# The Stacks Project

## Tag 07MI

### 54.24. Some further results

In this section we mention some results whose proof is missing. We will formulate these as a series of remarks and we will convert them into actual lemmas and propositions only when we add detailed proofs.

Remark 54.24.1 (Higher direct images). Let $p$ be a prime number. Let $(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ be a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Let $$\xymatrix{ X \ar[r]_f \ar[d] & X' \ar[d] \\ S_0 \ar[r] & S'_0 }$$ be a commutative diagram of morphisms of schemes and assume $p$ is locally nilpotent on $X$ and $X'$. Let $\mathcal{F}$ be an $\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then $Rf_{\text{cris}, *}\mathcal{F}$ can be computed as follows.

Given an object $(U', T', \delta')$ of $\text{Cris}(X'/S')$ set $U = X \times_{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote $(T_0, T, \delta)$ the divided power scheme over $S$ such that $$\xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' }$$ is cartesian in the category of divided power schemes, see Lemma 54.7.4. There is an induced morphism $U \to T_0$ and we obtain a morphism $(U/T)_{\text{cris}} \to (X/S)_{\text{cris}}$, see Remark 54.9.3. Let $\mathcal{F}_U$ be the pullback of $\mathcal{F}$. Let $\tau_{U/T} : (U/T)_{\text{cris}} \to T_{Zar}$ be the structure morphism. Then we have $$\tag{54.24.1.1} \left(Rf_{\text{cris}, *}\mathcal{F}\right)_{T'} = R(T \to T')_*\left(R\tau_{U/T, *} \mathcal{F}_U \right)$$ where the left hand side is the restriction (see Section 54.10).

Hints: First, show that $\text{Cris}(U/T)$ is the localization (in the sense of Sites, Lemma 7.29.3) of $\text{Cris}(X/S)$ at the sheaf of sets $f_{\text{cris}}^{-1}h_{(U', T', \delta')}$. Next, reduce the statement to the case where $\mathcal{F}$ is an injective module and pushforward of modules using that the pullback of an injective $\mathcal{O}_{X/S}$-module is an injective $\mathcal{O}_{U/T}$-module on $\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward.

Remark 54.24.2 (Mayer-Vietoris). In the situation of Remark 54.24.1 suppose we have an open covering $X = X' \cup X''$. Denote $X''' = X' \cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$ to $X'$, $X''$, and $X'''$. Moreover, Let $\mathcal{F}'$, $\mathcal{F}''$, and $\mathcal{F}'''$ be the restriction of $\mathcal{F}$ to the crystalline sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle $$Rf_{\text{cris}, *}\mathcal{F} \longrightarrow Rf'_{\text{cris}, *}\mathcal{F}' \oplus Rf''_{\text{cris}, *}\mathcal{F}'' \longrightarrow Rf'''_{\text{cris}, *}\mathcal{F}''' \longrightarrow Rf_{\text{cris}, *}\mathcal{F}[1]$$ in $D(\mathcal{O}_{X'/S'})$.

Hints: This is a formal consequence of the fact that the subcategories $\text{Cris}(X'/S)$, $\text{Cris}(X''/S)$, $\text{Cris}(X'''/S)$ correspond to open subobjects of the final sheaf on $\text{Cris}(X/S)$ and that the last is the intersection of the first two.

Remark 54.24.3 (Čech complex). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a separated1 scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.

Choose an affine open covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$. Write $U_\lambda = \mathop{\rm Spec}(C_\lambda)$. Choose a polynomial algebra $P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$. Having fixed these choices we can construct a Čech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

Given $n \geq 0$ and $\lambda_0, \ldots, \lambda_n \in \Lambda$ write $U_{\lambda_0 \ldots \lambda_n} = U_{\lambda_0} \cap \ldots \cap U_{\lambda_n}$. This is an affine scheme by assumption. Write $U_{\lambda_0 \ldots \lambda_n} = \mathop{\rm Spec}(C_{\lambda_0 \ldots \lambda_n})$. Set $$P_{\lambda_0 \ldots \lambda_n} = P_{\lambda_0} \otimes_A \ldots \otimes_A P_{\lambda_n}$$ which comes with a canonical surjection onto $C_{\lambda_0 \ldots \lambda_n}$. Denote the kernel $J_{\lambda_0 \ldots \lambda_n}$ and set $D_{\lambda_0 \ldots \lambda_n}$ the $p$-adically completed divided power envelope of $J_{\lambda_0 \ldots \lambda_n}$ in $P_{\lambda_0 \ldots \lambda_n}$ relative to $\gamma$. Let $M_{\lambda_0 \ldots \lambda_n}$ be the $P_{\lambda_0 \ldots \lambda_n}$-module corresponding to the restriction of $\mathcal{F}$ to $\text{Cris}(U_{\lambda_0 \ldots \lambda_n}/S)$ via Proposition 54.17.4. By construction we obtain a cosimplicial divided power ring $D(*)$ having in degree $n$ the ring $$D(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} D_{\lambda_0 \ldots \lambda_n}$$ (use that divided power envelopes are functorial and the trivial cosimplicial structure on the ring $P(*)$ defined similarly). Since $M_{\lambda_0 \ldots \lambda_n}$ is the ''value'' of $\mathcal{F}$ on the objects $\mathop{\rm Spec}(D_{\lambda_0 \ldots \lambda_n})$ we see that $M(*)$ defined by the rule $$M(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} M_{\lambda_0 \ldots \lambda_n}$$ forms a cosimplicial $D(*)$-module. Now we claim that we have $$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = s(M(*))$$ Here $s(-)$ denotes the cochain complex associated to a cosimplicial module (see Simplicial, Section 14.25).

Hints: The proof of this is similar to the proof of Proposition 54.21.1 (in particular the result holds for any module satisfying the assumptions of that proposition).

Remark 54.24.4 (Alternating Čech complex). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a separated quasi-compact scheme over $S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.

Choose a finite affine open covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$ and a total ordering on $\Lambda$. Write $U_\lambda = \mathop{\rm Spec}(C_\lambda)$. Choose a polynomial algebra $P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$. Having fixed these choices we can construct an alternating Čech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

We are going to use the notation introduced in Remark 54.24.3. Denote $\Omega_{\lambda_0 \ldots \lambda_n}$ the $p$-adically completed module of differentials of $D_{\lambda_0 \ldots \lambda_n}$ over $A$ compatible with the divided power structure. Let $\nabla$ be the integrable connection on $M_{\lambda_0 \ldots \lambda_n}$ coming from Proposition 54.17.4. Consider the double complex $M^{\bullet, \bullet}$ with terms $$M^{n, m} = \bigoplus\nolimits_{\lambda_0 < \ldots < \lambda_n} M_{\lambda_0 \ldots \lambda_n} \otimes^\wedge_{D_{\lambda_0 \ldots \lambda_n}} \Omega^m_{D_{\lambda_0 \ldots \lambda_n}}.$$ For the differential $d_1$ (increasing $n$) we use the usual Čech differential and for the differential $d_2$ we use the connection, i.e., the differential of the de Rham complex. We claim that $$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = \text{Tot}(M^{\bullet, \bullet})$$ Here $\text{Tot}(-)$ denotes the total complex associated to a double complex, see Homology, Definition 12.22.3.

