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Here Illusie mentions Grothendieck's observation that using Gauss-Manin connection one can give a non-canonical isomorphism between de Rham cohomology of smooth schemes over $W(k)$ with isomorphic reductions to $k$.

I understand that the formula written there is just the Taylor expansion of a given cohomology class. It seems reasonable that divided powers are involved.

Is there a reference for this observation in general and for the convergence of the series?

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    $\begingroup$ Grothendieck's article on deRham cohomology in the book "Dix Exposes..." includes his notion of the "infinitesimal site" that gives a site-theoretic interpretation of algebraic deRham cohomology in the smooth proper case over a field of characteristic zero. This was the first source of motivation for crystalline cohomology (as mentioned early in the book of Berthelot and Ogus). $\endgroup$
    – nfdc23
    Commented Oct 1, 2016 at 5:25
  • $\begingroup$ @nfdc23 Sure, I understand this motivation, I should have probably name the question differently. I am asking about another motivation which can be formulated as "Crystalline cohomology established a canonical isomorphism between de Rham cohomology of liftings" and the question is why a non-canonical isomorphism can be proen without crystalline cohomology. $\endgroup$
    – SashaP
    Commented Oct 1, 2016 at 7:38
  • $\begingroup$ Have you read the rest of Grothendieck's paper? He discusses Monsky-Washnitzer's canonical isomorphism (announced in unpublished work), and relates integrable connections to the notion of "stratification" -- a compatible system of "$n$-connections" for all $n \ge 1$ -- via his viewpoint of an infinitesimal descent-data formalism using iterated diagonals. The Appendix notes that integrable (1-)connections uniquely enhance to stratifications in char. 0 (see Theorem 2.15 in the Berthelot-Ogus book for a proof) but that this fails in residue char. $p>0$: see 3.5 in that paper, and 7.4 too. $\endgroup$
    – nfdc23
    Commented Oct 1, 2016 at 22:39

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