English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

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.

The Gauss–Manin connection on the periodic cyclic homology