On Tate Milnor-Witt Motives

TL;DR

The paper analyzes Tate Milnor-Witt motives over Euclidean fields, proving that every Tate MW-motive decomposes into blocks generated by $\mathbb{Z}(i)[2i]$, $\mathbb{Z}/2^t\eta(i)[2i]$, and $\mathbb{Z}/\mathbf{s}[i]$ with $\mathbf{s}$ odd, thereby reducing MW-motivic cohomology to Witt cohomology computations. The main technical tool is a carefully controlled Bockstein spectral sequence for Milnor-Witt motivic cohomology, built from exact couples and related to Witt and motivic cohomology via explicit differentials; degeneracy criteria are given in terms of torsion bounds. The authors also develop structural results for MW-motives of cellular varieties, including Kunneth-type properties, and study concrete applications: classification of rank-$n$ vector bundles on $\mathbb{HP}^1$ via Euler and second Chern classes, and a blow-up formula for MW-motives with even center codimension. These results jointly provide a framework for computing Chow-Witt groups through Witt cohomology and known Chow groups, with implications for examples such as Grassmannians, flag varieties, and moduli spaces, and offer a foundation for extending Tate–MW theory to integral settings. The work thus advances both the structural understanding and computational toolkit for Milnor-Witt motives in the Euclidean setting.

Abstract

Smooth projective $\mathbb{G}_m$-varieties with isolated rational fixed points admit Tate Milnor-Witt motives. Over Euclidean fields, we give a splitting formula of such motives, which reduces the computation of their Chow-Witt groups to that of their Chow groups and cohomologies of Witt sheaf.

Theorems & Definitions (64)

• Theorem 1
• Proposition 2
• Theorem 3
• Definition 4
• Proposition 5
• proof
• Corollary 6
• proof
• Lemma 7
• proof