal flag

Compositio Mathematica ◽

10.1112/s0010437x14008057 ◽

2015 ◽

Vol 151 (8) ◽

pp. 1416-1432 ◽

Cited By ~ 1

Author(s):

Alexander Merkurjev ◽

Alexander Neshitov ◽

Kirill Zainoulline

Keyword(s):

Algebraic Group ◽

Chow Group ◽

Linear Algebraic Group ◽

Factor Group ◽

Central Simple Algebras ◽

Cohomological Invariants ◽

Simple Algebras ◽

Central Simple ◽

Torsion Part

We prove that the group of normalized cohomological invariants of degree $3$ modulo the subgroup of semidecomposable invariants of a semisimple split linear algebraic group $G$ is isomorphic to the torsion part of the Chow group of codimension-$2$ cycles of the respective versal $G$-flag. In particular, if $G$ is simple, we show that this factor group is isomorphic to the group of indecomposable invariants of $G$. As an application, we construct nontrivial cohomological invariants for indecomposable central simple algebras.

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