A Hochschild-cyclic approach to additive higher Chow cycles

TL;DR

This paper introduces motivic operations on additive higher Chow cycles, establishing a differential graded algebra structure that connects additive Chow theory with additive K-theory and answers longstanding questions about Kähler differentials.

Contribution

It develops a Hochschild-cyclic framework for additive higher Chow cycles, defining new operations that form a differential graded algebra structure.

Findings

Connes boundary operator and shuffle product are constructed on additive Chow cycles.

The additive higher Chow groups form a commutative differential graded algebra.

On zero-cycles, the structure induces known operations on Kähler differentials.

Abstract

Over a field of characteristic zero, we introduce two motivic operations on additive higher Chow cycles: analogues of the Connes boundary $B$ operator and the shuffle product on Hochschild complexes. The former allows us to apply the formalism of mixed complexes to additive Chow complexes building a bridge between additive higher Chow theory and additive $K$-theory. The latter induces a wedge product on additive Chow groups for which we show that the Connes operator is a graded derivation for the wedge product using a variation of a Totaro's cycle. Hence, the additive higher Chow groups with the wedge product and the Connes operator form a commutative differential graded algebra. On zero-cycles, they induce the wedge product and the exterior derivation on the absolute K\"ahler differentials, answering a question of S. Bloch and H. Esnault.