Finiteness of Crystalline Cohomology

In my lectures on crystalline cohomology I have worked through the basic comparison theorem of crystalline cohomology with de Rham cohomology. This comparison allows one to prove that crystalline cohomology has some good properties exactly as is done by Berthelot in his thesis. To formulate these properties we introduce some notation

  • p is a prime number,
  • (A, I, γ) is a divided power ring over Z_{(p)},
  • S = Spec(A) and S_0 = Spec(A/I),
  • f : X —> Y is a quasi-compact, quasi-separated smooth morphism of schemes over S_0,
  • E = Rf_{cris, *}O_{X/S} on Cris(Y/S).

The comparison theorem is the main ingredient in showing that E has the following properties:

  1. the cohomology sheaves H^i(E) are locally quasi-coherent, i.e., for every object (V, T, δ) of Cris(Y/S) the restriction H^i(E)_T = H^i(E_T) of H^i(E) to the Zariski site of T is a quasi-coherent O_T-module,
  2. E is a “crystal”, i.e., for every morphism h : (V, T, δ) —> (V’, T’, δ’) of Cris(Y/S) the comparison map Lh^*E_{T’} —> E_T is a quasi-isomorphism,
  3. if (V, T, δ) is an object of Cris(Y/S) and if f is PROPER and T is a NOETHERIAN scheme, then E_T is a perfect complex of O_T-modules.

In case f is proper, I did not find a reference in the literature proving that E is a perfect complex of modules on Cris(Y/S), i.e., that E_T is a perfect complex of O_T-modules for every object T of the small crystalline site of Y. For those of you who are quickly scanning this web-page let me make the statement explicit as follows:

Let (A, I, γ) be a divided power ring with p nilpotent in A. Let X be proper smooth over A/I. Then RΓ(Cris(X/A), O_{X/A}) is a perfect complex of A-modules.

Again, far as I know this isn’t in the literature, but let me know if you have a reference [Edit: see comment by Bhargav below]. Here is an argument which I think works, but I haven’t written out all the details. By the base change theorem (part 2 above) it is enough if we can find a divided power ring (B, J, δ) with p nilpotent in B and a homomorphism of divided power rings B —> A such that X is the base change of a quasi-compact smooth scheme over B/J and such that the result holds over B. Arguing in this way and using standard limit arguments in algebraic geometry we reduce the question to the case where A/I is a finitely generated Z-algebra. Writing A/I = Z/p^NZ[x_1, …, x_n]/(f_1, …, f_m) we reduce to the case where A is the divided power envelope of (f_1, …, f_m) in Z/p^NZ[x_1, …, x_n]. In this case we see that we can lift X to a smooth scheme over A/(f_1, …, f_m)A because A/I maps to A/(f_1, …, f_m)A! Moreover, the ideal (f_1, …, f_m) is a nilpotent ideal in A! Now suppose we have a modified comparison theorem which reads as follows:

Modified comparsion. Let (A, I, γ) be a divided power ring with p nilpotent in A. Let J ⊂ I be an ideal. Let X’ be proper smooth over A’ = A/J and set X = X’ ⊗ A/I. Then RΓ(Cris(X/A), O_{X/A}) ⊗^L_A A’ = RΓ(X’, Ω^*_{X’/A’}).

The tricky bit is that J needn’t be a divided power ideal, but I think a Cech cover argument will work (this is a bit shaky). The usual arguments show that RΓ(X’, Ω^*_{X’/A’}) is a perfect complex of A’-modules. Proof of the statement above is finished by observing that a complex of A-modules K^* such that K^* ⊗^L_A A’ is perfect is perfect. (My proof of this uses that the kernel J of A —> A’ is a nilpotent ideal and the characterization of perfect complexes as those complexes such that hom in D(A) out of them commutes with direct sums.)

I’ll think this through more carefully in the next few days and if I find something wrong with this argument I’ll edit this post later. Let me know if you think there is a problem with this idea.

5 thoughts on “Finiteness of Crystalline Cohomology

  1. Pingback: Compact and perfect objects | Stacks Project Blog

  2. Let (A, I, γ) be a divided power ring with p nilpotent in A. Let X be proper smooth over A/I. Then RΓ(Cris(X/A), O_{X/A}) is a perfect complex of A-modules.

    I think this is just 7.24.3 of Berthelot-Ogus “Notes on Crystalline Cohomology”, unless I misunderstood some key point here. They said “RΓ(X/S, E) is perfect”, where E is a locally free, finitely generated crystal of O_{X/S^}-modules in (X/S^)_cris.

    • Nope, because there S is Noetherian! Of course, as is usual, the authors do not state the assumptions on S in the statement of 7.24.3… so it is hard to tell, but the proof I think uses that assumption.

  3. There is some discussion about the non-noetherian case in the appendix (see Theorem 14) to Faltings’ “Integral crystalline cohomology over very ramified valuation rings” paper (JAMS 1999).

    • Ah great! If I understand what he is saying correctly that is a very general and much stronger version of what I tried to say in the post (for example also dealing with the case where p is not nilpotent on the base). This probably means that I looked at that appendix at some time in the past and then I forgot having seen it. Thanks!

Comments are closed.