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Tag 07MX

Chapter 54: Crystalline Cohomology > Section 54.24: Some further results

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.

    The code snippet corresponding to this tag is a part of the file crystalline.tex and is located in lines 4879–4948 (see updates for more information).

    \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}

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