Lemma 14.1.1.
For any morphism \(R \to S\) in \(\Ring_{\FF_p}\text{,}\) the map \(\phi_S\colon \Omega^i_{S/R} \to \Omega^i_{S/R}\) is zero for all \(i \gt 0\text{.}\)
Section 14 Hodge-Tate comparison for crystalline prisms
Reference.
In this section, we prove the Hodge-Tate comparison theorem (Theorem 12.4.1) in the special case where the base prism \((A,I)\) is crystalline (meaning that \(I = (p)\)) and the ring \(R\) is a polynomial ring over \(\overline{A} = A/p\text{.}\) This simultaneously shows off some key ideas and provides a crucial base case for the general argument.
Subsection 14.1 de Rham cohomology in characteristic \(p\)
We first recall how de Rham cohomology works in characteristic \(p\text{,}\) focusing on the key case of an affine space (polynomial ring). The key point is that even in this case the cohomology is quite large, and in fact the cohomology groups reflect the structure of the original complex via the Cartier isomorphism.
Proof.
For any \(x \in S\text{,}\) we have
\begin{equation*}
\phi_S(dx) = d\phi_S(x) = d(x^p) = px^{p-1}\,dx = 0
\end{equation*}
because \(p=0\) in \(S\text{.}\) This proves the claim for \(i=1\text{,}\) from which the rest follows at once.
Definition 14.1.2.
For \(R \to S\) a morphism in \(\Ring_{\FF_p}\text{,}\) the map \(\phi_S\colon S \to S\) factors through an \(R\)-linear map \(\phi_{S/R}\colon S^{(1)} \to S\) where \(S^{(1)} = S \otimes_{R,\phi_R} R\text{.}\) We call \(\phi_{S/R}\) the relative Frobenius for the map \(R \to S\text{.}\)
In geometric language, \(\phi_{S/R}\) is the linearization of \(\phi^*_S\) over \(\Spec R\text{,}\) obtained by factoring \(\phi_S^*\) through a fiber product. See Figure 14.1.3.
Corollary 14.1.4.
For any morphism \(R \to S\) in \(\Ring_{\FF_p}\text{,}\) the map \(\phi_{S/R}\colon \Omega^i_{S^{(1)}/R} \to \Omega^i_{S/R}\) is zero for all \(i \gt 0\text{.}\)
Proof.
It suffices to check the claim for \(i = 1\text{.}\) Moreover, we may assume \(S = R[x_1,\dots,x_n]\text{,}\) as we may then take colimits to deduce the case where \(S\) is a polynomial ring in any number of variables, and then take quotients to deduce the case where \(S\) is arbitrary.
When \(S = R[x_1,\dots,x_n]\text{,}\) we may identify \(S^{(1)}\) with a second copy of \(R[x_1,\dots,x_n]\) with the map \(S^{(1)} \to S\) being given by the \(R\)-linear substitution \(x_i \mapsto x_i^p\text{.}\) In particular, \(\phi_{S/R}(dx_i) = d(x_i^p) = 0\) as per Lemma 14.1.1.
Remark 14.1.5.
As indicated in Figure 14.1.3, the construction of relative Frobenius extends to an arbitrary morphism of schemes \(f\colon Y \to X\) in characteristic \(p\text{.}\) The example of a polynomial ring, and its description in the proof of Corollary 14.1.4, may be misleading: in general \(Y\) and \(Y^{(1)}\) will not be isomorphic over \(X\text{.}\) For example, if \(X\) is the spectrum of an algebraically closed field \(k\) and \(Y\) is an elliptic curve over \(k\text{,}\) then the \(j\)-invariants \(j(Y)\) and \(j(Y^{(1)})\) will differ in general (the latter being the image of the former under \(\phi_k\)).
Lemma 14.1.6. Cartier isomorphism for affine space.
Choose \(R \in \Ring_{\FF_p}\text{,}\) put \(S = R[x_1,\dots,x_r]\text{,}\) and let \(\phi_{S/R}\colon S^{(1)} \to S\) be the relative Frobenius. Then there is a quasi-isomorphism
\begin{equation*}
(\Omega^\bullet_{S^{(1)}/R}, 0) \to (\Omega^\bullet_{S/R}, d_{\dR})
\end{equation*}
of \(S^{(1)}\)-dga's acting as \(\phi_{S/R}\) in degree \(0\) and taking \(dx_j\) to \(x_j^{p-1} dx_j\text{.}\)
Proof.