Hints: We have $$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet)$$ by Proposition 54.23.1. The right hand side of the formula is simply the alternating Čech complex for the covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$ (which induces an open covering of the final sheaf of $\text{Cris}(X/S)$) and the complex $\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet$, see Proposition 54.21.3. Now the result follows from a general result in cohomology on sites, namely that the alternating Čech complex computes the cohomology provided it gives the correct answer on all the pieces (insert future reference here).

Remark 54.24.5 (Quasi-coherence). In the situation of Remark 54.24.1 assume that $S \to S'$ is quasi-compact and quasi-separated and that $X \to S_0$ is quasi-compact and quasi-separated. Then for a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$ the sheaves $R^if_{\text{cris}, *}\mathcal{F}$ are locally quasi-coherent.

Hints: We have to show that the restrictions to $T'$ are quasi-coherent $\mathcal{O}_{T'}$-modules, where $(U', T', \delta')$ is any object of $\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine. We use the formula (54.24.1.1), the fact that $T \to T'$ is quasi-compact and quasi-separated (as $T$ is affine over the base change of $T'$ by $S \to S'$), and Cohomology of Schemes, Lemma 29.4.5 to see that it suffices to show that the sheaves $R^i\tau_{U/T, *}\mathcal{F}_U$ are quasi-coherent. Note that $U \to T_0$ is also quasi-compact and quasi-separated, see Schemes, Lemmas 25.21.15 and 25.21.15.

This reduces us to proving that $R^i\tau_{X/S, *}\mathcal{F}$ is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here $\tau_{X/S}$ is the structure morphism, see Remark 54.9.6. We may work locally on $S$, hence we may assume $S$ affine (see Lemma 54.9.5). Induction on the number of affines covering $X$ and Mayer-Vietoris (Remark 54.24.2) reduces the question to the case where $X$ is also affine (as in the proof of Cohomology of Schemes, Lemma 29.4.5). Say $X = \mathop{\rm Spec}(C)$ and $S = \mathop{\rm Spec}(A)$ so that $(A, I, \gamma)$ and $A \to C$ are as in Situation 54.5.1. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to C$ as in Section 54.17. Let $(M, \nabla)$ be the module corresponding to $\mathcal{F}$, see Proposition 54.17.4. Applying Proposition 54.21.3 we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is represented by $M \otimes_D \Omega_D^*$. Note that completion isn't necessary as $p$ is nilpotent in $A$! We have to show that this is compatible with taking principal opens in $S = \mathop{\rm Spec}(A)$. Suppose that $g \in A$. Then we conclude that similarly $R\Gamma(\text{Cris}(X_g/S_g), \mathcal{F})$ is computed by $M_g \otimes_{D_g} \Omega_{D_g}^*$ (again this uses that $p$-adic completion isn't necessary). Hence we conclude because localization is an exact functor on $A$-modules.

Remark 54.24.6 (Boundedness). In the situation of Remark 54.24.1 assume that $S \to S'$ is quasi-compact and quasi-separated and that $X \to S_0$ is of finite type and quasi-separated. Then there exists an integer $i_0$ such that for any crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$ we have $R^if_{\text{cris}, *}\mathcal{F} = 0$ for all $i > i_0$.

Hints: Arguing as in Remark 54.24.5 (using Cohomology of Schemes, Lemma 29.4.5) we reduce to proving that $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i \gg 0$ in the situation of Proposition 54.21.3 when $C$ is a finite type algebra over $A$. This is clear as we can choose a finite polynomial algebra and we see that $\Omega^i_D = 0$ for $i \gg 0$.

Remark 54.24.7 (Specific boundedness). In Situation 54.7.5 let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Assume that $S_0$ has a unique point and that $X \to S_0$ is of finite presentation.

1. If $\dim X = d$ and $X/S_0$ has embedding dimension $e$, then $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > d + e$.
2. If $X$ is separated and can be covered by $q$ affines, and $X/S_0$ has embedding dimension $e$, then $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > q + e$.

Hints: In case (1) we can use that $$H^i(\text{Cris}(X/S), \mathcal{F}) = H^i(X_{Zar}, Ru_{X/S, *}\mathcal{F})$$ and that $Ru_{X/S, *}\mathcal{F}$ is locally calculated by a de Rham complex constructed using an embedding of $X$ into a smooth scheme of dimension $e$ over $S$ (see Lemma 54.21.4). These de Rham complexes are zero in all degrees $>e$. Hence (1) follows from Cohomology, Proposition 20.21.7. In case (2) we use the alternating Čech complex (see Remark 54.24.4) to reduce to the case $X$ affine. In the affine case we prove the result using the de Rham complex associated to an embedding of $X$ into a smooth scheme of dimension $e$ over $S$ (it takes some work to construct such a thing).

Remark 54.24.8 (Base change map). In the situation of Remark 54.24.1 assume $S = \mathop{\rm Spec}(A)$ and $S' = \mathop{\rm Spec}(A')$ are affine. Let $\mathcal{F}'$ be an $\mathcal{O}_{X'/S'}$-module. Let $\mathcal{F}$ be the pullback of $\mathcal{F}'$. Then there is a canonical base change map $$L(S' \to S)^*R\tau_{X'/S', *}\mathcal{F}' \longrightarrow R\tau_{X/S, *}\mathcal{F}$$ where $\tau_{X/S}$ and $\tau_{X'/S'}$ are the structure morphisms, see Remark 54.9.6. On global sections this gives a base change map $$\tag{54.24.8.1} R\Gamma(\text{Cris}(X'/S'), \mathcal{F}') \otimes^\mathbf{L}_{A'} A \longrightarrow R\Gamma(\text{Cris}(X/S), \mathcal{F})$$ in $D(A)$.

Hint: Compose the very general base change map of Cohomology on Sites, Remark 21.20.3 with the canonical map $Lf_{\text{cris}}^*\mathcal{F}' \to f_{\text{cris}}^*\mathcal{F}' = \mathcal{F}$.

Remark 54.24.9 (Base change isomorphism). The map (54.24.8.1) is an isomorphism provided all of the following conditions are satisfied:

1. $p$ is nilpotent in $A'$,
2. $\mathcal{F}'$ is a crystal in quasi-coherent $\mathcal{O}_{X'/S'}$-modules,
3. $X' \to S'_0$ is a quasi-compact, quasi-separated morphism,
4. $X = X' \times_{S'_0} S_0$,
5. $\mathcal{F}'$ is a flat $\mathcal{O}_{X'/S'}$-module,
6. $X' \to S'_0$ is a local complete intersection morphism (see More on Morphisms, Definition 36.51.2; this holds for example if $X' \to S'_0$ is syntomic or smooth),
7. $X'$ and $S_0$ are Tor independent over $S'_0$ (see More on Algebra, Definition 15.57.1; this holds for example if either $S_0 \to S'_0$ or $X' \to S'_0$ is flat).

