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Die Brauersche Gruppe; ihre Verallgemeinerungen und Anwendungen in der arithmetischen Geometrie
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This is an unchanged version of an unpublished, ``state of the art'' survey given at a conference held in Stuttgart in 2001 to celebrate the 100th birthday of Richard Brauer. This text was recently quoted in several papers on the period-index problem for central simple algebras.
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Cited by 3 Pith papers
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The period-index conjecture is false
The period-index conjecture is disproved: for every d≥3 there is a variety with a 2-torsion Brauer class of index 2^{d-1}, exceeding the conjectural bound.
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Specialization and rigidity
A general specialization theorem shows that rigid properties of fibers in algebraic families define countable unions of closed loci, unifying and extending results in dynamics, rationality, and central simple algebras.
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The period-index problem for hyper-K\"ahler varieties via hyperholomorphic bundles
For K3^{[n]}-type hyper-Kähler varieties, the index of a Brauer class divides a power of its period, with exponent equal to the dimension in general and half the dimension for most classes in Picard rank at least two.
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