The map \(\phi_{S/R}\) induces a morphism of complexes thanks to Corollary 14.1.4. To check that this map is a quasi-isomorphism, we form a decomposition
\begin{equation}
\Omega^i_{S/R} = \bigoplus_{e_1,\dots,e_r \in \{0,\dots,p-1\}} \bigoplus_{1 \leq j_1 \lt \cdots \lt j_i \leq r} x_1^{e_1} \cdots x_r^{e_r}
S^{(1)} dx_{j_1} \wedge \cdots \wedge dx_{j_i}.\tag{14.1}
\end{equation}
We obtain a morphism \((\Omega^\bullet_{S/R}, d_{\dR}) \to (\Omega^\bullet_{S^{(1)}/R}, 0)\) of complexes (not respecting the multiplicative structure) taking \(x_1^{e_1} \cdots x_r^{e_r} dx_{j_1} \wedge \cdots \wedge dx_{j_i}\) to \(dx_{j_1} \wedge \cdots \wedge dx_{j_i}\) for
\begin{equation*}
e_j = \begin{cases} p-1 & j \in \{j_1,\dots,j_i\} \\ 0 & j \notin \{j_1,\dots,j_i\} \end{cases}.
\end{equation*}
We must show that the composition of these maps is homotopic to the identity on \(\Omega^\bullet_{S/R}\text{.}\) By proceeding by induction, we may reduce to the case \(r=1\text{.}\) In this case, for \(e_1 = 1,\dots,p-1\text{,}\) \(d_{\dR}\) maps \(x_1^{e_1} S^{(1)}\) to \(x_1^{e_1-1} S^{(1)}\,dx_1\) taking \(x_1^{e_1} f\) to \(e_1 x_1^{e_1-1} f\,dx\text{,}\) and this map is evidently invertible.
Remark 14.1.7.
While the Cartier map described in Lemma 14.1.6 is defined in terms of coordinates on the polynomial ring \(S\text{,}\) the construction is canonical up to homotopy in that the resulting map in \(D(S^{(1)})\) is well-defined independently of the way that \(S\) is expresesd as a polynomial ring. For example, making the change of variables \(x_1 \mapsto x_1 + x_2\) does not change the map because
\begin{equation*}
(x_1+x_2)^{p-1}\,d(x_1 + x_2) - x_1^{p-1}\,dx_1 - x_2^{p-2} \,dx_2 = \sum_{i=1}^{p-2} \binom{p-1}{i} x_1^i x_2^{p_1-i} (dx_1 + dx_2)
\end{equation*}
contributes only to summands in (14.1) that get killed off by the homotopy.
Yet another construction can be given (for \(R = \FF_p\text{,}\) then deducing the general case by base change) by lifting from \(\FF_p[x_1,\dots,x_r]\) to \(\ZZ_p[x_1,\dots,x_r]\text{.}\) Given an element \(f \in \ZZ_p[x_1^p,\dots,x_r^p]\) lifting \(\overline{f} \in \FF_p[x_1^p,\dots,x_r^p] \cong S^{(1)}\text{,}\) the element \(p^{-1} df\) reduces to an element of \(\Omega^1_{S/R}\) independent of the choice of \(f\text{,}\) and this is the image of \(d\overline{f}\) under the Cartier map.
This last construction is quite similar to how the Cartier isomorphism will appear in the proof of the Hodge-Tate comparison for crystalline prisms (Proposition 14.4.12). In fact that result will itself establish the canonicality of the Cartier isomorphism, so we don't need to worry much about it right now.
In any case, once canonicality is established by some means, we can easily promote Lemma 14.1.6 to a similar statement for any smooth morphism \(R \to S\) in \(\Ring_{\FF_p}\text{.}\) We omit the details here, as we will see the same argument again soon (Lemma 15.1.2).
Subsection 14.2 Divided powers
We next recall an algebraic construction that will help us study de Rham cohomology in mixed characteristic. See [15], section 3 for a detailed development, which also covers cases where the ring can have \(\ZZ\)-torsion.
Definition 14.2.1.
