Characteristic polynomials for symmetric forms

矩阵A的特征多项式是本科线性代数课程中经常介绍的一个基本数学对象，定义为行列式det（A-tI）。它的关键性质，即直和，张量积下的行为，以及对于三角矩阵，它只依赖于对角元素，可以通过从交换环R的循环K群到R的Witt向量环的环同态K ^cyc（R）->W（R）来优雅地编码，其元素由R生成的投射R-模的自同态表示，其元素是R上常系数为1的幂级数。Almkvist证明了这个映射是单射的，表明在某种意义上，特征多项式是自同态模扩张的完全不变量。这个项目将研究这个构造在多大程度上可以扩展到对称形式。加藤为特征为2的域上的对称形式的维特群构造了一个完全不变量，其值为基场在其平方子域上的二重张量积。我们建议把这个不变量看作是对称形式的秩或迹，这个项目的目标是把这个秩提升到一个Witt向量环上来构造Grothendieck-Witt群的不变量。我们将用于执行这个程序的技术是由Bökstedt，Hsiang和Madsen的Grothendieck-Witt谱的分圆迹映射的一个版本所告知的，并且自然会引导我们进一步研究真实的拓扑Hochschild同调、Witt向量和de Rham-Witt复形之间的关系。

The characteristic polynomial of a matrix A is a fundamental mathematical object which is usually introduced in an undergraduate linear algebra course, defined as the determinant det(A-tI). Its key properties, namely the behavior under direct sums, tensor products, and that for triangular matrices it depends only on the diagonal entries, can be elegantly encoded by a ring homomorphismK^cyc(R)--->W(R)from the cyclic K-group of a commutative ring R, whose elements are represented by endomorphisms of finitely generated projective R-modules, to the ring of Witt vectors of R, whose elements are power series on R with constant coefficient 1. Almkvist proves that this map is injective, showing that in a sense the characteristic polynomial is a complete invariant for endomorphisms modulo extensions.This project will investigate to which extent this construction can be extended to symmetric forms. Kato constructed a complete invariant for the Witt group of symmetric forms over a field of characteristic 2, valued in the two-fold tensor product of the base field over its subfield of squares. We propose to regard this invariant as the rank, or trace, of a symmetric form, and the goal of the project is to lift this rank to a ring of Witt vectors to construct an invariant for the Grothendieck-Witt group.The techniques we will use for carrying out this program are informed by a version of the cyclotomic trace map of Bökstedt, Hsiang and Madsen for the Grothendieck-Witt spectrum, and will naturally lead us to further investigate the relationship between real topological Hochschild homology, the Witt vectors, and the de Rham-Witt complex.