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The Gauss–Manin connection on the periodic cyclic homology
It is expected that the periodic cyclic homology of a DG algebra over C (and, more
generally, the periodic cyclic homology of a DG category) carries a lot of additional
structure similar to the mixedHodge structure on the deRhamcohomology 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 in [17] that under
some assumptions on aDGalgebra A over a perfect field k of characteristic p, a lifting
of A over the ring of secondWitt vectors W2(k) specifies the structure of a Fontaine–
Laffaille 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 Gauss–Manin connection on the relative periodic
cyclic homology constructed by Getzler in [13]. Our main result, Theorem 1, asserts
that, under some assumptions on A , the Gauss–Manin connection on its periodic
cyclic homology can be recovered from the Hochschild homology of A equipped
with the action of the Kodaira–Spencer operator as the inverse Cartier transform [29].
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
Gauss–Manin connection on its periodic cyclic homology is quasi-unipotent.