Open problems on central simple algebras
TLDR
A survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field intended both for experts and for beginners can be found in this paper.Abstract:
We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.read more
Citations
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Period-index and u -invariant questions for function fields over complete discretely valued fields
R. Parimala,V. Suresh +1 more
TL;DR: In this article, the Brauer p-dimension of F in terms of the p-rank of κ was shown to be at most d+2, where l is a prime not equal to p.
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Symbol length in the Brauer group of a field
TL;DR: In this paper, the symbol length of elements in the Brauer group of a field K containing a Cm field was shown to be bounded by (p) − 1, where p is the exponent p.
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On the local behaviour of specializations of function field extensions
TL;DR: In this paper, the authors investigated the local behavior at primes of $k$ of finite Galois extensions of primes arising as specializations of finite GAs, and provided a general result about decomposition groups at specializations, extending the result of Beckmann concerning inertia groups.
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$E_8$, the most exceptional group
TL;DR: The five exceptional simple Lie algebras over the complex number are included one within the other as $G_2, F_4, E_6 \subset E_7 and E_8 as mentioned in this paper.
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Refinements to patching and applications to field invariants
Abstract: We introduce a notion of refinements in the context of patching, in order to obtain new results about local-global principles and field invariants in the context of quadratic forms and central simple algebras. The fields we consider are finite extensions of the fraction fields of two-dimensional complete domains that need not be local. Our results in particular give the u-invariant and period-index bound for these fields, as consequences of more general abstract results.
References
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Book
The Book of Involutions
TL;DR: A comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups is given in this article, which provides the algebra-theoretic foundations for much of the recent work on linear algebraIC groups over arbitrary fields.
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Endomorphisms of Abelian Varieties over Finite Fields.
TL;DR: In this paper, it was shown that HOmk(A', A") is a free module of rank 2g over the ring Z l of l-adic integers, and the canonical map is Z-free.
Book
Structure of Algebras
TL;DR: The structure theorems of Wedderburn Simple algebras Crossed products and exponents Cyclic semi-fields Cyclic algeses and $p$ algebases Representations and Riemann matrices Rational division algesbras Involutions of algesbases Special results as discussed by the authors
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Complexe de de Rham-Witt et cohomologie cristalline
TL;DR: Gauthier-Villars as mentioned in this paper implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php).