A higher Hodge extension of the Feigin-Tsygan Theorem

TL;DR

This paper extends the Feigin-Tsygan theorem by establishing a Hodge filtration-based quasi-isomorphism between the noncommutative de Rham complex and cyclic complexes, with applications to Calabi-Yau categories.

Contribution

It introduces a Hodge filtration approach to extend the classical Feigin-Tsygan theorem, connecting noncommutative de Rham complexes with cyclic complexes.

Findings

The extended noncommutative de Rham complex is quasi-isomorphic to the periodic cyclic complex.

Each filtration piece is quasi-isomorphic to the negative cyclic complex.

The results have applications to the study of Calabi-Yau categories.

Abstract

We show that the extended noncommutative de Rham complex of a cofibrant resolution, when completed at a certain Hodge filtration, is (reduced) quasi-isomorphic to the periodic cyclic complex, while each of its filtration piece is quasi-isomorphic to the negative cyclic complex. This extends a classical result of Feigin and Tsygan, which corresponds to the Hodge degree $0$ part of our quasi-isomorphism. This result is applied to the study of Calabi-Yau categories in \cite{Yeu1, Yeu2}.