A Hochschild-Kostant-Rosenberg theorem and residue sequences for logarithmic Hochschild homology

TL;DR

This paper develops a logarithmic version of Hochschild homology, establishing a Hochschild-Kostant-Rosenberg theorem and residue sequences within the framework of log motives, with applications to log scheme blow-ups.

Contribution

It introduces a geometric definition of logarithmic Hochschild homology, constructs a spectral sequence, and proves a logarithmic HKR theorem, advancing the theory of log motives.

Findings

Degeneration of the spectral sequence for derived log smooth maps

Logarithmic Hochschild homology is representable in log motives

Derived residue sequences involving log scheme blow-ups

Abstract

This paper incorporates the theory of Hochschild homology into our program on log motives. We discuss a geometric definition of logarithmic Hochschild homology of derived pre-log rings and construct an Andr\'e-Quillen type spectral sequence. The latter degenerates for derived log smooth maps between discrete pre-log rings. We employ this to show a logarithmic version of the Hochschild-Kostant-Rosenberg theorem and that logarithmic Hochschild homology is representable in the category of log motives. Among the applications, we deduce a generalized residue sequence involving blow-ups of log schemes.