For \(R \in \Ring\) flat over \(\ZZ\text{,}\) the divided power operations \(\gamma_n\colon R \to R \otimes_{\ZZ} \QQ\) are the maps
\begin{equation*}
\gamma_n(x) = \frac{x^n}{n!} \qquad (x \in R, n \geq 0).
\end{equation*}
From the identities
\begin{equation*}
\gamma_n(x+y) = \sum_{i=0}^n \gamma_i(x) \gamma_{n-i}(y), \qquad \gamma_n(xy) = x^n \gamma_n(y),
\end{equation*}
we see that the set of \(x \in R\) for which \(\gamma_n(x) \in R\) for all \(n \geq 0\) is an ideal of \(R\text{.}\) For \(J\) an ideal contained in this ideal, we say that \(R\) admits divided powers on \(J\text{.}\)
Example 14.2.2.
The ring \(\ZZ_{(p)}\) admits divided powers on \((p)\) because
\begin{equation}
\frac{p^{n-1}}{n!} \in \ZZ_{(p)} \qquad (n \geq 1).\tag{14.2}
\end{equation}
Remark 14.2.3.
Our definition of “admits divided powers” is not quite the usual one: normally one also requires that \(\gamma_n\) maps \(J\) into \(J\) for each \(n \geq 1\text{.}\) For example, this occurs in Example 14.2.2 because (14.2) includes \(p^{n-1}\) rather than \(p^n\text{.}\) However, the last statement in Definition 14.2.4 ensures that this discrepancy doesn't affect anything later.
Definition 14.2.4.
The divided power envelope of \((R, J)\) is the subring \(D\) of \(R \otimes_{\ZZ} \QQ\) generated by \(R\) and \(\gamma_n(x)\) for all \(x \in J\text{.}\) Using the identities
\begin{gather}
\gamma_m(x) \gamma_n(x) = \binom{m+n}{n} \gamma_{m+n}(x)\tag{14.3}\\
\gamma_m(\gamma_n(x)) = \frac{(mn)!}{m!(n!)^m} \gamma_{mn}(x)\tag{14.4}
\end{gather}
and the fact that \((mn)!/(m!(n!)^m) \in \ZZ\) (it counts unordered partitions of \(\{1,\dots,mn\}\) into \(n\)-element subsets), we see that \(D\) admits divided powers on the ideal generated by \(\gamma_n(J)\) for all \(n \geq 1\) (even in the stronger sense of Remark 14.2.3).
Remark 14.2.5.
When studying divided powers, it is common to use the initialism pd for the French phrase puissances divisées. For example, the divided power envelope is also called the pd-envelope.
One key motivation for introducing divided powers is to formulate the Poincaré lemma.
Proposition 14.2.6.
Suppose that \(A \in \Ring\) is \(\ZZ\)-flat. Set \(P = A[x]\) and let \(D\) be the divided power envelope of \((P, (x))\text{.}\) Then the morphism
\begin{equation*}
d\colon D \to D \otimes_P \Omega^1_{P/A} = D\,dx
\end{equation*}
is surjective with kernel \(A\text{;}\) the same remains true if we replace \(D\) with its \(p\)-adic completion.
Proof.
Exercise (see Exercises 14.5).
Remark 14.2.7.
Proposition 14.2.6 amounts to the computation of the crystalline cohomology of a point. We will see in Subsection 14.4 that the proof of the Hodge-Tate comparison isomorphism for a crystalline prism naturally passes through crystalline cohomology.
One potentially confusing point is that unlike de Rham cohomology in characteristic 0, the crystalline cohomology of a higher-dimensional affine space is not the same as that of a point! In fact, the crystalline cohomology of \(\Spec \FF_p[x_1,\dots,x_r]\) is computed by the complex \(\widehat{\Omega}^{\bullet}_{P/\ZZ_p}\) where \(P = \ZZ_p[x_1,\dots,x_r]^\wedge_{(p)}\) (this ring already admits divided powers on \((p)\) by Example 14.2.2); taking the derived base change from \(\ZZ_p\) to \(\FF_p\) yields \(\Omega_{R/\FF_p}\text{,}\) whose cohomology we already know is quite large (Lemma 14.1.6). What Proposition 14.2.6 tells us is that the answer does not change if we include some extra “divided power variables”; see Proposition 14.2.8 for a concrete statement.
