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Complex of forms with log poles along a divisor

Another one bites the dust!
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aisejohan committed Oct 3, 2019
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  1. +349 −2 derham.tex
@@ -81,14 +81,38 @@ \section{The de Rham complex}
$$
be a cartesian diagram of schemes. Then the maps discussed
above induce isomorphisms
$f^*\Omega^p_{Y/X} \to \Omega^p_{Y'/X'}$.
$f^*\Omega^p_{X/S} \to \Omega^p_{X'/S'}$.
\end{lemma}

\begin{proof}
Combine Morphisms, Lemma \ref{morphisms-lemma-base-change-differentials}
with the fact that formation of exterior power commutes with base change.
\end{proof}

\begin{lemma}
\label{lemma-etale}
Consider a commutative diagram of schemss
$$
\xymatrix{
X' \ar[r]_f \ar[d] & X \ar[d] \\
S' \ar[r] & S
}
$$
If $X' \to X$ and $S' \to S$ are \'etale, then the maps discussed
above induce isomorphisms
$f^*\Omega^p_{X/S} \to \Omega^p_{X'/S'}$.
\end{lemma}

\begin{proof}
We have $\Omega_{S'/S} = 0$ and $\Omega_{X'/X} = 0$, see for example
Morphisms, Lemma \ref{morphisms-lemma-etale-at-point}. Then by
the short exact sequences of Morphisms, Lemmas
\ref{morphisms-lemma-triangle-differentials} and
\ref{morphisms-lemma-triangle-differentials-smooth}
we see that $\Omega_{X'/S'} = \Omega_{X'/S} = f^*\Omega_{X/S}$.
Taking exterior powers we conclude.
\end{proof}




@@ -1499,7 +1523,9 @@ \section{The spectral sequence for a smooth morphism}
\item $K^\bullet$ is a complex of $R$-modules whose terms are
$A$-modules,
\item $K^\bullet$ represents $R\Gamma(X, \Omega^\bullet_{X/S})$ in $D(R)$
(Cohomology, Lemma \ref{cohomology-lemma-cech-complex-complex-computes}),
(Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} and
Cohomology, Lemma \ref{cohomology-lemma-cech-complex-complex-computes}),
\item there is a natural map $\Omega^\bullet_{A/R} \to K^\bullet$
of complexes of $R$-modules which is $A$-linear on terms and
induces the pullback map $H^*_{dR}(Y/S) \to H^*_{dR}(X/S)$
@@ -1637,6 +1663,327 @@ \section{Projective space bundle formula}





\section{Log poles along a divisor}
\label{section-divisor}

\noindent
Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an
effective Cartier divisor. If $X$ \'etale locally along $Y$ looks
like $Y \times \mathbf{A}^1$, then there is a canonical short exact sequence
of complexes
$$
0 \to \Omega^\bullet_{X/S} \to
\Omega^\bullet_{X/S}(\log Y) \to
\Omega^\bullet_{Y/S}[-1] \to 0
$$
having many good properties we will discuss in this section.

\begin{definition}
\label{definition-local-product}
Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an
effective Cartier divisor. We say the
{\it de Rham complex of log poles is defined for $Y \subset X$ over $S$}
if for all $y \in Y$ and local equation $f \in \mathcal{O}_{X, y}$
of $Y$ we have
\begin{enumerate}
\item $\mathcal{O}_{X, x} \to \Omega_{X/S, x}$, $g \mapsto g \text{d}f$
is a split injection, and
\item $\Omega^p_{X/S, y}$ is $f$-torsion free for all $p$.
\end{enumerate}
\end{definition}

\noindent
An easy local calculation shows that it suffices to find one local
equation $f$ for which condition (1) holds.

