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 TPTP 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 ZZ. We denote the cofiber of the canonical map from gr(n)TC-(-)to gr(n)TP(-) by L Omega(<n)(-/S)[2n]. Let X be a regular connected scheme of dimension d proper over Spec(Z), and let n is an element of Z be an arbitrary integer. Together with Weil-& eacute;tale cohomology with compact support R Gamma(W,c)(X,Z(n)), the complex L Omega(<n)(X/S) is expected to give the zeta value +/-zeta(& lowast;)(X,n) on the nose. Following Deninger, we define the zeta function zeta(XR,s) of the R-scheme X-R in terms of zeta-regularized determinants. Our main result is a general special value formula for +/-zeta(& lowast;)(X-R,n) in terms of the Bloch conductor A(X)(d/2-n) and the determinants of R Gamma(X-R,X-W,Z(n)), L Omega(<n)(X/S), and L Omega(<d-n)(X/S). In particular, the passage from the base ring Z to the base E-infinity-ring S in derived de Rham cohomology is precisely quantified by the factorials in zeta(& lowast;)(X-R,n).