Hints: Condition (1) means that in the arguments below $p$-adic completion does nothing and can be ignored. Using condition (3) and Mayer Vietoris (see Remark 54.24.2) this reduces to the case where $X'$ is affine. In fact by condition (6), after shrinking further, we can assume that $X' = \mathop{\rm Spec}(C')$ and we are given a presentation $C' = A'/I'[x_1, \ldots, x_n]/(\bar f'_1, \ldots, \bar f'_c)$ where $\bar f'_1, \ldots, \bar f'_c$ is a Koszul-regular sequence in $A'/I'$. (This means that smooth locally $\bar f'_1, \ldots, \bar f'_c$ forms a regular sequence, see More on Algebra, Lemma 15.27.17.) We choose a lift of $\bar f'_i$ to an element $f'_i \in A'[x_1, \ldots, x_n]$. By (4) we see that $X = \mathop{\rm Spec}(C)$ with $C = A/I[x_1, \ldots, x_n]/(\bar f_1, \ldots, \bar f_c)$ where $f_i \in A[x_1, \ldots, x_n]$ is the image of $f'_i$. By property (7) we see that $\bar f_1, \ldots, \bar f_c$ is a Koszul-regular sequence in $A/I[x_1, \ldots, x_n]$. The divided power envelope of $I'A'[x_1, \ldots, x_n] + (f'_1, \ldots, f'_c)$ in $A'[x_1, \ldots, x_n]$ relative to $\gamma'$ is $$D' = A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c \rangle/(\xi_i - f'_i)$$ see Lemma 54.2.4. Then you check that $\xi_1 - f'_1, \ldots, \xi_n - f'_n$ is a Koszul-regular sequence in the ring $A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$. Similarly the divided power envelope of $IA[x_1, \ldots, x_n] + (f_1, \ldots, f_c)$ in $A[x_1, \ldots, x_n]$ relative to $\gamma$ is $$D = A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle/(\xi_i - f_i)$$ and $\xi_1 - f_1, \ldots, \xi_n - f_n$ is a Koszul-regular sequence in the ring $A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$. It follows that $D' \otimes_{A'}^\mathbf{L} A = D$. Condition (2) implies $\mathcal{F}'$ corresponds to a pair $(M', \nabla)$ consisting of a $D'$-module with connection, see Proposition 54.17.4. Then $M = M' \otimes_{D'} D$ corresponds to the pullback $\mathcal{F}$. By assumption (5) we see that $M'$ is a flat $D'$-module, hence $$M = M' \otimes_{D'} D = M' \otimes_{D'} D' \otimes_{A'}^\mathbf{L} A = M' \otimes_{A'}^\mathbf{L} A$$ Since the modules of differentials $\Omega_{D'}$ and $\Omega_D$ (as defined in Section 54.17) are free $D'$-modules on the same generators we see that $$M \otimes_D \Omega^\bullet_D = M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{D'} D = M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{A'}^\mathbf{L} A$$ which proves what we want by Proposition 54.21.3.

Remark 54.24.10 (Rlim). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ an algebra over $\mathbf{Z}_{(p)}$ with $p$ nilpotent in $A/I$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a scheme over $S_0$ with $p$ locally nilpotent on $X$. Let $\mathcal{F}$ be any $\mathcal{O}_{X/S}$-module. For $e \gg 0$ we have $(p^e) \subset I$ is preserved by $\gamma$, see Divided Power Algebra, Lemma 23.4.5. Set $S_e = \mathop{\rm Spec}(A/p^eA)$ for $e \gg 0$. Then $\text{Cris}(X/S_e)$ is a full subcategory of $\text{Cris}(X/S)$ and we denote $\mathcal{F}_e$ the restriction of $\mathcal{F}$ to $\text{Cris}(X/S_e)$. Then $$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\mathop{\rm lim}\nolimits_e R\Gamma(\text{Cris}(X/S_e), \mathcal{F}_e)$$

Hints: Suffices to prove this for $\mathcal{F}$ injective. In this case the sheaves $\mathcal{F}_e$ are injective modules too, the transition maps $\Gamma(\mathcal{F}_{e + 1}) \to \Gamma(\mathcal{F}_e)$ are surjective, and we have $\Gamma(\mathcal{F}) = \mathop{\rm lim}\nolimits_e \Gamma(\mathcal{F}_e)$ because any object of $\text{Cris}(X/S)$ is locally an object of one of the categories $\text{Cris}(X/S_e)$ by definition of $\text{Cris}(X/S)$.

Remark 54.24.11 (Comparison). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $Y$ be a smooth scheme over $S$ and set $X = Y \times_S S_0$. Let $\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Then

1. $\gamma$ extends to a divided power structure on the ideal of $X$ in $Y$ so that $(X, Y, \gamma)$ is an object of $\text{Cris}(X/S)$,
2. the restriction $\mathcal{F}_Y$ (see Section 54.10) comes endowed with a canonical integrable connection $\nabla : \mathcal{F}_Y \to \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S}$, and
3. we have $$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(Y, \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega^\bullet_{Y/S})$$ in $D(A)$.

Hints: See Divided Power Algebra, Lemma 23.4.2 for (1). See Lemma 54.15.1 for (2). For Part (3) note that there is a map, see (54.23.2.1). This map is an isomorphism when $X$ is affine, see Lemma 54.21.4. This shows that $Ru_{X/S, *}\mathcal{F}$ and $\mathcal{F}_Y \otimes \Omega^\bullet_{Y/S}$ are quasi-isomorphic as complexes on $Y_{Zar} = X_{Zar}$. Since $R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(X_{Zar}, Ru_{X/S, *}\mathcal{F})$ the result follows.

Remark 54.24.12 (Perfectness). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a proper smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules. Then $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is a perfect object of $D(A)$.

Hints: By Remark 54.24.9 we have $$R\Gamma(\text{Cris}(X/S), \mathcal{F}) \otimes_A^\mathbf{L} A/I \cong R\Gamma(\text{Cris}(X/S_0), \mathcal{F}|_{\text{Cris}(X/S_0)})$$ By Remark 54.24.11 we have $$R\Gamma(\text{Cris}(X/S_0), \mathcal{F}|_{\text{Cris}(X/S_0)}) = R\Gamma(X, \mathcal{F}_X \otimes \Omega^\bullet_{X/S_0})$$ Using the stupid filtration on the de Rham complex we see that the last displayed complex is perfect in $D(A/I)$ as soon as the complexes $$R\Gamma(X, \mathcal{F}_X \otimes \Omega^q_{X/S_0})$$ are perfect complexes in $D(A/I)$, see More on Algebra, Lemma 15.67.4. This is true by standard arguments in coherent cohomology using that $\mathcal{F}_X \otimes \Omega^q_{X/S_0}$ is a finite locally free sheaf and $X \to S_0$ is proper and flat (insert future reference here). Applying More on Algebra, Lemma 15.70.4 we see that $$R\Gamma(\text{Cris}(X/S), \mathcal{F}) \otimes_A^\mathbf{L} A/I^n$$ is a perfect object of $D(A/I^n)$ for all $n$. This isn't quite enough unless $A$ is Noetherian. Namely, even though $I$ is locally nilpotent by our assumption that $p$ is nilpotent, see Divided Power Algebra, Lemma 23.2.6, we cannot conclude that $I^n = 0$ for some $n$. A counter example is $\mathbf{F}_p\langle x \rangle$. To prove it in general when $\mathcal{F} = \mathcal{O}_{X/S}$ the argument of http://math.columbia.edu/ dejong/wordpress/?p=2227 works. When the coefficients $\mathcal{F}$ are non-trivial the argument of [Faltings-very] seems to be as follows. Reduce to the case $pA = 0$ by More on Algebra, Lemma 15.70.4. In this case the Frobenius map $A \to A$, $a \mapsto a^p$ factors as $A \to A/I \xrightarrow{\varphi} A$ (as $x^p = 0$ for $x \in I$). Set $X^{(1)} = X \otimes_{A/I, \varphi} A$. The absolute Frobenius morphism of $X$ factors through a morphism $F_X : X \to X^{(1)}$ (a kind of relative Frobenius). Affine locally if $X = \mathop{\rm Spec}(C)$ then $X^{(1)} = \mathop{\rm Spec}( C \otimes_{A/I, \varphi} A)$ and $F_X$ corresponds to $C \otimes_{A/I, \varphi} A \to C$, $c \otimes a \mapsto c^pa$. This defines morphisms of ringed topoi $$(X/S)_{\text{cris}} \xrightarrow{(F_X)_{\text{cris}}} (X^{(1)}/S)_{\text{cris}} \xrightarrow{u_{X^{(1)}/S}} \mathop{\textit{Sh}}\nolimits(X^{(1)}_{Zar})$$ whose composition is denoted $\text{Frob}_X$. One then shows that $R\text{Frob}_{X, *}\mathcal{F}$ is representable by a perfect complex of $\mathcal{O}_{X^{(1)}}$-modules(!) by a local calculation.