In any case, none of this has much meaning without an actual definition of crystalline cohomology itself. For that, see [15].
Proposition 14.2.8. Crystalline and de Rham cohomology of affine space.
Put \(R = \FF_p[x_1,\dots,x_r]\text{,}\) \(P_0 = \ZZ_p[x_1,\dots,x_r]\text{,}\) \(P = \ZZ_p[x_1,\dots,x_r,y_1,\dots,y_s]\text{,}\) and let \(D\) be the \(p\)-adic completion of the divided power envelope of \((P, (p,y_1,\dots,y_s))\text{.}\) Then there is a natural quasi-isomorphism
\begin{equation*}
\widehat{\Omega}^{\bullet}_{P_0/\ZZ_p} \cong D \widehat{\otimes}_P \widehat{\Omega}^{\bullet}_{P/\ZZ_p}
\end{equation*}
and hence a quasi-isomorphism
\begin{equation*}
\Omega_{R/\FF_p} \cong (D \widehat{\otimes}_P \widehat{\Omega}^{\bullet}_{P/\ZZ_p}) \otimes^L_{\ZZ_p} \FF_p.
\end{equation*}
Proof.
By repeated application of Proposition 14.2.6 we may reduce to the case \(s=0\text{,}\) in which case this is evident.
Subsection 14.3 Divided powers in \(\delta\)-rings
We next make a crucial link between \(\delta\)-rings and divided powers.
Remark 14.3.1.
If \(R\) is a \(\ZZ_{(p)}\)-algebra which admits a \(\delta\)-ring structure, then \(R\) admits divided powers on some ideal \(J\) if and only if \(\gamma_p(x) \in R\) for all \(x \in J\text{.}\) See Exercise 2.5.10.
Lemma 14.3.2.
For \(R = \ZZ_{(p)}\{x\}\) and \(J = xR\text{,}\) the map from \(R\) to the divided power envelope of \((R, J)\) promotes to a morphism of \(\delta\)-rings.
Proof.
Let \(D\) be the divided power envelope; it is the smallest subring of \(\ZZ_{(p)}\{x\}[p^{-1}]\) containing \(\ZZ_{(p)}\{x\}\) and \(\gamma_n(x)\) for all \(n \geq 1\text{.}\) The maximal ideal on which \(D\) admits divided powers includes both \(x\) (by construction) and \(p\) (by Example 14.2.2), and hence also \(\phi(x)\text{;}\) consequently, for all \(n \geq 1\text{,}\)
\begin{equation*}
\phi(\gamma_n(x)) = \gamma_n(\phi(x)) \in D.
\end{equation*}
Hence \(\phi\) induces an endomorphism of \(D\text{.}\)
We next check that \(\phi\) induces a Frobenius lift on \(D\text{;}\) this amounts to checking that for all \(n \geq 1\text{,}\)
\begin{equation*}
\phi(\gamma_n(x)) \equiv \gamma_n(x)^p \pmod{pD}.
\end{equation*}
We will see that in fact both sides are divisible by \(p\text{.}\) For \(\phi(\gamma_n(x)) = \gamma_n(\phi(x))\text{,}\) this holds by writing \(\phi(x) = p(x^p/p + \delta(x)) \in pD\) and invoking (14.2). For \(\gamma_n(x)^p\text{,}\) this holds by writing \(\gamma_n(x)^p = p! \gamma_p(\gamma_n(x))\) and applying (14.4).
Since \(D\) is \(p\)-torsion-free, by Lemma 2.1.3 we obtain a \(\delta\)-structure compatible with \(R\text{,}\) as desired.
Corollary 14.3.3.
In Lemma 14.3.2, the divided power envelope equals \(\ZZ_{(p)}\{x, \frac{\phi(x)}{p} \}\text{,}\) or more precisely the quotient of \(\ZZ_{(p)}\{x,z\}\) by the \(\delta\)-ideal generated by \(\phi(x) - pz\text{.}\)
Proof.
