[18], lecture X; [25], section 16. Some of this material was developed independently in the PhD thesis of Masullo [95]. However, we diverge significantly in form from these references; see below.
In this section, we introduce a -analogue of crystalline cohomology, derive a comparison with prismatic cohomology, and use this to establish a statement about the functoriality of -de Rham cohomology after -completion. We follow closely our analysis of the Hodge-Tate comparison for crystalline prisms (Section 14).
We will only present the affine part of the story, but one can globalize to obtain the “Wach module cohomology” of a smooth proper -scheme. This is a primary motivation for seeking a global analogue (Section 29).
In order to discuss -crystalline cohomology, we first need to define a -analogue of divided powers. It is not at all clear how to do this in general, but fortunately for our purposes it is sufficient to do this for -rings. In that case, we can use the fact that divided powers can be accounted for using Frobenius (Remark 14.3.1) to come up with a suitable analogue.
Throughout this section, view as a -ring in which is constant, and identify with via the map taking to . We will use frequently the fact that the ideals and of , although distinct, do define the same topology on ; keep in mind that is a prism but is not.
For ease of notation we treat only the case , identifying with . Consider the diagram as in Figure 27.1.4, in which the first row is given and the squares below are pushouts.
Figure27.1.4.
By inspection, the arrow is faithfully flat; we are thus reduced to checking that is -completely flat. This can be checked by inspection: it is clear that is faithfully flat, and the quotient of by the completion of -submodule is itself the completion of a free module on the basis consisting of products of the form , where are almost all zero. (Compare [25], Proposition 3.13, which gives a more general result.)
Definition27.1.5.
Recall that in the ordinary divided power setting, a -ring flat over admits divided powers on an ideal if and only if sends the ideal into the ring (Remark 14.3.1).
We need to show that if and , then . It will suffice to check this in the universal case .
Since is -torsion-free (by Lemma 27.1.3), to show that , it will suffice to show that . Moreover, by (27.1) we may replace with on the left-hand side. At this point we may proceed by direct computation:
where we use (27.1) again in the last line. (Compare [25], Lemma 16.7.)
Let be a -torsion-free, -complete -ring over in which . Let be an arbitrary ideal of . Then by Remark 14.3.1, is a -pd pair if and only if admits divided powers on in the (strong) classical sense, that is, the divided power operations carry into .
Point (1) is contained in Lemma 27.1.3. Point (2) follows from Remark 27.1.6 (applied to the indicated generators of ). Point (3) comes from Corollary 14.3.4. Point (4) is straightforward. For the rest, see [18], Lecture XI, Proposition 1.6.
vanishes in the homotopy category , and similarly after -adic completion. (More precisely, this is witnessed by a homotopy at the level of -cosimplicial modules.)
Put . Put , and form the map taking to 1 and taking to for . Let be the -completed Cech nerve of ; let be the kernel of where the first map is multiplication. We view as being generated by , the differences between copies of in different factors, and the images of copies of under for all .
Let be the -divided power envelope of the -pair as provided by Proposition 27.2.5. We refer to the Čech-Alexander complex as the -crystalline cohomology of .
To motivate the terminology in Definition 27.3.2, we give the definition of the -crystalline site of (relative to ) following [25], Definition 16.12. We take the opposite category to the category of -pd pairs over equipped with isomorphisms , in which the morphisms are morphisms of -pd pairs which respect the isomorphisms with . By Proposition 27.2.5, is a weakly final object in this category. We use the indiscrete Grothendieck topology; then by Lemma 11.1.7 the cohomology of the sheaf is computed by .
With notation as in Definition 27.3.2, the totalization of the double complex displayed in Figure 14.4.9 is quasi-isomorphic to both its first row and its first column via the inclusion maps.
We make the same argument as in Lemma 14.4.8: each row is homotopic to zero by Lemma 27.3.1, and all of the simplicial maps induce quasi-isomorphisms of columns by Lemma 27.3.4, so Corollary 13.3.8 yields the desired quasi-isomorphism. (Compare [18], Theorem 2.9.)
To link up with prismatic cohomology, we need a -analogue of the Cartier isomorphism.
Let be a -pd pair with . By assumption, and so we get an induced map which is linear over the Frobenius on . Linearizing yields an -algebra map , which then factors through .
The upshot of this is that for , there is a morphism from the -crystalline site of to the prismatic site of over , and hence a morphism of cohomology in the other direction.
Using Definition 27.3.7, we obtain a morphism from to the -crystalline cohomology (the top row of Figure 27.3.6. To check that it is an isomorphism, we may invoke derived Nakayama (Proposition 6.6.2) to reduce modulo , at which point we get back to the corresponding statement in the case of a crystalline prism (Remark 14.4.6).
Using Lemma 27.3.5, we obtain a quasi-isomorphism between the top row of Figure 27.3.6 and the left column of the same diagram. Using Lemma 27.3.4, we get a quasi-isomorphism of the left column with . (Compare [18], Lecture XI, Theorem 2.5 or [25], Theorem 16.22.)
This can be read off from the proof of Theorem 27.3.8, or by combining that result with Proposition 14.4.12. (Compare [18], Lecture XI, Corollary 2.6.)
Definition27.3.10.
Theorem 27.3.8 gives us a way to regard as a commutative ring object in functorially associated to . We can thus use left Kan extension (Proposition 16.4.6) to extend the definition of to any derived -complete -algebra .
It follows that for any derived -complete -algebra , the linearized Frobenius is an isogeny, in that it becomes an isomorphism after inverting . More precisely, because the action on cohomology in degree factors through multiplication by , one can apply the Berthelot-Ogus functor ([117], tag 0F7N); this is related to the discussion of the Nygaard filtration in [25].
One can establish a form of étale localization (Lemma 15.1.2) in order to extend the preceding discussion to the case where is a -completely smooth -algebra. In particular, for such rings the left Kan extension of Definition 27.3.10 can be computed by “naive” -de Rham complexes using local coordinate choices.
