[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 \(q\)-analogue of crystalline cohomology, derive a comparison with prismatic cohomology, and use this to establish a statement about the functoriality of \(q\)-de Rham cohomology after \(p\)-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 \(\ZZ_p\)-scheme. This is a primary motivation for seeking a global analogue (Section 29).
To simplify the presentation, we only consider \(q\)-crystalline cohomology relative to \(\ZZ_p\text{.}\) A more general relative setup is described in [25].
Subsection27.1\(q\)-divided powers
In order to discuss \(q\)-crystalline cohomology, we first need to define a \(q\)-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 \(\delta\)-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.
Definition27.1.1.
Throughout this section, view \(A = \ZZ_p\llbracket q-1 \rrbracket\) as a \(\delta\)-ring in which \(q\) is constant, and identify \(A/([p]_q)\) with \(\ZZ_p[\zeta_p]\) via the map taking \(q\) to \(\zeta_p\text{.}\) We will use frequently the fact that the ideals \((p,q-1)\) and \((p, [p]_q)\) of \(A\text{,}\) although distinct, do define the same topology on \(A\text{;}\) keep in mind that \((A, ([p]_q))\) is a prism but \((A, (q-1))\) is not.
We will also use on several occasions the congruence
For ease of notation we treat only the case \(r=1\text{,}\) identifying \(x_1\) with \(x\text{.}\) 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 \(A \to A'\) is faithfully flat; we are thus reduced to checking that \(A' \to C'\) is \((p,[p]_q)\)-completely flat. This can be checked by inspection: it is clear that \(A' \to A'\{\phi(x/[p]_q)\} \cong A'\{y\}\) is faithfully flat, and the quotient of \(C'\) by the completion of \(A'\)-submodule is itself the completion of a free module on the basis consisting of products of the form \(x^{e_0} \delta(x)^{e_1} \delta^2(x)^{e_2} \cdots\text{,}\) where \(e_0, e_1, \ldots \in \{0,\dots,p-1\}\) 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 \(\delta\)-ring flat over \(\ZZ_{(p)}\) admits divided powers on an ideal if and only if \(\gamma_p(x) = x^p/p!\) sends the ideal into the ring (Remark 14.3.1).
With this in mind, for \(D\) a \([p]_q\)-torsion-free \(\delta\)-ring over \(A\text{,}\) for any \(x \in D\) with \(\phi(x) \in [p]_qD\text{,}\) write
\begin{equation*}
\gamma(x) = \frac{\phi(x)}{[p]_q} - \delta(x) \in D.
\end{equation*}
Remark27.1.6.
With notation as in Definition 27.1.5, for \(x,y \in \phi^{-1}([p]_qD)\) we have
is itself an ideal of \(D\text{;}\) hence to check that \(J = I\text{,}\) it suffices to check that \(J\) contains a generating set of \(I\text{.}\) (Compare [25], Remark 16.6.)
We need to show that if \(x \in D\) and \(\phi(x) \in [p]_qD\text{,}\) then \(\phi(\gamma(x)) \in [p]_q D\text{.}\) It will suffice to check this in the universal case \(D = A \{ x, \phi(x)/[p]_q\}^\wedge_{(p,[p]_q)}\text{.}\)
Since \(D/[p]_q\) is \(p\)-torsion-free (by Lemma 27.1.3), to show that \(\phi(\gamma(x)) \in [p]_q D\text{,}\) it will suffice to show that \(p \phi(\gamma(x)) \in [p]_q D\text{.}\) Moreover, by (27.1) we may replace \(p\) with \(\phi([p]_q)\) 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.)
Subsection27.2\(q\)-divided power pairs and envelopes
We now define the \(q\)-analogue of divided power envelopes.
Definition27.2.1.
A \(q\)-pd pair is a pair \((D,I)\) in which \(D\) is a \(\delta\)-ring over \(A\) and \(I\) is an ideal of \(D\) satisfying the following conditions.
The rings \(D\) and \(D/I\) are derived \((p, [p]_q)\)-complete.
The ideal \(I\) contains \(q-1\) and satisfies \(\phi(I) \subseteq [p]_q D\) (so that \(\gamma\) is defined on \(I\)) and \(\gamma(I) \subseteq I\text{.}\)
The ring \(D\) is \([p]_q\)-torsion-free and the quotient \(D/[p]_q\) has bounded \(p\)-power torsion. Consequently, \((D, [p]_q))\) is a bounded prism over \((A, [p]_q)\text{.}\)
The ring \(D/(q-1)\) is \(p\)-torsion-free with finite \((p,[p]_q)\)-complete Tor amplitude over \(D\text{.}\)
These form a category in which a morphism \((D,I) \to (D',I')\) is a morphism \(D \to D'\) of \(\delta\)-rings carrying \(I\) into \(I'\text{.}\)
Example27.2.2.
