Logarithmic Hochschild co/homology via formality of derived intersections

TL;DR

This paper introduces log Hochschild co/homology for log schemes, establishing key isomorphisms, invariance properties, and a geometric framework via formality of derived intersections, advancing the understanding of log geometry.

Contribution

It defines log Hochschild co/homology, proves a Hochschild-Kostant-Rosenberg isomorphism, and demonstrates invariance under log alterations, using formality of derived intersections.

Findings

Hochschild-Kostant-Rosenberg isomorphism for log smooth schemes

Log Hochschild co/homology invariant under log alterations

Tropicalization of product of log schemes equals product of tropicalizations

Abstract

We define log Hochschild co/homology for log schemes that behaves well for simple normal crossing pairs $(X,D)$ or toroidal singularities. We prove a Hochschild-Kostant-Rosenberg isomorphism for log smooth schemes, as well as an equivariant version for log orbifolds. We define cyclic homology and compute it in simple cases. We show that log Hochschild co/homology is invariant under log alterations. Our main technical result in log geometry shows the tropicalization (Artin fan) of a product of log schemes $X \times Y$ is usually the product of the tropicalizations of $X$ and $Y$. This and the machinery of \emph{formality} of derived intersections facilitate a geometric approach to log Hochschild.