• Corpus ID: 227127688

Topological Hochschild homology and Zeta-values

@article{Morin2020TopologicalHH,
  title={Topological Hochschild homology and Zeta-values},
  author={Baptiste Morin},
  journal={arXiv: Number Theory},
  year={2020}
}

• B. Morin
• Published 23 November 2020
• Mathematics
• arXiv: Number Theory

Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants of quasi-lci schemes with bounded torsion. Then we compute the graded pieces of this filtration in terms of Hodge completed derived de Rham cohomology relative to the base ring $\mathbb{Z}$. We denote the cofiber of the canonical map from $\mathrm{gr}^{n}TC^-(-)$ to $\mathrm{gr}^{n}TP(-)$ by $L\Omega^{<n}_{-/\mathbb{S}}[2n]$. Let $\mathcal{X}$ be a regular connected scheme of dimension $d$ proper over $\mathrm{Spec}(\mathbb{Z})$ and let $n\in\mathbb{Z}$ be an arbitrary integer. Together with Weil-\'etale cohomology with compact support $R\Gamma_{W,c}(\mathcal{X},\mathbb{Z}(n))$, the complex $L\Omega^{<n}_{\mathcal{X}/\mathbb{S}}$ is expected to give the Zeta-value $\pm\zeta^*(\mathcal{X},n)$ on the nose. Combining the results proven here with a theorem recently proven in joint work with Flach, we obtain a formula relating $L\Omega^{<n}_{\mathcal{X}/\mathbb{S}}$, $L\Omega^{<d-n}_{\mathcal{X}/\mathbb{S}}$, Weil-\'etale cohomology of the archimedean fiber $\mathcal{X}_{\infty}$ with Tate twists $n$ and $d-n$, the Bloch conductor $A(\mathcal{X})$ and the special values of the archimedean Euler factor of the Zeta-function $\zeta(\mathcal{X},s)$ at $s=n$ and $s=d-n$. This formula is a shadow of the functional equation of Zeta-functions.