The pair \((A, (q-1))\) is the initial object in the category of \(q\)-pd pairs. More generally, if \(D\) is a \(\delta\)-ring over \(A\) which is \((p,[p]_q)\)-completely flat over \(A\text{,}\) then \((D, (q-1))\) is a \(q\)-pd pair.
Example27.2.3.
Let \(B\) be a perfect \(\delta\)-ring over \(A\) which is derived \((p, [p])_q)\)-complete. Since \([p]_q\) is distinguished in \(B\text{,}\) it is a non-zerodivisor (Theorem 7.2.2). Then \((B, (\phi^{-1}([p]_q))\) is a \(q\)-pd pair.
Example27.2.4.
Let \(D\) be a \(p\)-torsion-free, \(p\)-complete \(\delta\)-ring over \(A\) in which \(q=1\text{.}\) Let \(I\) be an arbitrary ideal of \(D\text{.}\) Then by Remark 14.3.1, \((D,I)\) is a \(\delta\)-pd pair if and only if \(D\) admits divided powers on \(I\) in the (strong) classical sense, that is, the divided power operations carry \(I\) into \(I \subset D[p^{-1}]\text{.}\)
Proposition27.2.5.
Let \(P\) be a \(\delta\)-ring over \(A\) equipped with a surjection \(P \to R = \ZZ_p [ x_1,\dots,x_r]^\wedge_{(p)}\) with kernel \(J = (q-1,y_1,y_2,\dots)\) where each initial segment \(y_1,\dots,y_s\) is a regular sequence on \(P/(p,q-1)\text{.}\) Define the \(\delta\)-ring
\begin{equation*}
D = P\{\phi(y_1)/[p]_q, \dots, \phi(y_s)/[p]_q \}^\wedge_{(p,[p]_q)}.
\end{equation*}
Let \(I\) be the kernel of \(D \to D/(q-1) \to R\text{.}\)
The ring \(D\) is \((p,[p]_q)\)-completely flat over \(A\text{.}\)
The map \(P \to D\) induces an isomorphism \(P/J \cong D/I\text{.}\)
The ring \(D/(q-1)\) identifies with the \(p\)-completed divided power envelope of the map \(P/(q-1) \to R\text{.}\)
The map \((P, J) \to (D,I)\) of \(\delta\)-pairs is universal for the target being a \(q\)-pd pair.
Point (1) is contained in Lemma 27.1.3. Point (2) follows from Remark 27.1.6 (applied to the indicated generators of \(J\)). Point (3) comes from Corollary 14.3.4. Point (4) is straightforward. For the rest, see [18], Lecture XI, Proposition 1.6.
Subsection27.3Comparison with prismatic cohomology
We now reprise the comparison of prismatic and Hodge-Tate cohomology in the crystalline case (Section 14).
Lemma27.3.1.
For \(P = R[x_1,\dots,x_r]\text{,}\) let \(P^i\) be the \((i+1)\)-fold tensor product of \(P\) over \(R\text{,}\) viewed as a polynomial ring whose generators are the various images of \(x_1,\dots,x_r\text{.}\) For every \(i \gt 0\text{,}\) the complex
vanishes in the homotopy category \(K(R)\text{,}\) and similarly after \(p\)-adic completion. (More precisely, this is witnessed by a homotopy at the level of \(P^\bullet\)-cosimplicial modules.)
Put \(R = \ZZ_p[x_1,\dots,x_r]^\wedge_{(p)}\text{.}\) Put \(P = A\{x_1,\dots,x_r\}^\wedge_{(p,[p]_q)}\text{,}\) and form the map \(P \to R\) taking \(q\) to 1 and taking \(\delta^m(x_i)\) to \(0\) for \(m \geq 0\text{.}\) Let \(P^\bullet\) be the \((p, [p]_q)\)-completed Cech nerve of \(A \to P\text{;}\) let \(J^n\) be the kernel of \(P^n \to P \to R\) where the first map is multiplication. We view \(J^n\) as being generated by \(p\text{,}\) the differences between copies of \(x_i\) in different factors, and the images of copies of \(x_i\) under \(\delta^m\) for all \(m \gt 0\text{.}\)
Let \(D_{J^n,q}(P^n)\) be the \(q\)-divided power envelope of the \(\delta\)-pair \((P^n, J^n)\) as provided by Proposition 27.2.5. We refer to the Čech-Alexander complex \(D_{J^\bullet,q}(P^\bullet)\) as the \(q\)-crystalline cohomology of \(R\text{.}\)
By viewing \(P^n\) as the completion of a polynomial ring with generators being the various images of \(\delta^m(x_1),\dots,\delta^m(x_r)\text{,}\) we may define the framed completed \(q\)-de Rham complex \(q\widehat{\Omega}^\bullet_{P^n/R, \square}\text{.}\)
Remark27.3.3.
