The $\gamma$-filtration on the Witt ring of a scheme

TL;DR

This paper explores the $\gamma$-filtration on the Grothendieck-Witt ring of a scheme, extending classical filtrations and connecting to Stiefel-Whitney classes, providing a deeper algebraic understanding of symmetric vector bundles.

Contribution

It introduces and studies the $\gamma$-filtration on the Grothendieck-Witt ring of a scheme, generalizing classical filtrations and relating to Stiefel-Whitney classes.

Findings

The $\gamma$-filtration generalizes the fundamental filtration on Witt rings.

It establishes a connection between the $\gamma$-filtration and classical Stiefel-Whitney class filtrations.

The structure of the $\lambda$-ring on symmetric vector bundles is clarified.

Abstract

The K-ring of symmetric vector bundles over a scheme X, the so-called Grothendieck-Witt ring of X, can be endowed with the structure of a (special) $\lambda$-ring. The associated $\gamma$-filtration generalizes the fundamental filtration on the (Grothendieck-)Witt ring of a field and is closely related to the "classical" filtration by the kernels of the first two Stiefel-Whitney classes.