Motivic structures in non-commutative geometry

TL;DR

This paper discusses the conjectural motivic structures in the periodic cyclic homology of smooth non-commutative algebraic varieties, linking it to classical Hodge and Dieudonne structures.

Contribution

It reviews recent results and conjectures suggesting that periodic cyclic homology encodes motivic structures analogous to classical cohomology theories.

Findings

Periodic cyclic homology may carry Hodge structures over R.

It may have filtered Dieudonne module structures over Z_p.

The paper summarizes ongoing research and conjectures in this area.

Abstract

We review some recent results and conjectures saying that, roughly speaking, periodic cyclic homology of a smooth non-commutative algebraic variety should carry all the additional "motivic" structures possessed by the usual de Rham cohomology of a smooth algebraic variety (specifically, an R-Hodge structure for varieties over R, and a filtered Dieudonne module structure for varieties over Z_p). To appear in Proc. ICM 2010.