Notes on Crystalline Cohomology. (MN-21)

Canonicalq-deformations in arithmetic geometry

“…The new observation in [5] was that it was possible to remove all (q − 1)-torsion in the previous result; the new idea was to use the following operation on the derived category (first defined by Berthelot-Ogus, [4,Chapter 8]). …”

“…If one fixes an étale map The first proof that the de Rham complex of a lift to characteristic zero is independent of the lift was through Grothendieck's formalism of the crystalline site, cf. [4].…”

The μ-ordinary Hasse invariant of\break unitary Shimura varieties

“…Let A be an abelian variety over an algebraically closed field of characteristic 0. Since the Hodge-de Rham spectral sequence of A degenerates at E 1 and since the crystalline cohomology of A torsion-free, the theorem is a special case of [BO78,8.26]. …”

Open problems: Descending cohomology, geometrically

“…The reason why it is reasonable to do this is that for all primes p of good reduction for X, the eigen-values of Frobenius in the original untwisted representation are algebraic integers divisible by p (since h 0,3 = 0: this is a consequence, for example, of a conjecture of Katz that is now a theorem; see specifically [4]; for explicit results regarding divisibility, see [6]; for related issues see also [15], [16]) and therefore the downward twisted representation has the property that these eigenvalues divided by p remain algebraic integers, and therefore-in the usual sense-are "Weil numbers" of absolute value p/2.…”