Let \(D\) be the divided power envelope and put \(D' = \ZZ_{(p)}\{x, \frac{\phi(x)}{p} \}\text{;}\) there is a natural map \(D' \to D[p^{-1}]\) which one checks is injective. Within \(D[p^{-1}]\text{,}\) we then have \(D' \subseteq D\) by Lemma 14.3.2 and \(D \subseteq D'\) by Remark 14.3.1. (Compare [25], Lemma 2.36.)
Corollary 14.3.4.
Let \(A \in \Ring_\delta\) be \(p\)-torsion-free. Choose \(f_1,\dots,f_r \in A\) which form a regular sequence in \(A/p\) and set \(I = (f_1,\dots,f_r)\text{.}\) Then the divided power envelope of \((A, I)\) is a \(\delta\)-ring, and can be written as \(A\{\phi(f_1)/p,\dots,\phi(f_r)/p\}\) (viewing the latter as a subring of \(A[p^{-1}]\)).
Proof.
By induction this reduces to the case \(r=1\text{,}\) in which case we write \(f\) for \(f_1\text{.}\) In this case, we may deduce the claim from Lemma 14.3.2 and Corollary 14.3.3 by base change. (Compare [25], Corollary 2.38.)
Remark 14.3.5.
Corollary 14.3.4 implies that the subring \(A\{\phi(f_1)/p,\dots,\phi(f_r)/p\}\) of \(A[p^{-1}]\) is independent of the choice of \(\delta\)-structure on \(A\text{,}\) as the characterization via the divided power envelope of \((A,I)\) makes no reference to \(\delta\) or \(\phi\text{.}\) By contrast, \(A\{f_1/p,\dots,f_r/p\}\) is not independent of this choice; see [25], Warning 2.40.
Remark 14.3.6.
Lemma 14.3.2 asserts that the divided power envelope of \((R, J)\) as a ring is also the divided power envelope as a \(\delta\)-ring. The corresponding statement for \(\lambda\)-rings is false: for \(A\) the free \(\lambda\)-ring on \(x\) (over \(\ZZ\)), the divided power envelope of \((A, (x))\) is not a \(\lambda\)-subring of \(A \otimes_{\ZZ} \QQ\text{.}\) The issue here is with the use of Example 14.2.2: the ring \(\ZZ\) does not admit divided powers on \((p)\) for any prime \(p\text{.}\)
This then leads to the question of describing the smallest \(\lambda\)-subring of \(A \otimes_{\ZZ} \QQ\) containing \(A\) which admits divided powers on \((x)\text{.}\) We do not know the answer, but as a partial result we note that this ring contains the elements \(\delta_p(x)^n/m\) where \(p\) is a prime, \(n\) is a positive integer, and \(m\) is the prime-to-\(p\) factor of \(n!\text{.}\)
A related question is whether Remark 14.3.5 admits a \(\lambda\)-ring analogue. That is, if \(A \in \Ring_\lambda\) is \(\ZZ\)-flat and \(f_1,\dots,f_r \in A\) form a regular sequence in \(A/p\) for every prime \(p\text{,}\) does the minimal \(\lambda\)-subring of \(A \otimes_{\ZZ} \QQ\) containing \(A\) which admits divided powers on \((f_1,\dots,f_r)\) depend only on the underlying ring structure of \(A\) and not its \(\lambda\)-structure?
Subsection 14.4 Prismatic cohomology for a crystalline prism
We next use divided powers to explicitly compute the cohomology of affine space over a crystalline prism. To begin with, we make the construction of weakly final objects of the prismatic site (Proposition 11.6.5) more explicit in some cases of interest.
Lemma 14.4.1.
For \(P\) a polynomial ring over \(\ZZ_p\text{,}\) for every \(i \gt 0\text{,}\) the complex
\begin{equation*}
\Omega^i_P \to \Omega^i_{P \otimes P} \to \Omega^i_{P \otimes P \otimes P} \to \cdots
\end{equation*}
vanishes in the homotopy category \(K(\ZZ_p)\text{.}\) (More precisely, this is witnessed by a homotopy at the level of \(P^\bullet\)-cosimplicial modules; see Definition 16.2.5 for the meaning of this statement.)
Proof.
