The Gauss-Manin connection on the periodic cyclic homology
Let R be the algebra of functions on a smooth affine irreducible curve S over a field k and let be a smooth and proper DG algebra over R. The relative periodic cyclic homology of over R is equipped with the Hodge filtration and the Gauss-Manin connection (Getzler, in: Quantum deformations of algebras and their representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), Israel mathematics conference proceedings, vol 7, Bar-Ilan University, Ramat Gan, pp 65-78, 1993; Kaledin, in: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin, vol II, pp 23-47, Progress in mathematics, vol 270, Birkhauser Inc., Boston, 2009) satisfying the Griffiths transversality condition. When k is a perfect field of odd characteristic p, we prove that, if the relative Hochschild homology vanishes in degrees , then a lifting of R over and a lifting of over determine the structure of a relative Fontaine-Laffaille module (Faltings, in: Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins University Press, Baltimore, MD, pp 25-80, 1989, A 2 (c); Ogus and Vologodsky in Publ Math Inst Hautes Aetudes Sci No 106:1-138, 2007 A 4.6) on . That is, the inverse Cartier transform of the Higgs R-module is canonically isomorphic to . This is non-commutative counterpart of Faltings' result (1989, Th. 6.2) for the de Rham cohomology of a smooth proper scheme over R. Our result amplifies the non-commutative Deligne-Illusie decomposition proven by Kaledin (Algebra, geometry and physics in the 21st century (Kontsevich Festschrift), Progress in mathematics, vol 324. Birkhauser, pp 99-129, 2017, Th. 5.1). As a corollary, we show that the p-curvature of the Gauss-Manin connection on is nilpotent and, moreover, it can be expressed in terms of the Kodaira-Spencer class [a similar result for the p-curvature of the Gauss-Manin connection on the de Rham cohomology is proven by Katz (Invent Math 18:1-118, 1972)]. As an application of the nilpotency of the p-curvature we prove, using a result from Katz (Inst Hautes Aetudes Sci Publ Math No 39:175-232, 1970), a version of "the local monodromy theorem" of Griffiths-Landman-Grothendieck for the periodic cyclic homology: if , is a smooth compactification of S, then, for any smooth and proper DG algebra over R, the Gauss-Manin connection on the relative periodic cyclic homology has regular singularities, and its monodromy around every point at is quasi-unipotent.