$q$-Hodge complexes over the Habiro ring

Peter Scholze has raised the question whether some variant of the $q$-de Rham complex is already defined over the Habiro ring $\mathcal H = \lim_{m\in\mathbb N}\mathbb Z[q]_{(q^m-1)}^\wedge$. We show that such a variant exists whenever the $q$-de Rham complex can be equipped with a "$q$-Hodge filtration": a $q$-deformation of the Hodge filtration, subject to some reasonable conditions. To any such $q$-Hodge filtration we associate a small modification of the $q$-de Rham complex, which we call the $q$-Hodge complex, and show that it descends canonically to the Habiro ring. This construction recovers and generalises the Habiro ring of a number field of Garoufalidis-Scholze-Wheeler-Zagier and is closely related to the $q$-de Rham--Witt complexes from previous work of the author as well as, conjecturally, to Scholze's analytic Habiro stack. While there's no canonical $q$-Hodge filtration in general, we show that it does exist in many cases of interest. For example, for a smooth scheme $X$ over $\mathbb Z$, the $q$-de Rham complex can be equipped with a canonical $q$-Hodge filtration as soon as one inverts all primes $p\leq \dim(X/\mathbb Z)$.