Witt and cohomological invariants of Witt classes

Abstract

We classify all Witt invariants of the functor $I^n$ (powers of the fundamental ideal of the Witt ring), that is functions $I^n(K)\rightarrow W(K)$ compatible with field extensions, and all mod 2 cohomological invariants, that is functions $I^n(K)\rightarrow H^*(K,\mu_2)$. 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.