Remark 54.24.13 (Complete perfectness). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $X$ be a proper smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules. Then $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is a perfect object of $D(A)$.

Hints: We know that $K = R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is the derived limit $K = R\mathop{\rm lim}\nolimits K_e$ of the cohomologies over $A/p^eA$, see Remark 54.24.10. Each $K_e$ is a perfect complex of $D(A/p^eA)$ by Remark 54.24.12. Since $A$ is $p$-adically complete the result follows from More on Algebra, Lemma 15.83.4.

Remark 54.24.14 (Complete comparison). Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power ring with $A$ a Noetherian $p$-adically complete ring and $p$ nilpotent in $A/I$. Set $S = \mathop{\rm Spec}(A)$ and $S_0 = \mathop{\rm Spec}(A/I)$. Let $Y$ be a proper smooth scheme over $S$ and set $X = Y \times_S S_0$. Let $\mathcal{F}$ be a finite type crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules. Then

1. there exists a coherent $\mathcal{O}_Y$-module $\mathcal{F}_Y$ endowed with integrable connection $$\nabla : \mathcal{F}_Y \longrightarrow \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S}$$ such that $\mathcal{F}_Y/p^e\mathcal{F}_Y$ is the module with connection over $A/p^eA$ found in Remark 54.24.11, and
2. we have $$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(Y, \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega^\bullet_{Y/S})$$ in $D(A)$.

Hints: The existence of $\mathcal{F}_Y$ is Grothendieck's existence theorem (insert future reference here). The isomorphism of cohomologies follows as both sides are computed as $R\mathop{\rm lim}\nolimits$ of the versions modulo $p^e$ (see Remark 54.24.10 for the left hand side; use the theorem on formal functions, see Cohomology of Schemes, Theorem 29.20.5 for the right hand side). Each of the versions modulo $p^e$ are isomorphic by Remark 54.24.11.

1. This assumption is not strictly necessary, as using hypercoverings the construction of the remark can be extended to the general case.

The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4385–5010 (see updates for more information).

\section{Some further results}
\label{section-missing}

\noindent
In this section we mention some results whose proof is missing.
We will formulate these as a series of remarks and we will convert
them into actual lemmas and propositions only when we add detailed
proofs.

\begin{remark}[Higher direct images]
\label{remark-compute-direct-image}
Let $p$ be a prime number. Let
$(S, \mathcal{I}, \gamma) \to (S', \mathcal{I}', \gamma')$ be
a morphism of divided power schemes over $\mathbf{Z}_{(p)}$. Let
$$\xymatrix{ X \ar[r]_f \ar[d] & X' \ar[d] \\ S_0 \ar[r] & S'_0 }$$
be a commutative diagram of morphisms of schemes and assume $p$ is
locally nilpotent on $X$ and $X'$. Let $\mathcal{F}$ be an
$\mathcal{O}_{X/S}$-module on $\text{Cris}(X/S)$. Then
$Rf_{\text{cris}, *}\mathcal{F}$ can be computed as follows.

\medskip\noindent
Given an object $(U', T', \delta')$ of $\text{Cris}(X'/S')$ set
$U = X \times_{X'} U' = f^{-1}(U')$ (an open subscheme of $X$). Denote
$(T_0, T, \delta)$ the divided power scheme over $S$ such that
$$\xymatrix{ T \ar[r] \ar[d] & T' \ar[d] \\ S \ar[r] & S' }$$
is cartesian in the category of divided power schemes, see
Lemma \ref{lemma-fibre-product}. There is an
induced morphism $U \to T_0$ and we obtain a morphism
$(U/T)_{\text{cris}} \to (X/S)_{\text{cris}}$, see
Remark \ref{remark-functoriality-cris}.
Let $\mathcal{F}_U$ be the pullback of $\mathcal{F}$.
Let $\tau_{U/T} : (U/T)_{\text{cris}} \to T_{Zar}$ be the structure morphism.
Then we have

\label{equation-identify-pushforward}
\left(Rf_{\text{cris}, *}\mathcal{F}\right)_{T'} =
R(T \to T')_*\left(R\tau_{U/T, *} \mathcal{F}_U \right)

where the left hand side is the restriction (see
Section \ref{section-sheaves}).

\medskip\noindent
Hints: First, show that $\text{Cris}(U/T)$ is the localization (in the sense
of Sites, Lemma \ref{sites-lemma-localize-topos-site}) of $\text{Cris}(X/S)$
at the sheaf of sets $f_{\text{cris}}^{-1}h_{(U', T', \delta')}$. Next, reduce
the statement to the case where $\mathcal{F}$ is an injective module
and pushforward of modules using that the pullback of an injective
$\mathcal{O}_{X/S}$-module is an injective $\mathcal{O}_{U/T}$-module on
$\text{Cris}(U/T)$. Finally, check the result holds for plain pushforward.
\end{remark}

\begin{remark}[Mayer-Vietoris]
\label{remark-mayer-vietoris}
In the situation of Remark \ref{remark-compute-direct-image}
suppose we have an open covering $X = X' \cup X''$. Denote
$X''' = X' \cap X''$. Let $f'$, $f''$, and $f''$ be the restriction of $f$
to $X'$, $X''$, and $X'''$. Moreover, Let $\mathcal{F}'$, $\mathcal{F}''$,
and $\mathcal{F}'''$ be the restriction of $\mathcal{F}$ to the crystalline
sites of $X'$, $X''$, and $X'''$. Then there exists a distinguished triangle
$$Rf_{\text{cris}, *}\mathcal{F} \longrightarrow Rf'_{\text{cris}, *}\mathcal{F}' \oplus Rf''_{\text{cris}, *}\mathcal{F}'' \longrightarrow Rf'''_{\text{cris}, *}\mathcal{F}''' \longrightarrow Rf_{\text{cris}, *}\mathcal{F}[1]$$
in $D(\mathcal{O}_{X'/S'})$.

