# Notes on Crystalline Cohomology. (MN-21)

Pierre Berthelot
Arthur Ogus
Pages: 256
https://www.jstor.org/stable/j.ctt130hk6f

1. Front Matter
(pp. None)
(pp. None)
3. Preface
(pp. i-vi)
P. Berthelot
4. §1. Introduction.
(pp. 1.1-1.14)

Let k be the field with q elements, X/k a smooth, projective, and geometrically connected scheme. One wants to know how many rational points X has, or more generally, the number cνof kν-valued points of X where kνis the extension of k of degree ν. The values of these numbers are conveniently summarized in the zeta function of X, given by${{\text{Z}}_{\text{X}}}(\text{t})=\exp \sum\limits_{\nu =1}^{\infty }{\frac{{{\text{c}}_{\nu }}}{\nu }{{\text{t}}^{\nu }}}$. This can also be written as$\underset{\text{x}}{\mathop{\prod }}\,\frac{1}{[1-{{\text{t}}^{\deg \text{x}}}]}$, the product being taken over the closed points x of X, where deg x means the degree of the residue field k(x) over k. It is clear...

5. §2. Calculus and Differential Operators.
(pp. 2.1-2.23)

In this chapter we develop Grothendieck's way of geometrizing the notions of calculus and differential geometry, and in particular the notion of a locally (or rather infinitesimally) constant sheaf. We begin by reviewing the formalism of differential operators.

If X → S is a morphism of schemes, and if F and G are OX-modules, then a differential operator from F to G, relative to S, will be an f−1(OS)-linear map h:F → G which is "almost" OX-linear. In order to make this precise, we begin by brutally linearizing h, i.e., by forming the obvious adjoint map:$\overline{\text{h}}:{\text{O}_{{{\text{X}}^{\otimes }}_{{{\text{f}}^{-1}}({\text{O}_{\text{S}}})}}}\text{F}\to \text{G}\quad.$

Using the OX-module...

6. §3. Divided Powers.
(pp. 3.1-3.23)

3.1 Definition. Let A be a commutative ring, I ⊂ A an idea1. By "divided powers on I" we mean a collection of maps γi:I → A, for all integers i ≥ 0, such that:

1) For all x ∈ I, γ0(x) = 1, γ1(x) = x, γi(x) ∈ I if i ≥ 1.

2) For x,y ∈ I,${{\gamma}_{\text{k}}}(\text{x}+\text{y})=\sum\limits_{\text{i}+\text{j}=\text{k}}{{{\gamma}_{\text{i}}}(\text{x}){{\gamma}_{\text{j}}}(\text{y})}$.

3) For λ ∈ A, x ∈ I, γk(λx) = λkγk(x).

4) For x ∈ I, γi(x)γj(x) = ((i,j))γi+j(x), where$((\text{i},\text{j}))=\frac{(\text{i}+\text{j})!}{(\text{i}!)(\text{j}!)}$.

5) γpq(x)) = Cp,qγpq(x), where${{\text{C}}_{\text{p},\text{q}}}=\frac{(\text{pq})!}{\text{p}!{{(\text{q}!)}^{\text{p}}}}$.

Note. By induction on p it is simple to prove...

7. §4. Calculus with Divided Powers.
(pp. 4.1-4.13)

Suppose (S,I,γ) is a PD scheme (with I a quasi-coherent ideal, as always), and suppose X is an S-scheme. Let X/S(ν+1)be the ν+1-fold Cartesian product of X with itself, computed over S, and let Δ⃥:X → X/S(ν+1)be the diagonal immersion. The immersion Δ̸ is locally closed and has ν+1 retractions to X. It follows from Remark (3.20.6) that if γ extends to X, the divided power envelope of X in X/S(ν+1)does not depend on γ.

4.1 Definition. Suppose γ extends to X and either mOX= 0 or X/S is separated. Then we can form the divided...

8. §5. The Crystalline Topos.
(pp. 5.1-5.29)

We are ready to assemble the constructions of the first four chapters into the notion of the "crystalline site", which will then give rise to the "crystalline topos". In this and the next section, all schemes will be killed by a power of a prime p, unless otherwise specified. This assumption will allow us to postpone the technical difficulties of inverse limits.

Let S = (S,I,γ) be a PD-scheme, which will play the role of the "base". For any S-scheme X to which γ extends (in the sense of (3.14)), we want to define the "crystalline site of X relative...

9. §6. Crystals.
(pp. 6.1-6.23)

A "crystal", says Grothendieck, is characterized by two properties: it is rigid, and it grows. Any sheaf F on Cris(X/S) "grows" over PD thickenings (U,T,δ) of open subsets of X by construction. In order for F to be a crystal, we impose a rigidity which we shall only make precise for sheaves of OX/S-modules. From now on, we shall write u−1for pull-back of a sheaves of sets, and u* for module pull-back (i.e. u−1followed by tensor product with the structure sheaf) – unless there is no danger of confusion.

6.1 Definition. A "crystal" of OX/S-modules is a sheaf...

10. §7. The Cohomology of a Crystal.
(pp. 7.1-7-35)

We are now ready to establish the fundamental property of crystalline cohomology, namely, its relation to de Rham cohomology. Our first goal is the following result, from which all the finiteness and base changing properties to come will be deduced:

7.1 Theorem. Suppose i: X↪Y is a closed immersion of S-schemes, with Y/S smooth. Let E be a DX,γ(Y)-module with HPD stratification, let E be the crystal on X obtained from E by (6.6), and let$\text{E}{{\otimes }_{\text{D}}}\Omega _{{{\text{D}}_{{\text {X}},\gamma }}}^{\centerdot }(\text{Y})/\text{S}$be the complex of sheaves on Xzarobtained from the connection on E. (Recall that the complex$\text{E}{{\otimes }_{\text{D}}}\Omega _{\text{D}/\text{S}}^{\centerdot }$is the complex$\text{E}{{\otimes }_{{\text {O}_{\text{Y}}}}}\Omega _{\text{Y}/\text{S}}^{\centerdot }$,...

11. §8. Frobenius and the Hodge Filtration.
(pp. 8-1-8-50)

Suppose W is the Witt ring of a perfect field k of characteristic p, and X is a smooth k-scheme. The Frobenius automorphism of W is a PD morphism, covered by the absolute Frobenius endomorphism FXof X, and it follows that FXacts on the crystalline cohomology of X relative to W. In this chapter we shall study this action, in particular, its relationship to the Hodge filtration on crystalline cohomology (as determined from the ideal JX/S). The main global applications are Mazur's theorem (8.26), which says that (with suitable hypotheses on X) the action of Frobenius determines the...

12. Appendix A The Construction of ΓAM
(pp. A-1-A-15)
13. Appendix B Finiteness of $\mathbb{R}\underleftarrow{\lim }$
(pp. B.1-B.16)
14. BIBLIOGRAPHY
(pp. 1.-3.)
15. Back Matter
(pp. None)