Operations on Milnor-Witt K-theory
Thor Wittich
TL;DR
This work extends the theory of stable operations from Milnor K-theory and quadratic forms to Milnor–Witt K-theory by constructing a rich family of operations $\underline{K}^{MW}_n \to M_*$ for any $\mathbb{N}$-graded homotopy algebra $M_*$ with a ring structure. The authors introduce divided-power-type operations $\lambda^n_l$ and their linear combinations $\sigma^n_l$, then prove a central isomorphism $\mathrm{Hom}(\underline{K}^{MW}_n, M_*) \cong M_*(k)^2 \times_{\delta_n h} M_*(k)^{\mathbb{N}\setminus\{0,1\}}$, equipped with a natural filtration, showing these generate all operations. They develop a robust shift/derivative formalism to systematically compute $\mathrm{Op}(\underline{K}^{MW}_n, M_*)$ and, via pullback squares, derive explicit descriptions for $\mathrm{Op}(\underline{K}^{MW}_n, \underline{K}^{MW}_m)$ and related maps between Milnor, Witt, and Milnor–Witt theories. The results unify and extend Garrel’s and Vial’s classical computations, recover known results as special cases, and provide a comprehensive operational calculus for Milnor–Witt K-theory in motivic homotopy theory.
Abstract
For all positive integers $n$ and all homotopy modules $M_*$, we define certain operations $\underline{\operatorname{K}}^{\operatorname{MW}}_n \rightarrow M_*$ and show that these generate the $M_*(k)$-module of all (in general non-additive) operations $\underline{\operatorname{K}}^{\operatorname{MW}}_n \rightarrow M_*$ in a suitable sense, if $M_*$ is $\mathbb{N}$-graded and has a ring structure. This also allows us to explicitly compute the abelian group $\operatorname{Op}(\underline{\operatorname{K}}^{\operatorname{MW}}_n,\underline{\operatorname{K}}^{\operatorname{MW}}_m)$ and all operations between related theories such as Milnor, Witt and Milnor-Witt K-theory.