Witt and Cohomological Invariants of Witt Classes

TL;DR

This paper classifies invariants of powers of the fundamental ideal in Witt rings and Galois cohomology, introducing new invariants that behave like divided powers and enabling the lifting of operations to $I^n$.

Contribution

It introduces a complete classification of invariants of $I^n$ using invariants $f_n^d$, providing a new framework for lifting operations from Milnor K-theory to Witt classes.

Findings

Invariants $f_n^d$ behave like divided powers.

Any invariant of $I^n$ can be expressed as a combination of $f_n^d$.

Properties under products, similitudes, and valuations are studied.

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.