\medskip\noindent
Hints: This is a formal consequence of the fact that the subcategories
$\text{Cris}(X'/S)$, $\text{Cris}(X''/S)$, $\text{Cris}(X'''/S)$ correspond
to open subobjects of the final sheaf on $\text{Cris}(X/S)$ and that the
last is the intersection of the first two.
\end{remark}

\begin{remark}[{\v C}ech complex]
\label{remark-cech-complex}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \Spec(A)$ and
$S_0 = \Spec(A/I)$. Let $X$ be a separated\footnote{This assumption is
not strictly necessary, as using hypercoverings the construction of the
remark can be extended to the general case.} scheme over
$S_0$ such that $p$ is locally nilpotent on $X$. Let $\mathcal{F}$ be a
crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.

\medskip\noindent
Choose an affine open covering
$X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$.
Write $U_\lambda = \Spec(C_\lambda)$. Choose a polynomial algebra
$P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$.
Having fixed these choices we can construct a {\v C}ech complex which
computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

\medskip\noindent
Given $n \geq 0$ and $\lambda_0, \ldots, \lambda_n \in \Lambda$
write $U_{\lambda_0 \ldots \lambda_n} = U_{\lambda_0} \cap \ldots \cap U_{\lambda_n}$. This is an affine scheme by assumption. Write
$U_{\lambda_0 \ldots \lambda_n} = \Spec(C_{\lambda_0 \ldots \lambda_n})$.
Set
$$P_{\lambda_0 \ldots \lambda_n} = P_{\lambda_0} \otimes_A \ldots \otimes_A P_{\lambda_n}$$
which comes with a canonical surjection onto $C_{\lambda_0 \ldots \lambda_n}$.
Denote the kernel $J_{\lambda_0 \ldots \lambda_n}$ and set
$D_{\lambda_0 \ldots \lambda_n}$
the $p$-adically completed divided power envelope of
$J_{\lambda_0 \ldots \lambda_n}$ in $P_{\lambda_0 \ldots \lambda_n}$
relative to $\gamma$. Let $M_{\lambda_0 \ldots \lambda_n}$ be the
$P_{\lambda_0 \ldots \lambda_n}$-module corresponding
to the restriction of $\mathcal{F}$ to
$\text{Cris}(U_{\lambda_0 \ldots \lambda_n}/S)$ via
Proposition \ref{proposition-crystals-on-affine}.
By construction we obtain a cosimplicial divided power ring $D(*)$
having in degree $n$ the ring
$$D(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} D_{\lambda_0 \ldots \lambda_n}$$
(use that divided power envelopes are functorial and the trivial
cosimplicial structure on the ring $P(*)$ defined similarly).
Since $M_{\lambda_0 \ldots \lambda_n}$ is the value'' of $\mathcal{F}$
on the objects $\Spec(D_{\lambda_0 \ldots \lambda_n})$ we see that
$M(*)$ defined by the rule
$$M(n) = \prod\nolimits_{\lambda_0 \ldots \lambda_n} M_{\lambda_0 \ldots \lambda_n}$$
forms a cosimplicial $D(*)$-module. Now we claim that we have
$$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = s(M(*))$$
Here $s(-)$ denotes the cochain complex associated to a cosimplicial
module (see
Simplicial, Section \ref{simplicial-section-dold-kan-cosimplicial}).

\medskip\noindent
Hints: The proof of this is similar to the proof of
Proposition \ref{proposition-compute-cohomology} (in particular
the result holds for any module satisfying the assumptions of
that proposition).
\end{remark}

\begin{remark}[Alternating {\v C}ech complex]
\label{remark-alternating-cech-complex}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $A$ a $\mathbf{Z}_{(p)}$-algebra. Set $S = \Spec(A)$ and
$S_0 = \Spec(A/I)$. Let $X$ be a separated quasi-compact scheme
over $S_0$ such that $p$ is locally nilpotent on $X$. Let
$\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.

\medskip\noindent
Choose a finite affine open covering
$X = \bigcup_{\lambda \in \Lambda} U_\lambda$ of $X$
and a total ordering on $\Lambda$.
Write $U_\lambda = \Spec(C_\lambda)$. Choose a polynomial algebra
$P_\lambda$ over $A$ and a surjection $P_\lambda \to C_\lambda$.
Having fixed these choices we can construct an alternating
{\v C}ech complex which computes $R\Gamma(\text{Cris}(X/S), \mathcal{F})$.

\medskip\noindent
We are going to use the notation introduced in
Remark \ref{remark-cech-complex}.
Denote $\Omega_{\lambda_0 \ldots \lambda_n}$
the $p$-adically completed module of differentials of
$D_{\lambda_0 \ldots \lambda_n}$ over $A$ compatible with the divided power
structure. Let $\nabla$ be the integrable connection on
$M_{\lambda_0 \ldots \lambda_n}$ coming from
Proposition \ref{proposition-crystals-on-affine}.
Consider the double complex $M^{\bullet, \bullet}$ with
terms
$$M^{n, m} = \bigoplus\nolimits_{\lambda_0 < \ldots < \lambda_n} M_{\lambda_0 \ldots \lambda_n} \otimes^\wedge_{D_{\lambda_0 \ldots \lambda_n}} \Omega^m_{D_{\lambda_0 \ldots \lambda_n}}.$$
For the differential $d_1$ (increasing $n$) we use the usual
{\v C}ech differential and for the differential $d_2$ we use
the connection, i.e., the differential of the de Rham complex.
We claim that
$$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = \text{Tot}(M^{\bullet, \bullet})$$
Here $\text{Tot}(-)$ denotes the total complex associated to a
double complex, see
Homology, Definition \ref{homology-definition-associated-simple-complex}.

\medskip\noindent
Hints: We have
$$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(\text{Cris}(X/S), \mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet)$$
by Proposition \ref{proposition-compare-with-de-Rham}.
The right hand side of the formula is simply the alternating {\v C}ech complex
for the covering $X = \bigcup_{\lambda \in \Lambda} U_\lambda$
(which induces an open covering of the final sheaf of $\text{Cris}(X/S)$)
and the complex $\mathcal{F} \otimes_{\mathcal{O}_{X/S}} \Omega_{X/S}^\bullet$,
see Proposition \ref{proposition-compute-cohomology-crystal}.
Now the result follows from a general result in cohomology on sites,
namely that the alternating {\v C}ech complex computes the cohomology
provided it gives the correct answer on all the pieces (insert future
reference here).
\end{remark}

\begin{remark}[Quasi-coherence]
\label{remark-quasi-coherent}
In the situation of Remark \ref{remark-compute-direct-image}
assume that $S \to S'$ is quasi-compact and quasi-separated and
that $X \to S_0$ is quasi-compact and quasi-separated. Then for a crystal
in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$
the sheaves $R^if_{\text{cris}, *}\mathcal{F}$ are locally quasi-coherent.

