Topological Hochschild homology and Zeta-values

Abstract

Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $T{C}^{-}$ of quasi-lci rings with bounded torsion, which recovers the BMS-filtration after $p$-adic completion. 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 ${\mathrm{gr}}^{n}T{C}^{-}(-)$ to ${\mathrm{gr}}^{n}TP(-)$ by $L{\Omega }_{-/\mathbb{S}}^{<n}[2n]$. 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-étale cohomology with compact support $R{\Gamma }_{W,c}(\mathcal{X},\mathbb{Z}(n))$, the complex $L{\Omega }_{\mathcal{X}/\mathbb{S}}^{<n}$ is expected to give the Zeta-value $\pm {\zeta }^{\ast }(\mathcal{X},n)$ on the nose. Combining the results proven here with a theorem recently proven in joint work with Flach, we obtain a formula relating $L{\Omega }_{\mathcal{X}/\mathbb{S}}^{<n}$, $L{\Omega }_{\mathcal{X}/\mathbb{S}}^{<d-n}$, Weil-étale 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.

References in corpus (1)

• Higher Topos Theory

Cited by in corpus (1)

• Zeta-values of one-dimensional arithmetic schemes at strictly negative integers