We may reduce to the case of a polynomial ring in finitely many variables by taking colimits. We may further reduce the case \(i = 1\) using exterior powers. We may further reduce to the case \(P = \ZZ_p[x]\) using base change and induction on the number of variables. At this point, we can write down the homotopy \(h\) explicitly: if we write the \((n+1)\)-fold tensor product of \(P\) as \(P^n = \ZZ[x_{n0}, \dots, x_{nn}]\text{,}\) then the homotopy carries \(\Omega^1_{P^n}\) to \(\Omega_{P^{n-1}}\) taking \(dx_{ni}\) to \(dx_{(n-1)i}\) for \(i=0,\dots,n-1\) and to 0 for \(i=n\text{.}\) We leave it to the reader to check that \(h\) is a homotopy for the identity map (Exercise 14.5.2). (Compare [19], Example 2.16.)
Lemma 14.4.2.
Let \((A,I)\) be the prism \((\ZZ_p, (p))\) and put \(R = \FF_p[x_1,\dots,x_r]\text{.}\) Let \(P\) be the classical \(p\)-completion of \(\ZZ_p\{x_1,\dots,x_r\}\text{.}\) Let \(J\) be the kernel of the map \(P \to R\) taking \(x_i\) to \(x_i\) and \(\delta^m(x_i)\) to \(0\) for all \(m \gt 0\text{.}\) Write \(P\{J/p\}^\wedge_{(p)}\) for the classical \(p\)-completion of \(P\{f/p\colon f \in J\}\text{.}\) Then \((P\{J/p\}^\wedge_{(p)}, (p))\) is a weakly final object of \((R/A)_{\Prism}\text{.}\)
Proof.
By Exercise 2.5.8, \((P, J)\) is a \(\delta\)-pair. As in the proof of Proposition 11.6.5. we may apply Lemma 11.6.1 to \((P, J)\) to obtain a weakly final object of \((R/A)_{\Prism}\text{.}\) To identify the result explicitly, we step through the proof of Lemma 11.6.1. We first take the derived \(p\)-completion of \(P\{f/p\colon f \in J\}\text{;}\) as this object is \(p\)-adically separated this is in fact a classical \(p\)-completion. In addition the result is \(p\)-torsion-free, so there is no need to iterate the construction. (Compare [18], Lecture VI, Corollary 2.3.)
Remark 14.4.3.
Note that \(P^\bullet\) is itself a Čech-Alexander complex, namely the one associated to the covering \(\Spf P \to \Spf A\) in the category of \(p\)-adic formal schemes. In particular, \(A \to P^\bullet\) is an isomorphism in \(K(A)\) by the Čech-Alexander construction in the category of \(p\)-formal schemes (and even a homotopy equivalence of cosimplicial \(A\)-algebras, as per Definition 16.2.5).
Corollary 14.4.4.
With notation as in Lemma 14.4.2, for \(n \geq 0\text{,}\) identify
\begin{equation*}
P^n = \ZZ_p\{x_{ij}\colon i=1,\dots,r; j=0,\dots,n\}^\wedge_{(p)}
\end{equation*}
with the \((n+1)\)-fold completed tensor product of \(P\) over \(\ZZ_p\text{.}\) Let \(J^n\) be the kernel of the morphism \(P^n \to R\) in \(\Ring_\delta\) taking \(x_{ij}\) to \(x_i\) and \(\delta^m(x_{ij})\) to \(0\) for all \(m \gt 0\text{,}\) and write \(P^n\{J^n/p\}^\wedge_{(p)}\) for the classical \(p\)-completion of \(P^n\{f/p\colon f \in J^n\}\text{.}\) Then \(\Prism_{R/A}\) is quasi-isomorphic to the Čech-Alexander complex
\begin{equation*}
0 \to P^0\{J^0/p\}^\wedge_{(p)} \to P^1\{J^1/p\}^\wedge_{(p)} \to P^2\{J^2/p\}^\wedge_{(p)} \to \cdots.
\end{equation*}
Proof.
By Lemma 14.4.2, \(P^0\{J^0/p\}^\wedge_{(p)}\) is a weakly final object of \((R/A)_{\Prism}\text{.}\) Now note that \((P^n\{J^n/p\}^\wedge_{(p)}, (p))\) is the \((n+1)\)-fold product of \((P^0\{J^0/p\}^\wedge_{(p)}, (p))\) in \(\Prism_{R/A}\text{.}\) Hence we are in the setting described in Remark 11.6.7.
Remark 14.4.5.
