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，首席调查员建议研究在其Brauer群中支持除法代数的领域，这些领域被视为奇异的。例如，在该项目的一个部分中，PI建议研究非交叉积除法代数以及p-进曲线函数域上的不可分解除法代数的存在性。PI提出的一般方法是研究函数域完备上的除法代数，并利用从完备函数域的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中，国际和平研究所建议继续研究定义阿贝尔交叉积的矩阵的简并性与阿贝尔交叉积的可分解性之间的联系。PI将跟进她所做的一项观察，该观察表明矩阵的简并性、阿贝尔交叉积的可分解性和相关Severi-Brauer变种的第二Chow群中的扭转之间的联系尚未完全理解。最后，PI建议研究那些域上的低次除法代数可以通过它们的分离域来区分的域。特别是，PI将在K3曲面的函数域上考虑这个问题。除法代数的起源可以追溯到1843年哈密尔顿发现四元数。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中的力学。自从这个发现以来，四元数代数被推广到域上的有限维除代数，环上的Azumaya代数，甚至方案上的Azumaya代数。在每种情况下，对象的同构类与一个群--布劳尔群--1-1对应。一直以来，对这些代数的研究涉及到许多数学工具，包括伽罗华上同调、赋值理论、数论和代数几何，仅举几例。存在于给定域上的除法代数的类型可以被视为域的复杂性或稳健性的度量。在这个项目中，PI建议通过考虑特定域上存在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.