\medskip\noindent
Hints: We have to show that the restrictions to $T'$ are quasi-coherent
$\mathcal{O}_{T'}$-modules, where $(U', T', \delta')$ is any object of
$\text{Cris}(X'/S')$. It suffices to do this when $T'$ is affine.
We use the formula (\ref{equation-identify-pushforward}),
the fact that $T \to T'$ is quasi-compact and quasi-separated (as $T$
is affine over the base change of $T'$ by $S \to S'$), and
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}
to see that it suffices to show that the sheaves
$R^i\tau_{U/T, *}\mathcal{F}_U$ are quasi-coherent.
Note that $U \to T_0$ is also quasi-compact and quasi-separated, see
Schemes, Lemmas \ref{schemes-lemma-quasi-compact-permanence} and
\ref{schemes-lemma-quasi-compact-permanence}.

\medskip\noindent
This reduces us to proving that $R^i\tau_{X/S, *}\mathcal{F}$
is quasi-coherent on $S$ in the case that $p$ locally nilpotent on $S$. Here
$\tau_{X/S}$ is the structure morphism, see
Remark \ref{remark-structure-morphism}.
We may work locally on $S$, hence we may assume $S$ affine
(see Lemma \ref{lemma-localize}). Induction on the number
of affines covering $X$ and Mayer-Vietoris
(Remark \ref{remark-mayer-vietoris}) reduces the question to
the case where $X$ is also affine (as in the proof of
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images}).
Say $X = \Spec(C)$ and $S = \Spec(A)$ so that $(A, I, \gamma)$ and
$A \to C$ are as
in Situation \ref{situation-affine}. Choose a polynomial algebra
$P$ over $A$ and a surjection $P \to C$ as in
Section \ref{section-quasi-coherent-crystals}.
Let $(M, \nabla)$ be the module corresponding to $\mathcal{F}$, see
Proposition \ref{proposition-crystals-on-affine}.
Applying
Proposition \ref{proposition-compute-cohomology-crystal}
we see that $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is represented by
$M \otimes_D \Omega_D^*$. Note that completion isn't necessary
as $p$ is nilpotent in $A$! We have to show that this is compatible
with taking principal opens in $S = \Spec(A)$. Suppose that $g \in A$.
Then we conclude that similarly $R\Gamma(\text{Cris}(X_g/S_g), \mathcal{F})$
is computed by $M_g \otimes_{D_g} \Omega_{D_g}^*$ (again this uses that
$p$-adic completion isn't necessary). Hence we conclude because localization
is an exact functor on $A$-modules.
\end{remark}

\begin{remark}[Boundedness]
\label{remark-bounded-cohomology}
In the situation of Remark \ref{remark-compute-direct-image}
assume that $S \to S'$ is quasi-compact and quasi-separated and
that $X \to S_0$ is of finite type and quasi-separated. Then there exists
an integer $i_0$ such that for any crystal
in quasi-coherent $\mathcal{O}_{X/S}$-modules $\mathcal{F}$
we have $R^if_{\text{cris}, *}\mathcal{F} = 0$ for all $i > i_0$.

\medskip\noindent
Hints: Arguing as in Remark \ref{remark-quasi-coherent} (using
Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherence-higher-direct-images})
we reduce to proving that $H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i \gg 0$
in the situation of Proposition \ref{proposition-compute-cohomology-crystal}
when $C$ is a finite type algebra over $A$. This is clear as we can
choose a finite polynomial algebra and we see that $\Omega^i_D = 0$
for $i \gg 0$.
\end{remark}

\begin{remark}[Specific boundedness]
\label{remark-bounded-cohomology-over-point}
In Situation \ref{situation-global} let $\mathcal{F}$ be a crystal in
quasi-coherent $\mathcal{O}_{X/S}$-modules. Assume that $S_0$
has a unique point and that $X \to S_0$ is of finite presentation.
\begin{enumerate}
\item If $\dim X = d$ and $X/S_0$ has embedding dimension $e$, then
$H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > d + e$.
\item If $X$ is separated and can be covered by $q$ affines, and
$X/S_0$ has embedding dimension $e$, then
$H^i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i > q + e$.
\end{enumerate}
Hints: In case (1) we can use that
$$H^i(\text{Cris}(X/S), \mathcal{F}) = H^i(X_{Zar}, Ru_{X/S, *}\mathcal{F})$$
and that $Ru_{X/S, *}\mathcal{F}$ is locally calculated by a de Rham
complex constructed using an embedding of $X$ into a smooth scheme
of dimension $e$ over $S$
(see Lemma \ref{lemma-compute-cohomology-crystal-smooth}).
These de Rham complexes are zero in all degrees $>e$. Hence (1)
follows from Cohomology, Proposition
\ref{cohomology-proposition-vanishing-Noetherian}.
In case (2) we use the alternating {\v C}ech complex (see
Remark \ref{remark-alternating-cech-complex}) to reduce to the case
$X$ affine. In the affine case we prove the result using the de Rham complex
associated to an embedding of $X$ into a smooth scheme of dimension $e$
over $S$ (it takes some work to construct such a thing).
\end{remark}

\begin{remark}[Base change map]
\label{remark-base-change}
In the situation of Remark \ref{remark-compute-direct-image}
assume $S = \Spec(A)$ and $S' = \Spec(A')$ are affine.
Let $\mathcal{F}'$ be an $\mathcal{O}_{X'/S'}$-module.
Let $\mathcal{F}$ be the pullback of $\mathcal{F}'$.
Then there is a canonical base change map
$$L(S' \to S)^*R\tau_{X'/S', *}\mathcal{F}' \longrightarrow R\tau_{X/S, *}\mathcal{F}$$
where $\tau_{X/S}$ and $\tau_{X'/S'}$ are the structure morphisms, see
Remark \ref{remark-structure-morphism}. On global sections this
gives a base change map

\label{equation-base-change-map}
R\Gamma(\text{Cris}(X'/S'), \mathcal{F}') \otimes^\mathbf{L}_{A'} A
\longrightarrow
R\Gamma(\text{Cris}(X/S), \mathcal{F})

in $D(A)$.

