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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, 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 vanishes in degrees , then a lifting of R over and a lifting of over determine the structure of a relative Fontaine–Laffaille module (Faltings …