Topological Hochschild homology and Zeta-values

Using work of Antieau and Bhatt-Morrow-Scholze, we define a filtration on topological Hochschild homology and its variants $TP$ and $TC^-$ 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 $\mathrm{gr}^{n}TC^-(-)$ 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.