Measures of complexity in Brauer groups

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。从广义上讲，主要研究者建议研究支持Brauer群中的除法代数的领域。 作为一个例子，在项目的一部分中，PI提出研究非交叉积除代数的存在性以及p-adic曲线的函数域上的不可分解除代数。 PI提出的一般技术是研究存在于函数域的完备上的除法代数，并使用从完备函数域的Brauer群到函数域的Brauer群的分裂映射来提升除法代数。 在该项目的另一部分，PI建议继续研究定义阿贝尔交叉积的矩阵的退化与阿贝尔交叉积的可分解性之间的联系。 PI将跟进一个观察，她提出了这表明一个尚未完全理解之间的联系退化的矩阵，分解的阿贝尔交叉产品和扭转在第二周组的相关塞维里-布劳尔品种。 最后，PI建议研究领域，其中该领域上的小度除法代数可以通过其分裂领域来区分。特别是，PI将考虑这个问题的功能领域的K3曲面。起源的司代数可以追溯到汉密尔顿的发现，四元数在1843年。汉密尔顿构造了他的四元数代数来推广复数，并将其应用于三维空间的力学。自从这个发现，四元数代数已经被推广到域上的有限维除代数，环上的Azumaya代数，甚至是概形上的Azumaya代数层。在每种情况下，对象的同构类与一个群一一对应，即布劳尔群。沿着的方式研究这些代数涉及许多数学工具，包括伽罗瓦上同调，估值理论，数论和代数几何，仅举几例。 存在于给定域上的除代数的类型可以被看作是域的复杂性或鲁棒性的度量。 在这个项目中，PI建议通过考虑存在于特定域上的除法代数的类型来研究这种复杂性。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).In broad terms the principle investigator proposes to study fields which support division algebras in their Brauer groups that are seen as exotic. As an example, in one part of the project the PI proposes to study the existence of non-crossed product division algebras as well as indecomposable division algebras over the function field of a p-adic curve. The general technique that the PI proposes is to study the division algebras that exist over completions of the function field and use a splitting map from the Brauer group of the completed function field to the Brauer group of the function field to lift the division algebras. In another part of the project the PI proposes to continue to study the connection between degeneracy of a matrix defining an abelian crossed product and decomposability of the abelian crossed product. The PI will follow up on an observation which she made which shows a not yet fully understood connection between degeneracy of the matrix, decomposability of the abelian crossed product and torsion in the 2nd Chow group of the associated Severi-Brauer variety. Lastly, the PI proposes to study fields for which division algebras of small degree over that field can be distinguished by their splitting fields. In particular, the PI will consider this question over function fields of K3 surfaces.The origins of division algebras can be traced back to Hamilton's discovery of the quaternions in 1843. Hamilton constructed his quaternion algebra to generalize the complex numbers and apply it to mechanics in three dimensional space. Since this discovery, quaternion algebras have been generalized to finite dimensional division algebras over a field, Azumaya algebras over a ring, and even sheaves of Azumaya algebras over a scheme. In each case the isomorphism classes of the objects are in 1-1 correspondence with a group, the Brauer group. Along the way the study of these algebras has involved many mathematical tools including Galois cohomology, valuation theory, number theory and algebraic geometry, just to name a few. The types of division algebras that exist over a given field can be seen as a measure of complexity or robustness of the field. In this project the PI proposes to study this complexity by considering the types of division algebras that exist over particular fields.