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Spectral sequence for a smooth morphism

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aisejohan committed Sep 26, 2019
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  1. +65 −0 cohomology.tex
  2. +208 −1 derham.tex
  3. +1 −1 injectives.tex
  4. +21 −34 perfect.tex
@@ -6341,6 +6341,22 @@ \section{Cohomology of unbounded complexes}
coming from the counit $Lf^* \circ Rf_* \to \text{id}$.
\end{remark}






\section{Cohomology of filtered complexes}
\label{section-cohomology-filtered-object}

\noindent
Filtered complexes of sheaves frequently come up in a natural fashion
when studying cohomology of algebraic varieties, for example the de Rham
complex comes with its Hodge filtration. In this sectionwe use the very general
Injectives, Lemma \ref{injectives-lemma-K-injective-embedding-filtration}
to find construct spectral sequences on cohomology and we relate these to
previously constructed spectral sequences.

\begin{lemma}
\label{lemma-spectral-sequence-filtered-object}
Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a
@@ -6444,6 +6460,55 @@ \section{Cohomology of unbounded complexes}
(constructed using Cartan-Eilenberg resolutions).
\end{example}

\begin{example}
\label{example-spectral-sequence-bis}
Let $(X, \mathcal{O}_X)$ be a ringed space. Let $\mathcal{F}^\bullet$ be a
complex of $\mathcal{O}_X$-modules. We can apply
Lemma \ref{lemma-spectral-sequence-filtered-object}
with $F^p\mathcal{F}^\bullet = \sigma_{\geq p}\mathcal{F}^\bullet$.
(If $\mathcal{F}^\bullet$ is bounded below we can use
Remark \ref{remark-spectral-sequence-filtered-object}.)
Then we get a spectral sequence
$$
E_1^{p, q} = H^{p + q}(X, \mathcal{F}^p[-p]) = H^q(X, \mathcal{F}^p)
$$
If $\mathcal{F}^\bullet$ is bounded below, then
\begin{enumerate}
\item we can use
Remark \ref{remark-spectral-sequence-filtered-object}
to construct this spectral sequence,
\item the spectral sequence is bounded and converges to
$H^{i + j}(X, \mathcal{F}^\bullet)$, and
\item the spectral sequence is equal to the first spectral sequence of
Derived Categories, Lemma \ref{derived-lemma-two-ss-complex-functor}
(constructed using Cartan-Eilenberg resolutions).
\end{enumerate}
\end{example}

\begin{lemma}
\label{lemma-relative-spectral-sequence-filtered-object}
Let $f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y)$ be a morphism of
ringed spaces. Let $\mathcal{F}^\bullet$ be a filtered complex of
$\mathcal{O}_X$-modules. There exists a canonical spectral sequence
$(E_r, \text{d}_r)_{r \geq 1}$ of bigraded
$\mathcal{O}_Y$-modules with $d_r$ of bidegree $(r, -r + 1)$ and
$$
E_1^{p, q} = R^{p + q}f_*\text{gr}^p\mathcal{F}^\bullet
$$
If for every $n$ we have
$$
R^nf_*F^p\mathcal{F}^\bullet = 0 \text{ for }p \gg 0
\quad\text{and}\quad
R^nf_*F^p\mathcal{F}^\bullet = R^nf_*\mathcal{F}^\bullet \text{ for }p \ll 0
$$
then the spectral sequence is bounded and converges to
$Rf_*\mathcal{F}^\bullet$.
\end{lemma}

\begin{proof}
The proof is exactly the same as the proof of
Lemma \ref{lemma-spectral-sequence-filtered-object}.
\end{proof}