To motivate the terminology in Definition 27.3.2, we give the definition of the \(q\)-crystalline site of \(R\) (relative to \(A\)) following [25], Definition 16.12. We take the opposite category to the category of \(q\)-pd pairs \((D,I)\) over \((A, [p]_q)\) equipped with isomorphisms \(D/I \cong R\text{,}\) in which the morphisms are morphisms of \(q\)-pd pairs which respect the isomorphisms with \(R\text{.}\) By Proposition 27.2.5, \(D_{J^n,q}(P^n)\) 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 \((D,I) \mapsto D\) is computed by \(D_{J^\bullet,q}(P^\bullet)\text{.}\)
Lemma27.3.4.\(q\)-Poincaré lemma.
With notation as in Definition 27.3.2, for any \(n\text{,}\) each of the maps
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 \(q\)-analogue of the Cartier isomorphism.
Definition27.3.7.
For \(R\) an \(\ZZ_p\)-algebra, define \(R^{(1)} = R \otimes_{\ZZ_p} \ZZ_p[\zeta_p]\text{;}\) this will play the role of the “Frobenius twist” in this setting.
Let \((D,I)\) be a \(q\)-pd pair with \(D/I \cong R\text{.}\) By assumption, \(\phi(I) \subseteq [p]_q D\) and so we get an induced map \(R \cong D/I \to D/[p]_q\) which is linear over the Frobenius on \(A\text{.}\) Linearizing yields an \(A\)-algebra map \(A \otimes_{\phi,A} R \to D/[p]_q\text{,}\) which then factors through \((A \otimes_{\phi,A} R)/[p]_q \cong R^{(1)}\text{.}\)
The upshot of this is that for \(R = \ZZ_p[x_1,\dots,x_r]^\wedge_{(p)}\text{,}\) there is a morphism from the \(q\)-crystalline site of \(R\) to the prismatic site of \(R^{(1)}\) over \(A\text{,}\) and hence a morphism of cohomology in the other direction.
Theorem27.3.8.
Put \(R = \ZZ_p[x_1,\dots,x_r]^\wedge_{(p)}\) and \(R^{(1)} = R \otimes_{\ZZ_p} \ZZ_p[\zeta_p]\text{.}\) We then have an isomorphism
Using Definition 27.3.7, we obtain a morphism from \(\Prism_{R^{(1)}/A}\) to the \(q\)-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 \(q-1\text{,}\) 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 \(q\Omega^\bullet_{R/\ZZ_p,\square}\text{.}\) (Compare [18], Lecture XI, Theorem 2.5 or [25], Theorem 16.22.)
Corollary27.3.9.\(q\)-Hodge-Tate comparison.
Put \(R = \ZZ_p[x_1,\dots,x_r]^\wedge_{(p)}\text{.}\) Then there is a natural identification
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 \(q \Omega^\bullet_{R/\ZZ_p, \square}\) as a commutative ring object in \(D_{\comp}(A)\) functorially associated to \(\ZZ_p[x_1,\dots,x_r]\text{.}\) We can thus use left Kan extension (Proposition 16.4.6) to extend the definition of \(q \Omega^\bullet_{R/\ZZ_p, \square}\) to any derived \(p\)-complete \(\ZZ_p\)-algebra \(R\text{.}\)
Subsection27.4Frobenius is an isogeny
Remark27.4.1.
Put \(R = \ZZ_p[x]^\wedge_{(p)}\) and view \(R \llbracket q-1 \rrbracket \) as a \(\delta\)-ring in which \(q\) and \(x\) are constant. Then \(q\Omega^\bullet_{R/\ZZ_p,\square}\) carries an action of \(\phi\) given in degree \(0\) by \(f \mapsto \phi(f)\) and in degree \(1\) by \(g\,dx \mapsto \phi(g) x^{p-1} [p]_q\,dx\text{.}\) A similar statement applies to \(R = \ZZ_p[x_1,\dots,x_r]^\wedge_{(p)}\text{.}\)
It follows that for any derived \(p\)-complete \(\ZZ_p\)-algebra \(R\text{,}\) the linearized Frobenius \(\phi_R\colon \phi_A^* \Prism_{R^{(1)}/A} \to \Prism_{R^{(1)}/A}\) is an isogeny, in that it becomes an isomorphism after inverting \([p]_q\text{.}\) More precisely, because the action on cohomology in degree \(i\) factors through multiplication by \([p]_q^i\text{,}\) one can apply the Berthelot-Ogus functor \(\eta_{[p]_q}\) ([117], tag 0F7N); this is related to the discussion of the Nygaard filtration in [25].
Subsection27.5Étale localization
Remark27.5.1.
One can establish a form of étale localization (Lemma 15.1.2) in order to extend the preceding discussion to the case where \(R\) is a \(p\)-completely smooth \(\ZZ_p\)-algebra. In particular, for such rings the left Kan extension of Definition 27.3.10 can be computed by “naive” \(q\)-de Rham complexes using local coordinate choices.