\medskip\noindent
Hint: Compose the very general base change map of
Cohomology on Sites, Remark \ref{sites-cohomology-remark-base-change}
with the canonical map
$Lf_{\text{cris}}^*\mathcal{F}' \to f_{\text{cris}}^*\mathcal{F}' = \mathcal{F}$.
\end{remark}

\begin{remark}[Base change isomorphism]
\label{remark-base-change-isomorphism}
The map (\ref{equation-base-change-map}) is an isomorphism provided
all of the following conditions are satisfied:
\begin{enumerate}
\item $p$ is nilpotent in $A'$,
\item $\mathcal{F}'$ is a crystal in quasi-coherent
$\mathcal{O}_{X'/S'}$-modules,
\item $X' \to S'_0$ is a quasi-compact, quasi-separated morphism,
\item $X = X' \times_{S'_0} S_0$,
\item $\mathcal{F}'$ is a flat $\mathcal{O}_{X'/S'}$-module,
\item $X' \to S'_0$ is a local complete intersection morphism (see
More on Morphisms, Definition \ref{more-morphisms-definition-lci}; this
holds for example if $X' \to S'_0$ is syntomic or smooth),
\item $X'$ and $S_0$ are Tor independent over $S'_0$ (see
More on Algebra, Definition \ref{more-algebra-definition-tor-independent};
this holds for example if either $S_0 \to S'_0$ or $X' \to S'_0$ is flat).
\end{enumerate}
Hints: Condition (1) means that in the arguments below $p$-adic completion
does nothing and can be ignored.
Using condition (3) and Mayer Vietoris (see
Remark \ref{remark-mayer-vietoris}) this reduces to the case
where $X'$ is affine. In fact by condition (6), after shrinking
further, we can assume that $X' = \Spec(C')$ and we are given a presentation
$C' = A'/I'[x_1, \ldots, x_n]/(\bar f'_1, \ldots, \bar f'_c)$
where $\bar f'_1, \ldots, \bar f'_c$ is a Koszul-regular sequence in $A'/I'$.
(This means that smooth locally $\bar f'_1, \ldots, \bar f'_c$ forms
a regular sequence, see More on Algebra,
Lemma \ref{more-algebra-lemma-Koszul-regular-flat-locally-regular}.)
We choose a lift of
$\bar f'_i$ to an element $f'_i \in A'[x_1, \ldots, x_n]$. By (4) we see that
$X = \Spec(C)$ with $C = A/I[x_1, \ldots, x_n]/(\bar f_1, \ldots, \bar f_c)$
where $f_i \in A[x_1, \ldots, x_n]$ is the image of $f'_i$.
By property (7) we see that $\bar f_1, \ldots, \bar f_c$ is a Koszul-regular
sequence in $A/I[x_1, \ldots, x_n]$. The divided power envelope of
$I'A'[x_1, \ldots, x_n] + (f'_1, \ldots, f'_c)$ in $A'[x_1, \ldots, x_n]$
relative to $\gamma'$ is
$$D' = A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c \rangle/(\xi_i - f'_i)$$
see Lemma \ref{lemma-describe-divided-power-envelope}. Then you check that
$\xi_1 - f'_1, \ldots, \xi_n - f'_n$ is a Koszul-regular sequence in the
ring $A'[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$.
Similarly the divided power envelope of
$IA[x_1, \ldots, x_n] + (f_1, \ldots, f_c)$ in $A[x_1, \ldots, x_n]$
relative to $\gamma$ is
$$D = A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle/(\xi_i - f_i)$$
and $\xi_1 - f_1, \ldots, \xi_n - f_n$ is a Koszul-regular sequence in the
ring $A[x_1, \ldots, x_n]\langle \xi_1, \ldots, \xi_c\rangle$.
It follows that $D' \otimes_{A'}^\mathbf{L} A = D$. Condition (2)
implies $\mathcal{F}'$ corresponds to a pair $(M', \nabla)$
consisting of a $D'$-module with connection, see
Proposition \ref{proposition-crystals-on-affine}.
Then $M = M' \otimes_{D'} D$ corresponds to the pullback $\mathcal{F}$.
By assumption (5) we see that $M'$ is a flat $D'$-module, hence
$$M = M' \otimes_{D'} D = M' \otimes_{D'} D' \otimes_{A'}^\mathbf{L} A = M' \otimes_{A'}^\mathbf{L} A$$
Since the modules of differentials $\Omega_{D'}$ and $\Omega_D$
(as defined in Section \ref{section-quasi-coherent-crystals})
are free $D'$-modules on the same generators we see that
$$M \otimes_D \Omega^\bullet_D = M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{D'} D = M' \otimes_{D'} \Omega^\bullet_{D'} \otimes_{A'}^\mathbf{L} A$$
which proves what we want by
Proposition \ref{proposition-compute-cohomology-crystal}.
\end{remark}

\begin{remark}[Rlim]
\label{remark-rlim}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $A$ an algebra over $\mathbf{Z}_{(p)}$ with $p$ nilpotent
in $A/I$. Set $S = \Spec(A)$ and $S_0 = \Spec(A/I)$.
Let $X$ be a scheme over $S_0$ with $p$ locally
nilpotent on $X$. Let $\mathcal{F}$ be any
$\mathcal{O}_{X/S}$-module. For $e \gg 0$ we have $(p^e) \subset I$
is preserved by $\gamma$, see
Divided Power Algebra, Lemma \ref{dpa-lemma-extend-to-completion}.
Set $S_e = \Spec(A/p^eA)$ for $e \gg 0$.
Then $\text{Cris}(X/S_e)$ is a full subcategory of $\text{Cris}(X/S)$
and we denote $\mathcal{F}_e$ the restriction of $\mathcal{F}$ to
$\text{Cris}(X/S_e)$. Then
$$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\lim_e R\Gamma(\text{Cris}(X/S_e), \mathcal{F}_e)$$

\medskip\noindent
Hints: Suffices to prove this for $\mathcal{F}$ injective.
In this case the sheaves $\mathcal{F}_e$ are injective
modules too, the transition maps
$\Gamma(\mathcal{F}_{e + 1}) \to \Gamma(\mathcal{F}_e)$ are
surjective, and we have
$\Gamma(\mathcal{F}) = \lim_e \Gamma(\mathcal{F}_e)$ because
any object of $\text{Cris}(X/S)$ is locally an object of one
of the categories $\text{Cris}(X/S_e)$ by definition of
$\text{Cris}(X/S)$.
\end{remark}

\begin{remark}[Comparison]
\label{remark-comparison}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $p$ nilpotent in $A$. Set $S = \Spec(A)$ and
$S_0 = \Spec(A/I)$. Let $Y$ be a smooth scheme over $S$ and set
$X = Y \times_S S_0$. Let
$\mathcal{F}$ be a crystal in quasi-coherent $\mathcal{O}_{X/S}$-modules.
Then
\begin{enumerate}
\item $\gamma$ extends to a divided power structure on the ideal
of $X$ in $Y$ so that $(X, Y, \gamma)$ is an object of $\text{Cris}(X/S)$,
\item the restriction $\mathcal{F}_Y$ (see Section \ref{section-sheaves})
comes endowed with a canonical integrable connection
$\nabla : \mathcal{F}_Y \to \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S}$, and
\item we have
$$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(Y, \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega^\bullet_{Y/S})$$
in $D(A)$.
\end{enumerate}
Hints: See Divided Power Algebra, Lemma \ref{dpa-lemma-gamma-extends} for (1).
See Lemma \ref{lemma-automatic-connection} for (2).
For Part (3) note that there is a map, see
(\ref{equation-restriction}). This map is an isomorphism when
$X$ is affine, see
Lemma \ref{lemma-compute-cohomology-crystal-smooth}.
This shows that $Ru_{X/S, *}\mathcal{F}$ and
$\mathcal{F}_Y \otimes \Omega^\bullet_{Y/S}$ are quasi-isomorphic
as complexes on $Y_{Zar} = X_{Zar}$.
Since $R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(X_{Zar}, Ru_{X/S, *}\mathcal{F})$ the result follows.
\end{remark}

\begin{remark}[Perfectness]
\label{remark-perfect}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $p$ nilpotent in $A$. Set $S = \Spec(A)$ and
$S_0 = \Spec(A/I)$. Let $X$ be a proper smooth scheme over $S_0$.
Let $\mathcal{F}$ be a crystal in finite locally free
quasi-coherent $\mathcal{O}_{X/S}$-modules.
Then $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is a
perfect object of $D(A)$.

