The Stacks project

60.24 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 60.24.1 (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 60.7.4. 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 60.9.3. 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

60.24.1.1
\begin{equation} \label{crystalline-equation-identify-pushforward} \left(Rf_{\text{cris}, *}\mathcal{F}\right)_{T'} = R(T \to T')_*\left(R\tau _{U/T, *} \mathcal{F}_ U \right) \end{equation}

where the left hand side is the restriction (see Section 60.10).

Hints: First, show that $\text{Cris}(U/T)$ is the localization (in the sense of Sites, Lemma 7.30.3) 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 60.24.2 (Mayer-Vietoris). In the situation of Remark 60.24.1 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 60.24.3 (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 $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. 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 $U_\lambda = \mathop{\mathrm{Spec}}(C_\lambda )$. 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 $U_{\lambda _0 \ldots \lambda _ n} = \mathop{\mathrm{Spec}}(C_{\lambda _0 \ldots \lambda _ n})$. 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 60.17.4. 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 $\mathop{\mathrm{Spec}}(D_{\lambda _0 \ldots \lambda _ n})$ 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 60.21.1 (in particular the result holds for any module satisfying the assumptions of that proposition).

Remark 60.24.4 (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 $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. 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 $U_\lambda = \mathop{\mathrm{Spec}}(C_\lambda )$. 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 60.24.3. 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 60.17.4. 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 12.18.3.

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 60.23.1. 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 60.21.3. 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 60.24.5 (Quasi-coherence). In the situation of Remark 60.24.1 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 (60.24.1.1), 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 30.4.5 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 26.21.14 and 26.21.14.

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 60.9.6. We may work locally on $S$, hence we may assume $S$ affine (see Lemma 60.9.5). Induction on the number of affines covering $X$ and Mayer-Vietoris (Remark 60.24.2) reduces the question to the case where $X$ is also affine (as in the proof of Cohomology of Schemes, Lemma 30.4.5). Say $X = \mathop{\mathrm{Spec}}(C)$ and $S = \mathop{\mathrm{Spec}}(A)$ so that $(A, I, \gamma )$ and $A \to C$ are as in Situation 60.5.1. Choose a polynomial algebra $P$ over $A$ and a surjection $P \to C$ as in Section 60.17. Let $(M, \nabla )$ be the module corresponding to $\mathcal{F}$, see Proposition 60.17.4. Applying Proposition 60.21.3 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 $S = \mathop{\mathrm{Spec}}(A)$. 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 60.24.6 (Boundedness). In the situation of Remark 60.24.1 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 60.24.5 (using Cohomology of Schemes, Lemma 30.4.5) we reduce to proving that $H^ i(\text{Cris}(X/S), \mathcal{F}) = 0$ for $i \gg 0$ in the situation of Proposition 60.21.3 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 60.24.7 (Specific boundedness). In Situation 60.7.5 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 60.21.4). These de Rham complexes are zero in all degrees $> e$. Hence (1) follows from Cohomology, Proposition 20.20.7. In case (2) we use the alternating Čech complex (see Remark 60.24.4) 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 60.24.8 (Base change map). In the situation of Remark 60.24.1 assume $S = \mathop{\mathrm{Spec}}(A)$ and $S' = \mathop{\mathrm{Spec}}(A')$ 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 60.9.6. On global sections this gives a base change map

60.24.8.1
\begin{equation} \label{crystalline-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}) \end{equation}

in $D(A)$.

Hint: Compose the very general base change map of Cohomology on Sites, Remark 21.19.3 with the canonical map $Lf_{\text{cris}}^*\mathcal{F}' \to f_{\text{cris}}^*\mathcal{F}' = \mathcal{F}$.

Remark 60.24.9 (Base change isomorphism). The map (60.24.8.1) 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 37.59.2; 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 15.61.1; 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 60.24.2) this reduces to the case where $X'$ is affine. In fact by condition (6), after shrinking further, we can assume that $X' = \mathop{\mathrm{Spec}}(C')$ 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 15.30.17.) 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 $X = \mathop{\mathrm{Spec}}(C)$ 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 60.2.4. 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 60.17.4. 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 60.17) 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 60.21.3.

Remark 60.24.10 (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 $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. 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 $S_ e = \mathop{\mathrm{Spec}}(A/p^ eA)$ 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

\[ R\Gamma (\text{Cris}(X/S), \mathcal{F}) = R\mathop{\mathrm{lim}}\nolimits _ e R\Gamma (\text{Cris}(X/S_ e), \mathcal{F}_ e) \]

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 $\Gamma (\mathcal{F}) = \mathop{\mathrm{lim}}\nolimits _ e \Gamma (\mathcal{F}_ e)$ 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 60.24.11 (Comparison). Let $p$ be a prime number. Let $(A, I, \gamma )$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. 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 60.10) 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 60.15.1 for (2). For Part (3) note that there is a map, see (60.23.2.1). This map is an isomorphism when $X$ is affine, see Lemma 60.21.4. 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 60.24.12 (Perfectness). Let $p$ be a prime number. Let $(A, I, \gamma )$ be a divided power ring with $p$ nilpotent in $A$. Set $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. 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 60.24.9 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 60.24.11 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 15.74.4. 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 15.78.4 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 https://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 15.78.4. 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 $X = \mathop{\mathrm{Spec}}(C)$ then $X^{(1)} = \mathop{\mathrm{Spec}}( C \otimes _{A/I, \varphi } A)$ 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

\[ (X/S)_{\text{cris}} \xrightarrow {(F_ X)_{\text{cris}}} (X^{(1)}/S)_{\text{cris}} \xrightarrow {u_{X^{(1)}/S}} \mathop{\mathit{Sh}}\nolimits (X^{(1)}_{Zar}) \]

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 60.24.13 (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 $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. 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 $K = R\mathop{\mathrm{lim}}\nolimits K_ e$ of the cohomologies over $A/p^ eA$, see Remark 60.24.10. Each $K_ e$ is a perfect complex of $D(A/p^ eA)$ by Remark 60.24.12. Since $A$ is $p$-adically complete the result follows from More on Algebra, Lemma 15.97.4.

Remark 60.24.14 (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 $S = \mathop{\mathrm{Spec}}(A)$ and $S_0 = \mathop{\mathrm{Spec}}(A/I)$. 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 60.24.11, 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{\mathrm{lim}}\nolimits $ of the versions modulo $p^ e$ (see Remark 60.24.10 for the left hand side; use the theorem on formal functions, see Cohomology of Schemes, Theorem 30.20.5 for the right hand side). Each of the versions modulo $p^ e$ are isomorphic by Remark 60.24.11.

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

Comments (2)

Comment #2478 by on

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

Comment #2511 by on

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


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