\begin{lemma}
\label{lemma-log-complex}
Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an
effective Cartier divisor.
Assume the de Rham complex of log poles is defined for $Y \subset X$ over $S$.
There is a canonical short exact sequence
of complexes
$$
0 \to \Omega^\bullet_{X/S} \to
\Omega^\bullet_{X/S}(\log Y) \to
\Omega^\bullet_{Y/S}[-1] \to 0
$$
\end{lemma}

\begin{proof}
Our assumption is that for every $y \in Y$ and local equation
$f \in \mathcal{O}_{X, y}$ of $Y$ we have
$$
\Omega_{X/S, y} = \mathcal{O}_{X, y}\text{d}f \oplus M
\quad\text{and}\quad
\Omega^p_{X/S, y} = \wedge^{p - 1}(M)\text{d}f \oplus \wedge^p(M)
$$
for some module $M$ with $f$-torsion free exterior powers $\wedge^p(M)$.
It follows that
$$
\Omega^p_{Y/S, y} = \wedge^p(M/fM) = \wedge^p(M)/f\wedge^p(M)
$$
Below we will tacitly use these facts.
In particular the sheaves $\Omega^p_{X/S}$ have no nonzero torsion
sections supported on $Y$ and we have a canonical inclusion
$$
\Omega^p_{X/S} \subset \Omega^p_{X/S}(Y)
$$
see More on Flatness, Section \ref{flat-section-eta}. Let $U = \Spec(A)$
be an affine open subscheme such that $Y \cap U = V(f)$ for some
nonzerodivisor $f \in A$. Let us consider the $\mathcal{O}_U$-submodule
of $\Omega^p_{X/S}(Y)|_U$ generated by
$\Omega^p_{X/S}|_U$ and $\text{d}\log(f) \wedge \Omega^{p - 1}_{X/S}$
where $\text{d}\log(f) = f^{-1}\text{d}(f)$.
This is independent of the choice of $f$ as another generator of the
ideal of $Y$ on $U$ is equal to $uf$ for a unit $u \in A$ and we get
$$
\text{d}\log(uf) - \text{d}\log(f) = \text{d}\log(u) = u^{-1}\text{d}u
$$
which is a section of $\Omega_{X/S}$ over $U$. Obviously, these local
sheaves glue to give a quasi-coherent submodule
$$
\Omega^p_{X/S} \subset \Omega^p_{X/S}(\log Y) \subset \Omega^p_{X/S}(Y)
$$
Let us agree to think of $\Omega^p_{Y/S}$ as a quasi-coherent
$\mathcal{O}_X$-module. There is a unique surjective
$\mathcal{O}_X$-linear map
$$
\Omega^p_{X/S}(\log Y) \to \Omega^{p - 1}_Y
$$
which over $U$ sends a local section $\text{d}\log(f) \wedge \eta$ to
the image $\eta$ in $\Omega^{p - 1}_{Y/S}$ and
annihilates the submodule $\Omega^p_{X/S}$.
If a form $\eta$ over $U$ restricts to zero on $\Omega_{Y/S}$, then
$\eta = \text{d}f \wedge \eta' + f\eta''$ for some forms $\eta'$ and $\eta''$
over $U$. We conclude that
we have a short exact sequence
$$
0 \to \Omega^p_{X/S} \to \Omega^p_{X/S}(\log Y) \to \Omega^{p - 1}_{Y/S} \to 0
$$
for all $p$.

\medskip\noindent
We still have to define the differentials
$\Omega^p_{X/S}(\log Y) \to \Omega^{p + 1}_{X/S}(\log Y)$.
On the subsheaf $\Omega^p_{X/S}$ we use the differential of
the de Rham complex of $X$ over $S$. Finally, we define
$\text{d}(\text{d}\log(f) \wedge \eta) = -\text{d}\log(f) \wedge \text{d}\eta$.
It is a pleasant exercise to show that we obtain a short exact
sequence of complexes as stated in the lemma.
\end{proof}

\begin{lemma}
\label{lemma-log-complex-consequence}
Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective
Cartier divisor. Assume
\begin{enumerate}
\item the diagonal of $X$ is affine, and
\item the de Rham complex of log poles is defined for
$Y \subset X$ over $S$.
\end{enumerate}
Let $b \in H^m_{dR}(X/S)$ be a de Rham cohomology class whose restriction
to $Y$ is zero. Then $c_1^{dR}(\mathcal{O}_X(Y)) \cup b = 0$ in
$H^{m + 2}_{dR}(X/S)$.
\end{lemma}

\noindent
This lemma is true without the assumption on the diagonal of $X$.
It can be proven by replacing open coverings in the proof below
by hypercoverings.

\begin{proof}
The short exact sequence of complexes of Lemma \ref{lemma-log-complex}
gives a boundary map
$$
H^m_{dR}(Y/S) =
H^{m + 1}(X, \Omega^\bullet_{Y/S}[-1]) \to
H^{m + 2}_{dR}(X, \Omega^\bullet_{X/S}) = H^{m + 2}_{dR}(X/S)
$$
We claim that $b \cup c_1^{dR}(\mathcal{O}_X(-Y))$ is equal to
the boundary of the restriction of $b$ to $Y$. The claim proves the lemma.

