Crystalline cohomology and \(p\)-adic Galois-representations.

*(English)*Zbl 0805.14008
Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 25-80 (1989).

Summary: [For the entire collection see Zbl 0747.00038.]

The paper proceeds as follows: We first review relevant facts from commutative algebra, étale and crystalline cohomology. These are mostly somehow known, or at least dwell on well known ideas, but they cannot be found in the literature in such a form that we can use them directly. We also use the occasion to generalise many results. Some of the main new features are:

We extend the comparison-theory of J.-M. Fontaine and G. Laffaille [cf. Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 547-608 (1982; Zbl 0579.14037)] to families of \(F\)-crystals.

We construct logarithmic versions of crystalline cohomology, for non proper varieties with a good compactification (hidden behind this there is a whole theory of “logarithmic commutative algebra”, which however we choose not to develop in detail).

We make precise the relation between étale cohomology and Galois- cohomology.

We allow nontrivial systems of coefficients.

After that we prove the comparison results, first in the absolute case, and then more general for direct images under “log-smooth” maps. We conclude by an application to the theory of finite flat group schemes, giving a complete description in terms of “semilinear algebra”. Finally we show that our method also allows to settle the “de Rham conjecture”.

The paper proceeds as follows: We first review relevant facts from commutative algebra, étale and crystalline cohomology. These are mostly somehow known, or at least dwell on well known ideas, but they cannot be found in the literature in such a form that we can use them directly. We also use the occasion to generalise many results. Some of the main new features are:

We extend the comparison-theory of J.-M. Fontaine and G. Laffaille [cf. Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 547-608 (1982; Zbl 0579.14037)] to families of \(F\)-crystals.

We construct logarithmic versions of crystalline cohomology, for non proper varieties with a good compactification (hidden behind this there is a whole theory of “logarithmic commutative algebra”, which however we choose not to develop in detail).

We make precise the relation between étale cohomology and Galois- cohomology.

We allow nontrivial systems of coefficients.

After that we prove the comparison results, first in the absolute case, and then more general for direct images under “log-smooth” maps. We conclude by an application to the theory of finite flat group schemes, giving a complete description in terms of “semilinear algebra”. Finally we show that our method also allows to settle the “de Rham conjecture”.