Hodge cycles and crystalline cohomology. (English) Zbl 0538.14010

Hodge cycles, motives, and Shimura varieties, Lect. Notes Math. 900, 357-414 (1982).
[For the entire collection see Zbl 0465.00010.]
The author presents a survey of his research on crystalline cohomology. The paper is divided into four sections. In the first section the author gives the geometric interpretation of the ”conjugate” spectral sequence \[ E_ 2^{pq}=H^ p(X,H^ q(\Omega_{X/k})\Rightarrow F_{con}H^ i(X,\Omega_{X/k}), \] where X is a smooth scheme over a field k of positive characteristic. In particular it is proved that the associated filtration of the abutment is the filtration by codimension of support. - The second section is devoted to reduction mod p. - In the third section the author constructs a new invariant of a variety in characteristic p called the crystalline discriminant and gives a conjectural formula for this invariant. - At last the fourth section is devoted to a crystalline analogue of Deligne’s notion of ”absolute Hodge cycles”. The paper contains many interesting conjectures.
Reviewer: A.M.҆ermenev


14F30 \(p\)-adic cohomology, crystalline cohomology
14F40 de Rham cohomology and algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)


Zbl 0465.00010