\medskip\noindent
To prove this, choose an affine open covering
$\mathcal{U} : X = \bigcup_{i \in I} U_i$ such that
$Y \cap U_i$ is the vanishing scheme of a nonzerodivisor
$f_i \in \mathcal{O}_X(U_i)$.
Then $U_{i_0 \ldots i_p} = U_{i_0} \cap \ldots \cap U_{i_p}$
is affine for all $i_0, \ldots, i_p \in I$ by our assumption on
the diagonal of $X$. Now the short exact sequence of complexes of
Lemma \ref{lemma-log-complex}
is termwise given by a short exact sequence of quasi-coherent modules.
Hence taking sections over any affine open produces a short exact
sequence. Thus we obtain a short exact sequence
$$
0 \to
\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{X/S}^\bullet)) \to
\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U},
\Omega_{X/S}^\bullet(\log Y)) \to
\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet[-1]))
\to 0
$$
of {\v C}ech complexes. By Cohomology of Schemes, Lemma
\ref{coherent-lemma-quasi-coherent-affine-cohomology-zero} and
Cohomology, Lemma
\ref{cohomology-lemma-cech-complex-complex-computes}
the associated long exact sequence of cohomology groups is the
long exact sequence of cohomology associated to the short exact
sequence of complexes in Lemma \ref{lemma-log-complex}
(small detail omitted).

\medskip\noindent
Using notation as in
Cohomology, Section \ref{cohomology-section-cech-cohomology-of-complexes}.
let $\beta = \{\beta_{i_0 \ldots i_p}\}$ be a cocycle representing $b$
which has degree $m$.
Let us compute the de Rham chern class of the invertible $\mathcal{O}_X$-module
$\mathcal{O}_X(-Y)$. On $U_i$ the section $f_i$ is a trivialization.
On $U_{i_0i_1}$ the regular functions $f_{i_0}$ and $f_{i_1}$
differ by a unit $f_{i_0i_1}$, like so
$$
f_{1_1}|_{U_{i_0i_1}} = f_{i_0i_1} \cdot f_{i_0}|_{U_{i_0i_1}}
$$
By the discussion in Section \ref{section-first-chern-class}
the de Rham cohomology class $c_1^{dR}(\mathcal{O}_X(-Y))$
is the class of the cocycle $\alpha = \{\alpha_{i_0 \ldots i_p}\}$
with $\alpha_{i_0i_1} = f_{i_0i_1}^{-1} \text{d}f_{i_0i_1}$
and zero in other degrees. The cup product is given by
$$
(\alpha \cup \beta)_{i_0 \ldots i_p} =
\sum\nolimits_{r = 0}^p
\epsilon(2, m, p, r)
\alpha_{i_0 \ldots i_r} \wedge \beta_{i_r \ldots i_p} =
(-1)^{p + 1} \alpha_{i_0i_1} \wedge \beta_{i_1 \ldots i_p}
$$
because $\epsilon(n, m, p, r)$ is $-1$ to the power
$(p + r)n + rp + r$.

