The Gauss–Manin connection on the periodic cyclic homology

Abstract

Let R be the algebra of functions on a smooth affine irreducible curve S over a field k and let $${A_{\bullet }}$$ be a smooth and proper DG algebra over R. The relative periodic cyclic homology $$HP_* ({A_{\bullet }})$$ of $${A_{\bullet }}$$ over R is equipped with the Hodge filtration $${\mathcal F}^{\cdot }$$ and the Gauss–Manin connection $$\nabla $$ (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 $$HH_m({A_{\bullet }}, {A_{\bullet }})$$ vanishes in degrees $$|m| \ge p-2$$ , then a lifting $$\tilde{R}$$ of R over $$W_2(k)$$ and a lifting of $${A_{\bullet }}$$ over $$\tilde{R}$$ 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, §2 (c); Ogus and Vologodsky in Publ Math Inst Hautes Etudes Sci No 106:1–138, 2007 §4.6) on $$HP_* ({A_{\bullet }})$$ . That is, the inverse Cartier transform of the Higgs R-module $$(Gr^{\mathcal F}HP_* ({A_{\bullet }}), Gr^{\mathcal F}\nabla )$$ is canonically isomorphic to $$ (HP_* ({A_{\bullet }}), \nabla )$$ . 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 $$HP_* ({A_{\bullet }})$$ is nilpotent and, moreover, it can be expressed in terms of the Kodaira–Spencer class $$\kappa \in HH^2({A_{\bullet }}, {A_{\bullet }}) \otimes _R \Omega ^1_R$$ [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 Etudes 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 $$k={\mathbb C}$$ , $$\overline{S}$$ is a smooth compactification of S, then, for any smooth and proper DG algebra $${A_{\bullet }}$$ over R, the Gauss–Manin connection on the relative periodic cyclic homology $$HP_* ({A_{\bullet }})$$ has regular singularities, and its monodromy around every point at $$\overline{S}-S$$ is quasi-unipotent.