Prismatic logarithm and prismatic Hochschild homology via norm

TL;DR

This paper introduces a new prismatic logarithm refinement for line bundle Chern classes, relates it to prismatic Witt vectors, and proposes prismatic Hochschild homology as a noncommutative analogue of prismatic de Rham complex.

Contribution

It presents an elementary construction of the first Chern class using a refined prismatic logarithm and introduces prismatic Hochschild homology as a novel noncommutative framework.

Findings

Refinement of prismatic logarithm for line bundle Chern classes

Relation of this construction to prismatic Witt vectors

Proposal of prismatic Hochschild homology as a noncommutative analogue

Abstract

In this brief note, we present an elementary construction of the first Chern class of Hodge--Tate crystals in line bundles using a refinement of the prismatic logarithm, which should be comparable to the one considered by Bhargav Bhatt. The key to constructing this refinement is Yuri Sulyma's norm on (animated) prisms. We explain the relation of this construction to prismatic Witt vectors, as a generalization of Kaledin's polynomial Witt vectors. We also propose the prismatic Hochschild homology as a noncommutative analogue of prismatic de Rham complex.