@@ -172,6 +172,52 @@ \section{de Rham cohomology}
This is a special case of Lemma \ref{lemma-coherence-relative}.
\end{proof}

\begin{lemma}
\label{lemma-proper-smooth-de-Rham}
Let $f : X \to S$ be a proper smooth morphism of schemes. Then
$Rf_*\Omega^p_{X/S}$, $p \geq 0$ and $Rf_*\Omega^\bullet_{X/S}$ are
perfect objects of $D(\mathcal{O}_S)$ whose formation commutes
with arbitrary change of base.
\end{lemma}

\begin{proof}
Since $f$ is smooth the modules $\Omega^p_{X/S}$ are finite locally
free $\mathcal{O}_X$-modules, see Morphisms, Lemma
\ref{morphisms-lemma-smooth-omega-finite-locally-free}. Their
formation commutes with arbitrary change of base by
Lemma \ref{lemma-base-change-de-rham}. Hence
$Rf_*\Omega^p_{X/S}$ is a perfect object of $D(\mathcal{O}_S)$
whose formation commutes with abitrary base change, see
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-flat-proper-perfect-direct-image-general}.
This proves the first assertion of the lemma.

\medskip\noindent
To prove that $Rf_*\Omega^\bullet_{X/S}$ is perfect on $S$ we may work
locally on $S$. Thus we may assume $S$ is quasi-compact. This means
we may assume that $\Omega^n_{X/S}$ is zero for $n$ large enough.
For every $p \geq 0$ we claim that
$Rf_*\sigma_{\geq p}\Omega^\bullet_{X/S}$ is a
perfect object of $D(\mathcal{O}_S)$ whose formation commutes
with arbitrary change of base. By the above we see that
this is true for $p \gg 0$. Suppose the claim holds for $p$
and consider the distinguished triangle
$$
\sigma_{\geq p}\Omega^\bullet_{X/S} \to
\sigma_{\geq p - 1}\Omega^\bullet_{X/S} \to
\Omega^{p - 1}_{X/S}[-(p - 1)] \to
(\sigma_{\geq p}\Omega^\bullet_{X/S})[1]
$$
in $D(f^{-1}\mathcal{O}_S)$.
Applying the exact functor $Rf_*$ we obtain a distinguished triangle
in $D(\mathcal{O}_S)$.
Since we have the 2-out-of-3 property for being perfect
(Cohomology, Lemma \ref{cohomology-lemma-two-out-of-three-perfect})
we conclude $Rf_*\sigma_{\geq p - 1}\Omega^\bullet_{X/S}$ is a
perfect object of $D(\mathcal{O}_S)$. Similarly for the
commutation with arbitrary base change.
\end{proof}




@@ -513,7 +559,7 @@ \section{K\"unneth formula}
\end{lemma}

\begin{proof}
By Derived Categories of Schemes, Equation (\ref{perfect-equation-silly})
By Derived Categories of Schemes, Remark \ref{perfect-remark-silly}
we have
$$
p^{-1}\Omega^i_{X/S} \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\Omega^j_{Y/S} =
@@ -1053,6 +1099,167 @@ \section{de Rham cohomology of projective space}





\section{The spectral sequence for a smooth morphism}
\label{section-relative-spectral-sequence}

