Witt and cohomological invariants of Witt classes
Witt and cohomological invariants of Witt classes
复制标题
维特和维特类的上同调不变量
作者：
N. Garrel
我们对函子$I^n$（Witt环的基本理想的幂）的所有Witt不变量进行分类，即函数$I^n（K）
与域扩张相容的八箭头W（K）$和所有模2上同调不变量，即函数$I^n（K）
八箭头H^*（K，mu_2）$.这在两种情况下都是根据某些运算（分别表示为$pi_n^{d}$和$u_{nd}^{（n）}$）来完成的，这些运算看起来像是被分割的幂，它们被证明是独立的，并生成所有不变量。这可以看作是模2 Milnor K-理论（或等价模2伽罗瓦上同调）上定义的运算的提升。
我们还研究了这些不变量的各种性质，包括相似性下的行为，离散赋值的留数，以及从$I^n$到$I^{n+1}$的限制。我们的目标是在以后的文章中使用它来研究具有对合的代数的不变量。
We classify all Witt invariants of the functor $I^n$ (powers of the fundamental ideal of the Witt ring), that is functions $I^n(K)
ightarrow W(K)$ compatible with field extensions, and all mod 2 cohomological invariants, that is functions $I^n(K)
ightarrow H^*(K,mu_2)$. This is done in both cases in terms of certain operations (denoted $pi_n^{d}$ and $u_{nd}^{(n)}$ respectively) looking like divided powers, which are shown to be independent and generate all invariants. This can be seen as a lifting of operations defined on mod 2 Milnor K-theory (or equivalently mod 2 Galois cohomology).
We also study various properties of these invariants, including behaviour under similitudes, residues for discrete valuations, and restriction from $I^n$ to $I^{n+1}$. The goal is to use this to study invariants of algebras with involutions in future articles.