Stable operations and cooperations in derived Witt theory with rational coefficients

TL;DR

This paper computes the stable operations and cooperations in derived Witt theory with rational coefficients, providing an additive description and drawing parallels to real K-theory, highlighting the role of the Bott element.

Contribution

It offers the first explicit computation of stable operations and cooperations in derived Witt theory with rational coefficients, paralleling classical K-theory results.

Findings

Stable operations are determined by values on powers of the Bott element.

An explicit additive description of cooperations is provided.

Results parallel the known case of real K-theory.

Abstract

The algebras of stable operations and cooperations in derived Witt theory with rational coefficients are computed and an additive description of cooperations in derived Witt theory is given. The answer is parallel to the well-known case of K-theory of real vector bundles in topology. In particular, we show that stable operations in derived Witt theory with rational coefficients are given by the values on the powers of Bott element.