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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, Birkhäuser 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 Études 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. Birkhäuser, 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 Études 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.

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Correspondence to Vadim Vologodsky.

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To Sasha Beilinson on his 60th birthday, with admiration.

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Petrov, A., Vaintrob, D. & Vologodsky, V. The Gauss–Manin connection on the periodic cyclic homology. Sel. Math. New Ser. 24, 531–561 (2018). https://doi.org/10.1007/s00029-018-0388-0

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Mathematics Subject Classification

  • 16E40
  • 14G17