\noindent
Let $S$ be a scheme. Let $f : X \to Y$ be a quasi-compact and
quasi-separated smooth morphism of schemes over $S$.
Then we obtain a locally split short exact sequence
$$
0 \to f^*\Omega_{Y/S} \to \Omega_{X/S} \to \Omega_{X/Y} \to 0
$$
by Morphisms, Lemma \ref{morphisms-lemma-triangle-differentials-smooth}.
Let us think of this as a descending filtration on $\Omega_{X/S}$.
Then for every $p$ we obtain an induced filtration on $\Omega^p_{X/S}$
whose succesive quotients are
$$
f^*\Omega^k_{Y/S} \otimes_{\mathcal{O}_X} \Omega^{p - k}_{X/Y} =
f^{-1}\Omega^k_{Y/S} \otimes_{f^{-1}\mathcal{O}_Y} \Omega^{p - k}_{X/Y}
$$
for $k = 0, \ldots, p$. In fact, these filtrations give $\Omega^\bullet_{X/S}$
the structure of a filtered complex. We can also directly define the
filtration by setting
$$
F^k\Omega^\bullet_{X/S} =
\Im\left(\wedge :
\text{Tot}(
f^{-1}\sigma_{\geq k}\Omega^\bullet_{Y/S} \otimes_{f^{-1}\mathcal{O}_Y}
\Omega^\bullet_{X/S})
\longrightarrow
\Omega^\bullet_{X/S}\right)
$$
as complexes on $X$. Thus we have
$$
\Omega^\bullet_{X/S} = F^0\Omega^\bullet_{X/S} \supset
F^1\Omega^\bullet_{X/S} \supset F^2\Omega^\bullet_{X/S} \supset \ldots
$$
and we have the identification of complexes
$$
\text{gr}^k\Omega^\bullet_{X/S} =
f^{-1}\Omega^k_{Y/S}[-k] \otimes_{f^{-1}\mathcal{O}_Y} \Omega^\bullet_{X/Y}
$$
by the discussion above. Let us study the spectral sequence associated
to this filtered complex of Cohomology, Lemma
\ref{cohomology-lemma-relative-spectral-sequence-filtered-object}.
By Lemma \ref{lemma-cohomology-de-rham-base-change} we have
$$
Rf_*\text{gr}^k\Omega^\bullet_{X/S} =
\Omega^k_{Y/S}[-k] \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\Omega^\bullet_{X/Y}
$$
All in all we conclude that there is a spectral sequence
$$
E_1^{p, q} =
H^q(\Omega^p_{Y/S} \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\Omega^\bullet_{X/Y})
$$
converging to $R^{p + q}f_*\Omega^\bullet_{X/S}$.

\begin{lemma}
\label{lemma-cohomology-de-rham-base-change}
Let $f : X \to Y$ be a quasi-compact and quasi-separated morphism
of schemes. Let $\mathcal{F}^\bullet$ be a locally bounded complex of
$f^{-1}\mathcal{O}_Y$-modules. Assume for all $n \in \mathbf{Z}$
the sheaf $\mathcal{F}^n$ is a flat $f^{-1}\mathcal{O}_Y$-module and
$\mathcal{F}^n$ has the structure of a quasi-coherent $\mathcal{O}_X$-module
compatible with the given $p^{-1}\mathcal{O}_Y$-module structure (but the
differentials in the complex $\mathcal{F}^\bullet$ need not
be $\mathcal{O}_X$-linear). Then
$$
\mathcal{G} \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf_*\mathcal{F}^\bullet =
Rf_*(f^{-1}\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{F}^\bullet)
$$
for any quasi-coherent $\mathcal{O}_Y$-module $\mathcal{G}$.
\end{lemma}

\begin{proof}
This lemma is a variant of
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-cohomology-base-change}
and we urge the reader to read the proof of that lemma first.
Denote $f' : (X, f^{-1}\mathcal{O}_Y) \to (Y, \mathcal{O}_Y)$
the obvious flat morphism of ringed spaces.
We view $\mathcal{F}^\bullet$ as a complex of modules on the ringed space
$(X, f^{-1}\mathcal{O}_Y)$. Since $\mathcal{F}^\bullet$ is
a locally bounded complex of flat $f^{-1}\mathcal{O}_Y$-modules
we see that the complex
$f^{-1}\mathcal{G} \otimes_{f^{-1}\mathcal{O}_Y} \mathcal{F}^\bullet$
represents $L(f')^*\mathcal{G}
\otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L}
\mathcal{F}^\bullet$ in $D(f^{-1}\mathcal{O}_Y)$. Hence the
statement of the lemma is really that
$$
\mathcal{G} \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf'_*\mathcal{F}^\bullet =
Rf'_*(L(f')^*\mathcal{G}
\otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L}
\mathcal{F}^\bullet)
$$
Formulated in this manner we can generalize the statement to
the statement that the canonical, functorial arrow
Cohomology, Equation (\ref{cohomology-equation-projection-formula-map})
$$
G \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf'_*\mathcal{F}^\bullet
\longrightarrow
Rf'_*(L(f')^*G \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet)
$$
is an isomorphism for all $G$ in $D_\QCoh(\mathcal{O}_Y)$.
Formulated in this manner the problem is
local on $Y$ and we may assume $Y$ is affine. Moreover, the source and
target of the arrow are exact functors
$D_\QCoh(\mathcal{O}_Y) \to D(\mathcal{O}_Y)$
of triangulated categories. Hence if $G_1 \to G_2 \to G_3 \to G_1[1]$
is a distinguished triangle in $D_\QCoh(\mathcal{O}_Y)$ and the
result holds for two out of $G_1, G_2, G_3$, then the result
holds for the third. Finally, both sides of the arrow
commute with arbitrary direct sums (see below for the right hand side).
Thus, exactly as in the proof of Derived Categories of Schemes, Lemma
\ref{perfect-lemma-cohomology-base-change} it suffices to prove that
$$
\mathcal{O}_Y \otimes_{\mathcal{O}_Y}^\mathbf{L} Rf'_*\mathcal{F}^\bullet
\longrightarrow
Rf'_*(L(f')^*\mathcal{O}_Y
\otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet)
$$
which is obvious.