Note that while \(J^0\) is generated by \(\delta^m(x_{i})\) for all \(m \gt 0\text{,}\) \(J^n\) is not generated by \(\delta^m(x_{ij})\) for all \(m \gt 0\text{;}\) we must also add the generators \(x_{ij} - x_{ij'}\) for \(j \neq j'\text{.}\)
Remark 14.4.6.
With notation as in Lemma 14.4.2 and Corollary 14.4.4, the map \(\phi\) on \(A = \ZZ_p\) is an isomorphism. By Remark 14.4.3, \(\phi_{P*}\colon P^\bullet \to P^\bullet\) is a quasi-isomorphism (and a homotopy equivalence), yielding isomorphisms in \(K(\ZZ_p)\) of the form
\begin{equation*}
P^\bullet\{J^\bullet/p\}^\wedge_{(p)} \to \phi_{P^\bullet}^*(P^\bullet\{J^\bullet/p\})^\wedge_{(p)} = P^\bullet\{\phi(J^\bullet)/p\}^\wedge_{(p)}.
\end{equation*}
By Corollary 14.3.4, the latter coincides with the \(p\)-completed divided power envelope \(D_{J^\bullet}(P^\bullet)\) of \((P^\bullet, J^\bullet)\text{.}\) (More precisely, these are homotopy equivalences of cosimplicial \(\ZZ_p\)-algebras; see again Definition 16.2.5.)
To summarize, the rows of Figure 14.4.7 are quasi-isomorphic to each other and to \(\Prism_{R/A}\text{.}\)
Lemma 14.4.8.
Let \((A,I)\) be the prism \((\ZZ_p, (p))\) and put \(R = \FF_p[x_1,\dots,x_r]\text{.}\) Let \((P, IP)\) be the weakly final object of \((R/A)_{\Prism}\) given by Lemma 14.4.2. Then 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.
Proof.
We can compute the cohomology of the total complex using the “first” spectral sequence, in which we first compute the cohomology of the columns. In this case, \(H^m(D_{J^n}(P^n) \widehat{\otimes}_{P^n} \widehat{\Omega}^\bullet_{P^n/\ZZ_p})\) is independent of \(n\text{:}\) by Proposition 14.2.8 each column computes the crystalline cohomology of \(\Spec \FF_p[x_1,\dots,x_r]\text{,}\) and for this identification each map \(P^{n-1} \to P^n\) represents the identity map on cohomology. Hence the horizontal differentials between the columns (which are alternating sums of the induced maps) are represented by
\begin{equation*}
H^m \stackrel{0}{\to} H^m \stackrel{1}{\to} H^m \stackrel{0}{\to} H^m \stackrel{1}{\to} \cdots.
\end{equation*}
At the next page of the spectral sequence, we end up with the groups \(H^m(D_{J^0}(P^0) \widehat{\otimes}_{P^0} \widehat{\Omega}^\bullet_{P^0/\ZZ_p})\) in column 0 and zeroes elsewhere. By Corollary 13.3.8, the map from the first column to the totalization is a quasi-isomorphism. (Compare Example 16.2.4 for a similar phenomenon.)
Meanwhile, by Lemma 14.4.1, each row except the first is homotopic to zero. Consequently, by Corollary 13.3.8 again, the natural map from the first row to the totalization is also a quasi-isomorphism. (Compare [19], Theorem 2.12.)
Corollary 14.4.10.
Let \((A,I)\) be the prism \((\ZZ_p, (p))\) and put \(R = \FF_p[x_1,\dots,x_r]\text{.}\) Then \(\phi^* \Prism_{R/A}\) is quasi-isomorphic to the crystalline cohomology of the affine space \(\Spec \FF_p[x_1,\dots,x_r]\) in the sense of Remark 14.2.7, i.e., to \(\widehat{\Omega}^\bullet_{P/\ZZ_p}\) for \(P = \ZZ_p[x_1,\dots,x_r]^\wedge_{(p)}\text{.}\)
Proof.
