Topological Hochschild homology and zeta values

Using work of Antieau and Bhatt, Morrow, and Scholze, we define a filtration on topological Hochschild homology and its variants TP and TC− of quasi-lci rings 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 Z. We denote the cofiber of the canonical map from grnTC−(−) to grnTP(−) by LΩ−∕S<n[2n]. Let X be a regular connected scheme of dimension d proper over Spec(Z), and let n∈Z be an arbitrary integer. Together with Weil-étale cohomology with compact support RΓW,c(X,Z(n)), the complex LΩX∕S<n is expected to give the zeta value ±ζ∗(X,n) on the nose. Following Deninger, we define the zeta function ζ(XR,s) of the R-scheme XR in terms of zeta-regularized determinants. Our main result is a general special value formula for ±ζ∗(XR,n) in terms of the Bloch conductor A(X)d∕2−n and the determinants of RΓ(XR,W,Z(n)), LΩX∕S<n, and LΩX∕S<d−n. In particular, the passage from the base ring Z to the base E∞-ring S in derived de Rham cohomology is precisely quantified by the factorials in ζ∗(XR,n).