\medskip\noindent
We will have to show that
$G \mapsto
Rf'_*(L(f')^*G \otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet)$
commutes with direct sums on $D_\QCoh(\mathcal{O}_Y)$.
This is where we will use $\mathcal{F}^n$ has the structure
of a quasi-coherent $\mathcal{O}_X$-module. First, observe that
$G \mapsto L(f')^*G
\otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet$
commutes with arbitrary direct sums. Next, if
$\mathcal{F}^\bullet$ consists of a single quasi-coherent
$\mathcal{O}_X$-module $\mathcal{F}^\bullet = \mathcal{F}^n[-n]$
then we have $L(f')^*G
\otimes_{f^{-1}\mathcal{O}_Y}^\mathbf{L} \mathcal{F}^\bullet =
Lf^*G \otimes_{\mathcal{O}_X}^\mathbf{L} \mathcal{F}^n[-n]$, see
Cohomology, Lemma \ref{cohomology-lemma-variant-derived-pullback}.
Hence in this case the commutation with direct sums follows from
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-pushforward-direct-sums}.
Now, in general, since $Y$ is affine and $\mathcal{F}^\bullet$
is locally bounded, we see that
$$
\mathcal{F}^\bullet = (\mathcal{F}^a \to \ldots \to \mathcal{F}^b)
$$
is bounded. Arguing by induction on $b - a$ and considering the
distinguished triangle
$$
\mathcal{F}^b[-b] \to (\mathcal{F}^a \to \ldots \to \mathcal{F}^b)
\to (\mathcal{F}^a \to \ldots \to \mathcal{F}^{b - 1}) \to
\mathcal{F}^b[-b + 1]
$$
the proof is finished. Some details omitted.
\end{proof}




\section{Comparing sheaves of differential forms}
\label{section-quasi-finite-syntomic}

@@ -2147,7 +2147,7 @@ \section{Additional remarks on Grothendieck abelian categories}
$$
such that $\alpha^p \circ \alpha^{p' p} = \alpha^{p'}$
and $\alpha^{p'p''} \circ \alpha^{pp'} = \alpha^{pp''}$.
The problem is that the maps $\alpha^P : I^{p, \bullet} \to I^\bullet$
The problem is that the maps $\alpha^p : I^{p, \bullet} \to I^\bullet$
need not be injective. For each $p$ we choose an injection
$t^p : I^{p, \bullet} \to J^{p, \bullet}$ into an acyclic K-injective
complex $J^{p, \bullet}$ whose terms are injective objects of $\mathcal{A}$
@@ -4650,46 +4650,35 @@ \section{K\"unneth formula}
and the cup product constructed in
Cohomology, Section \ref{cohomology-section-cup-product}.