By Remark 14.4.6, we obtain a quasi-isomorphism of \(\phi^* \Prism_{R/A}\) with the first row of Figure 14.4.9. By Lemma 14.4.8, this is in turn quasi-isomorphic to the left column of Figure 14.4.9. By Proposition 14.2.8, the latter is quasi-isomorphic to \(\widehat{\Omega}^\bullet_{P/\ZZ_p}\) for \(P = \ZZ_p[x_1,\dots,x_r]^\wedge_{(p)}\) (note that this amounts to the same use of the Poincaré lemma as was needed to compare columns in Lemma 14.4.8).
Remark 14.4.11.
In Corollary 14.4.10, we write \(\phi^*\Prism_{R/A}\) instead of \(\Prism_{R/A}\) to keep track of the fact that prismatic cohomology computes not crystalline cohomology per se, but rather a canonical Frobenius descent of it.
Proposition 14.4.12.
Let \((A,I)\) be the prism \((\ZZ_p, (p))\) and put \(R = \FF_p[x_1,\dots,x_r]\text{.}\) Then the Hodge-Tate comparison map (12.1) is an isomorphism.
Proof.
By Lemma 14.4.2, we may compute the object \(\Prism_{R/A} \in D(A)\) using the Čech-Alexander complex associated to the weakly final object \((P, IP)\) as described in Corollary 14.4.4, and then obtain \(\overline{\Prism}_{R/A} \in D(\overline{A})\) by applying \(\otimes^L_A A/p\text{.}\) By Remark 14.4.6, the object \(\Prism_{R/A}\) (or more correctly \(\phi^*\Prism_{R/A}\)) is represented by the top row of the double complex Figure 14.4.9, which by Lemma 14.4.8 is isomorphic in \(D(A)\) to the first column of the double complex. By Corollary 14.4.10, that column computes the crystalline cohomology of affine space. By applying \(\otimes^L_{\ZZ_p} \FF_p\text{,}\) we obtain an isomorphism
\begin{equation}
\phi^* \overline{\Prism}_{R/A} \cong (\Omega^\bullet_{R/\FF_p}, d_{\dR})\tag{14.5}
\end{equation}
of \(\FF_p\)-dga's which in degree \(0\) is the identity map on \(R\text{.}\)
To check that \(\eta_R\) is an isomorphism, it will suffice to deduce from (14.5) that \(\phi^*(\eta_R)\) corresponds to the Cartier isomorphism; it suffices to do this in degree 1. As in Definition 12.3.1, consider the exact sequence
\begin{equation*}
0 \to p T / p^2 T\to T / p^2 T \to T / p T \to 0, \qquad T = P^0\{J^0/p\}^\wedge_{(p)}.
\end{equation*}
Viewing the element \(x_i \in P^0\{J^0/p\}^\wedge_{(p)}\) as representing a class in \(H^0(\overline{\Prism}_{R/A})\text{,}\) we find that its image under the Bockstein differential is represented by \((x_{i0} - x_{i1})/p\text{.}\) This is then the image of \(dx_i \in \Omega_{R/\FF_p}\) under \(\eta_R\text{,}\) and it remains to transfer the answer via (14.5).
Applying \(\phi\) to \((x_{i0} - x_{i1})/p\) yields \((x_{i0}^p - x_{i1}^p)/p \in D_{J^1}(P^1)\text{.}\) Going down the vertical arrow \(D_{J^1}(P^1) \to D_{J^1}(P^1) \widehat{\otimes}_{P^1} \widehat{\Omega}^1_{P^1/\ZZ_p}\) in Figure 14.4.9 yields
\begin{equation*}
d((x_{i0}^p - x_{i1}^p)/p) = x_{i0}^{p-1} dx_{i0} - x_{i1}^{p-1} dx_{i1}.
\end{equation*}
This is the image of \(x_i^{p-1}\,dx_i\) along the horizontal arrow \(D_{J^0}(P^0) \widehat{\otimes}_{P^0} \widehat{\Omega}^1_{P^0/\ZZ_p} \to D_{J^1}(P^1) \widehat{\otimes}_{P^1} \widehat{\Omega}^1_{P^1/\ZZ_p}\) in Figure 14.4.9. When we reduce mod \(p\text{,}\) we get exactly the image of \(dx_i\) under the Cartier map, proving the claim.
Exercises 14.5 Exercises
1.
Prove Proposition 14.2.6.
2.
Complete the proof of Lemma 14.4.1 by confirming that \(h\) is indeed a homotopy for the identity map.
