Witt and Cohomological Invariants of Witt Classes

Abstract

We classify all invariants of the functor $I^n$ (powers of the fundamental ideal of the Witt ring) with values in $A$, that it to say functions $I^n(K)\rightarrow A(K)$ compatible with field extensions, in the cases where $A(K)=W(K)$ is the Witt ring and $A(K)=H^*(K,\mu_2)$ is mod 2 Galois cohomology. This is done in terms of some invariants $f_n^d$ that behave like divided powers with respect to sums of Pfister forms, and we show that any invariant of $I^n$ can be written uniquely as a (possibly infinite) combination of those $f_n^d$. This in particular allows to lift operations defined on mod 2 Milnor K-theory (or equivalently mod 2 Galois cohomology) to the level of $I^n$. We also study various properties of these invariants, including behaviour under products, 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.