$q$-Witt vectors and $q$-Hodge complexes

TL;DR

This paper introduces a $q$-variant of Witt vectors and de Rham-Witt complexes, establishing foundational concepts that connect to various advanced topics in number theory and algebraic geometry, setting the stage for future research.

Contribution

It develops the technical foundation for $q$-Witt vectors and $q$-Hodge complexes, linking them to the Habiro ring, $q$-Hodge cohomology, and THH, which are novel connections in the field.

Findings

Defined a $q$-variant of Witt vectors and de Rham-Witt complexes.

Established foundational properties and relations to existing structures.

Laid groundwork for future exploration of connections to Habiro ring and $q$-Hodge cohomology.

Abstract

In this article, we'll introduce a $q$-variant of Witt vectors and de Rham-Witt complexes. This variant is closely related to the Habiro ring of a number field constructed by Garoufalidis, Scholze, Wheeler, and Zagier, to $q$-Hodge cohomology, and to $\operatorname{THH}(-/\mathrm{ku})$. While most of these connections will only be explored in forthcoming work, the goal of this article is to provide the necessary technical foundation.