The Gauss–Manin connection on the periodic cyclic homology

• Alexander Petrov ^[1] ; Dmitry Vaintrob ^[2] ; Vadim Vologodsky ^[1]
  1. [1] National Research University “Higher School of Economics”
  2. [2] Institute for Advanced Study
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 24, Nº. 1, 2018, págs. 531-561
• Idioma: inglés
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• Resumen

  • Let R be the algebra of functions on a smooth affine irreducible curve S over a field k and let A∙ be a smooth and proper DG algebra over R. The relative periodic cyclic homology HP∗(A∙) of A∙ over R is equipped with the Hodge filtration F⋅ 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, 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 HHm(A∙,A∙) vanishes in degrees |m|≥p−2 , then a lifting R~ of R over W2(k) and a lifting of A∙ over 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∙) . That is, the inverse Cartier transform of the Higgs R-module (GrFHP∗(A∙),GrF∇) is canonically isomorphic to (HP∗(A∙),∇) . 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∙) is nilpotent and, moreover, it can be expressed in terms of the Kodaira–Spencer class κ∈HH2(A∙,A∙)⊗RΩ1R [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=C , S¯¯¯ is a smooth compactification of S, then, for any smooth and proper DG algebra A∙ over R, the Gauss–Manin connection on the relative periodic cyclic homology HP∗(A∙) has regular singularities, and its monodromy around every point at S¯¯¯−S is quasi-unipotent.