\medskip\noindent
Hints: By Remark \ref{remark-base-change-isomorphism} we have
$$R\Gamma(\text{Cris}(X/S), \mathcal{F}) \otimes_A^\mathbf{L} A/I \cong R\Gamma(\text{Cris}(X/S_0), \mathcal{F}|_{\text{Cris}(X/S_0)})$$
By Remark \ref{remark-comparison} we have
$$R\Gamma(\text{Cris}(X/S_0), \mathcal{F}|_{\text{Cris}(X/S_0)}) = R\Gamma(X, \mathcal{F}_X \otimes \Omega^\bullet_{X/S_0})$$
Using the stupid filtration on the de Rham complex we see that
the last displayed complex is perfect in $D(A/I)$ as soon as the complexes
$$R\Gamma(X, \mathcal{F}_X \otimes \Omega^q_{X/S_0})$$
are perfect complexes in $D(A/I)$, see
More on Algebra, Lemma \ref{more-algebra-lemma-two-out-of-three-perfect}.
This is true by standard arguments
in coherent cohomology using that $\mathcal{F}_X \otimes \Omega^q_{X/S_0}$
is a finite locally free sheaf and $X \to S_0$ is proper and flat
(insert future reference here). Applying
More on Algebra, Lemma \ref{more-algebra-lemma-perfect-modulo-nilpotent-ideal}
we see that
$$R\Gamma(\text{Cris}(X/S), \mathcal{F}) \otimes_A^\mathbf{L} A/I^n$$
is a perfect object of $D(A/I^n)$ for all $n$. This isn't quite enough
unless $A$ is Noetherian. Namely, even though $I$ is locally nilpotent
by our assumption that $p$ is nilpotent, see
Divided Power Algebra, Lemma \ref{dpa-lemma-nil},
we cannot conclude that $I^n = 0$ for some $n$. A counter example
is $\mathbf{F}_p\langle x \rangle$. To prove it in general when
$\mathcal{F} = \mathcal{O}_{X/S}$ the argument of
\url{http://math.columbia.edu/~dejong/wordpress/?p=2227}
works. When the coefficients $\mathcal{F}$ are non-trivial the
argument of \cite{Faltings-very} seems to be as follows. Reduce to the
case $pA = 0$ by More on Algebra, Lemma
\ref{more-algebra-lemma-perfect-modulo-nilpotent-ideal}.
In this case the Frobenius map $A \to A$, $a \mapsto a^p$ factors
as $A \to A/I \xrightarrow{\varphi} A$ (as $x^p = 0$ for $x \in I$). Set
$X^{(1)} = X \otimes_{A/I, \varphi} A$. The absolute Frobenius morphism
of $X$ factors through a morphism $F_X : X \to X^{(1)}$ (a kind of
relative Frobenius). Affine locally if $X = \Spec(C)$ then
$X^{(1)} = \Spec( C \otimes_{A/I, \varphi} A)$
and $F_X$ corresponds to $C \otimes_{A/I, \varphi} A \to C$,
$c \otimes a \mapsto c^pa$. This defines morphisms of ringed topoi
$$(X/S)_{\text{cris}} \xrightarrow{(F_X)_{\text{cris}}} (X^{(1)}/S)_{\text{cris}} \xrightarrow{u_{X^{(1)}/S}} \Sh(X^{(1)}_{Zar})$$
whose composition is denoted $\text{Frob}_X$. One then shows that
$R\text{Frob}_{X, *}\mathcal{F}$ is representable by a
perfect complex of $\mathcal{O}_{X^{(1)}}$-modules(!)
by a local calculation.
\end{remark}

\begin{remark}[Complete perfectness]
\label{remark-complete-perfect}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $A$ a $p$-adically complete ring and $p$ nilpotent in $A/I$. Set
$S = \Spec(A)$ and $S_0 = \Spec(A/I)$. Let $X$ be a proper
smooth scheme over $S_0$. Let $\mathcal{F}$ be a crystal in
finite locally free quasi-coherent $\mathcal{O}_{X/S}$-modules.
Then $R\Gamma(\text{Cris}(X/S), \mathcal{F})$ is a
perfect object of $D(A)$.

\medskip\noindent
Hints: We know that $K = R\Gamma(\text{Cris}(X/S), \mathcal{F})$
is the derived limit $K = R\lim K_e$ of the cohomologies over $A/p^eA$,
see Remark \ref{remark-rlim}.
Each $K_e$ is a perfect complex of $D(A/p^eA)$ by
Remark \ref{remark-perfect}.
Since $A$ is $p$-adically complete the result
follows from
More on Algebra, Lemma \ref{more-algebra-lemma-Rlim-perfect-gives-complete}.
\end{remark}

\begin{remark}[Complete comparison]
\label{remark-complete-comparison}
Let $p$ be a prime number. Let $(A, I, \gamma)$ be a divided power
ring with $A$ a Noetherian $p$-adically complete ring and $p$ nilpotent
in $A/I$. Set $S = \Spec(A)$ and
$S_0 = \Spec(A/I)$. Let $Y$ be a proper smooth scheme over $S$ and set
$X = Y \times_S S_0$. Let $\mathcal{F}$ be a finite type crystal in
quasi-coherent $\mathcal{O}_{X/S}$-modules. Then
\begin{enumerate}
\item there exists a coherent $\mathcal{O}_Y$-module $\mathcal{F}_Y$
endowed with integrable connection
$$\nabla : \mathcal{F}_Y \longrightarrow \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega_{Y/S}$$
such that $\mathcal{F}_Y/p^e\mathcal{F}_Y$ is the module with connection
over $A/p^eA$ found in Remark \ref{remark-comparison}, and
\item we have
$$R\Gamma(\text{Cris}(X/S), \mathcal{F}) = R\Gamma(Y, \mathcal{F}_Y \otimes_{\mathcal{O}_Y} \Omega^\bullet_{Y/S})$$
in $D(A)$.
\end{enumerate}
Hints: The existence of $\mathcal{F}_Y$ is Grothendieck's existence theorem
(insert future reference here). The isomorphism of cohomologies follows
as both sides are computed as $R\lim$ of the versions modulo $p^e$
(see Remark \ref{remark-rlim} for the left hand side; use the theorem
on formal functions, see
Cohomology of Schemes, Theorem \ref{coherent-theorem-formal-functions}
for the right hand side).
Each of the versions modulo $p^e$ are isomorphic by
Remark \ref{remark-comparison}.
\end{remark}

Comment #2478 by Daniel Litt (site) on April 11, 2017 a 5:41 am UTC

There is a missing tilde in the link to: http://math.columbia.edu/ dejong/wordpress/?p=2227

Comment #2511 by Johan (site) on April 14, 2017 a 12:38 am UTC

THis is a problem with the parsing of the pages... Not something I know how to fix right now. Pieter?

## Add a comment on tag 07MI

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the lower-right corner).