Timeline for Grothendieck's motivation of crystalline cohomology
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| Oct 1, 2016 at 22:39 | comment | added | nfdc23 | 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 He discusses Monsky-Washnitzer's canonical isomorphism (announced in unpublished work), and relates integrable connections to the notion of "stratification" -- a compatible system of "$n$-connections" for all $n \ge 1$ -- via his viewpoint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. | |
| Oct 1, 2016 at 7:38 | comment | added | SashaP | @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. | |
| Oct 1, 2016 at 5:25 | comment | added | nfdc23 | 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). | |
| Sep 30, 2016 at 23:00 | history | asked | SashaP | CC BY-SA 3.0 |