The GaussManin connection on the periodic cyclic homology
Abstract
It is expected that the periodic cyclic homology of a DG algebra over the field of complex numbers (and, more generally, the periodic cyclic homology of a DG category) carries a lot of additional structure similar to the mixed Hodge structure on the de Rham cohomology of algebraic varieties. Whereas a construction of such a structure seems to be out of reach at the moment its counterpart in finite characteristic is much better understood thanks to recent groundbreaking works of Kaledin. In particular, it is proven by Kaledin that under some assumptions on a DG algebra $A$ over a perfect field $k$ of characteristic $p$, a lifting of $A$ over the ring of second Witt vectors $W_2(k)$ specifies the structure of a FontaineLaffaille module on the periodic cyclic homology of $A$. The purpose of this paper is to develop a relative version of Kaledin's theory for DG algebras over a base $k$algebra $R$ incorporating in the picture the GaussManin connection on the relative periodic cyclic homology constructed by Getzler. Our main result asserts that, under some assumptions on $A$, the GaussManin connection on its periodic cyclic homology can be recovered from the Hochschild homology of $A$ equipped with the action of the KodairaSpencer operator as the inverse Cartier transform (in the sense of OgusVologodsky). As an application, we prove, using the reduction modulo $p$ technique, that, for a smooth and proper DG algebra over a complex punctured disk, the monodromy of the GaussManin connection on its periodic cyclic homology is quasiunipotent.
 Publication:

arXiv eprints
 Pub Date:
 November 2017
 arXiv:
 arXiv:1711.02802
 Bibcode:
 2017arXiv171102802P
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Algebraic Topology;
 Mathematics  KTheory and Homology
 EPrint:
 To Alexander Beilinson on his 60th birthday, with admiration