$q$-Hodge complexes over the Habiro ring
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915538030034944 |
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| author | Wagner, Ferdinand |
| author_facet | Wagner, Ferdinand |
| contents | 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 $p\leq \dim(X/\mathbb Z)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04782 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $q$-Hodge complexes over the Habiro ring Wagner, Ferdinand Algebraic Geometry Number Theory 14F20, 14F40 (primary), 13F35 (secondary) 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 $p\leq \dim(X/\mathbb Z)$. |
| title | $q$-Hodge complexes over the Habiro ring |
| topic | Algebraic Geometry Number Theory 14F20, 14F40 (primary), 13F35 (secondary) |
| url | https://arxiv.org/abs/2510.04782 |