\medskip\noindent
This cup product has an application to the de Rham cohomology. In this
case (as in the next lemma) the sheaves
$\mathcal{F}^n$ have the structure of quasi-coherent $\mathcal{O}_X$-modules
compatible with the given $p^{-1}\mathcal{O}_S$-module structures (but the
differentials in the complex $\mathcal{F}^\bullet$ need not
be $\mathcal{O}_X$-linear) and the sheaves $\mathcal{G}^m$ have the structure
of a quasi-coherent $\mathcal{O}_Y$-modules compatible with the given
$q^{-1}\mathcal{O}_S$-module structures (but the
differentials in the complex $\mathcal{G}^\bullet$ need not
be $\mathcal{O}_Y$-linear). If so then we have
\begin{equation}
\label{equation-silly}
\begin{aligned}
& p^{-1}\mathcal{F}^n \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}^m \\
\begin{remark}
\label{remark-silly}
In the situation above suppose we have an $\mathcal{O}_X$-module
$\mathcal{F}$ and an $\mathcal{O}_Y$-module $\mathcal{G}$.
Then we have
\begin{align*}
& p^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G} \\
& =
p^{-1}(\mathcal{F}^n \otimes_{\mathcal{O}_X} \mathcal{O}_X)
p^{-1}(\mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_X)
\otimes_{f^{-1}\mathcal{O}_S}
q^{-1}(\mathcal{O}_Y \otimes_{\mathcal{O}_Y} \mathcal{G}^m) \\
q^{-1}(\mathcal{O}_Y \otimes_{\mathcal{O}_Y} \mathcal{G}) \\
& =
p^{-1}\mathcal{F}^n \otimes_{p^{-1}\mathcal{O}_X} p^{-1}\mathcal{O}_X
p^{-1}\mathcal{F} \otimes_{p^{-1}\mathcal{O}_X} p^{-1}\mathcal{O}_X
\otimes_{f^{-1}\mathcal{O}_S}
q^{-1}\mathcal{O}_Y \otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G}^m \\
q^{-1}\mathcal{O}_Y \otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G} \\
& =
p^{-1}\mathcal{F}^n \otimes_{q^{-1}\mathcal{O}_X}
p^{-1}\mathcal{F} \otimes_{q^{-1}\mathcal{O}_X}
\mathcal{O}_{X \times_S Y}
\otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G}^m \\
\otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G} \\
& =
p^{-1}\mathcal{F}^n \otimes_{q^{-1}\mathcal{O}_X}
p^{-1}\mathcal{F} \otimes_{q^{-1}\mathcal{O}_X}
\mathcal{O}_{X \times_S Y}
\otimes_{\mathcal{O}_{X \times_S Y}}
\mathcal{O}_{X \times_S Y}
\otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G}^m \\
\otimes_{q^{-1}\mathcal{O}_Y} q^{-1}\mathcal{G} \\
& =
p^*\mathcal{F}^n \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}^m
\end{aligned}
\end{equation}
The target of the cup product (\ref{equation-de-rham-kunneth}) is
$R\Gamma(X \times_S Y, \text{Tot}(p^*\mathcal{F}^\bullet
\otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}^\bullet))$ in this case.
p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G}
\end{align*}
\end{remark}

\begin{lemma}
\label{lemma-kunneth-special}
@@ -4906,13 +4895,11 @@ \section{K\"unneth formula}
\otimes_A
\check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{G}))$
represents the derived tensor product
$R\Gamma(X, \mathcal{F}^\bullet)
\otimes_A^\mathbf{L}
R\Gamma(Y, \mathcal{G}^\bullet)$ as claimed.
Finally, we have
$R\Gamma(X, \mathcal{F}) \otimes_A^\mathbf{L} R\Gamma(Y, \mathcal{G})$
as claimed. Finally, we have
$p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times_S Y}} q^*\mathcal{G} =
p^{-1}\mathcal{F} \otimes_{f^{-1}\mathcal{O}_S} q^{-1}\mathcal{G}$
by (\ref{equation-silly}).
by Remark \ref{remark-silly}.

\medskip\noindent
We still have to show that (\ref{equation-kunneth-on-cech})

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