\medskip\noindent
On the other hand, the restriction of $b$ to $Y$ is represented by the
cocycle
$\beta|_Y = \{\beta_{i_0 \ldots i_p}|_{U_{i_0 \ldots i_p} \cap Y}\}$
of the complex
$\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet))$.
This means that the cocycle in
$$
\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet[-1]))
$$
corresponding to $\beta|_Y$ has is given by
$\{(-1)^p\beta_{i_0 \ldots i_p}|_{U_{i_0 \ldots i_p} \cap Y}\}$
by the commutative diagram in
Cohomology, Remark \ref{cohomology-remark-shift-complex-cech-complex}
and the construction of the map
$$
\gamma :
\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet))[-1]
\longrightarrow
\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U}, \Omega_{Y/S}^\bullet[-1]))
$$
in Homology, Remark \ref{homology-remark-shift-double-complex}.
By our construction of the complex with log poles in the proof of
Lemma \ref{lemma-log-complex} this cocycle lifts to the cycle
$$
\gamma = \{(-1)^p\text{d}\log(f_{i_0}) \wedge \beta_{i_0 \ldots i_p}\}
$$
in the complex $\text{Tot}(\check{\mathcal{C}}^\bullet(\mathcal{U},
\Omega_{X/S}^\bullet(\log Y))$. We have
\begin{align*}
\text{d}(\gamma)_{i_0 \ldots i_p} = &
(-1)^{p - 1} \text{d}\log(f_{i_1}) \wedge \beta_{i_1 \ldots i_p} \\
& +
\sum\nolimits_{j = 1, \ldots, p}
(-1)^{j + p - 1}
\text{d}\log(f_{i_0}) \wedge \beta_{i_0 \ldots \hat i_j \ldots i_p} \\
& +
(-1)^{p + p} \text{d}(\text{d}\log(f_{i_0}) \wedge \beta_{i_0 \ldots i_p}) \\
= &
(-1)^{p - 1} (\text{d}\log(f_{i_1} - \text{d}\log(f_{i_0}))
\wedge \beta_{i_1 \ldots i_p} \\
& +
\sum\nolimits_{j = 0, \ldots, p}
(-1)^{j + p - 1}
\text{d}\log(f_{i_0}) \wedge \beta_{i_0 \ldots \hat i_j \ldots i_p} \\
& -
\text{d}\log(f_{i_0}) \wedge \text{d}(\beta_{i_0 \ldots i_p}) \\
= &
(-1)^{p - 1} f_{i_0i_1}^{-1} \text{d}f_{i_0i_1} \wedge \beta_{i_1 \ldots i_p}
\end{align*}
The last equality because the remaining terms sum up to
$(-1)^{p - 1}\text{d}\log(f_{i_0}) \wedge d(\beta)_{i_0 \ldots i_p}$ which
is zero as $\beta$ is a cocycle. This is the same result we got above
and the proof is complete.
\end{proof}

\begin{lemma}
\label{lemma-check-log-smooth}
Let $X \to S$ be a morphism of schemes. Let $Y \subset X$ be an effective
Cartier divisor. If both $X \to S$ and $Y \to S$ are smooth, then
the de Rham complex of log poles is defined for $Y \subset X$ over $S$.
\end{lemma}

\begin{proof}
In this case the modules $\Omega^p_{X/S}$ are locally free, see
Morphisms, Lemma \ref{morphisms-lemma-smooth-at-point}, and hence the second
condition of Definition \ref{definition-local-product} holds.
On the other hand, for $s \in S$ the fibre $Y_s \subset X_s$ is
an effective Cartier divisor
(Divisors, Lemma \ref{divisors-lemma-relative-Cartier}).
Hence if $y \in Y$ maps to $s \in S$ we have
$\dim \mathcal{O}_{Y_s, y} = \dim \mathcal{O}_{X_s, y} - 1$
for example by Algebra, Lemma \ref{algebra-lemma-one-equation}.
Thus $\dim_y(Y_s) = \dim_y(X_s) - 1$ by
Morphisms, Lemma \ref{morphisms-lemma-dimension-fibre-at-a-point}.
Now $\Omega_{X/S, x}$ is free of rank $\dim_y(X_s)$
and $\Omega_{Y/S, y}$ is free of rank $\dim_y(Y_s) = \dim_y(X_s) - 1$ by
the already used Morphisms, Lemma \ref{morphisms-lemma-smooth-at-point}.
Since $\Omega_{Y/S, y}$ is the quotient of
$\Omega_{X/S, x}$ by the submodule $f\Omega_{X/S, x}$ and
$\mathcal{O}_{X, x}\text{d}f$ we conclude that $\text{d}f$
must map to a nonzero element of $\Omega_{X/S, x} \otimes \kappa(y)$.
Hence $\text{d}f$ generates a direct summand and the proof is complete.
\end{proof}

\begin{remark}
\label{remark-check-log-completion-1}
Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite
type over $S$. Let $Y \subset X$ be an effective Cartier divisor.
If the map
$$
\mathcal{O}_{X, y}^\wedge \longrightarrow \mathcal{O}_{Y, y}^\wedge
$$
has a section for all $y \in Y$, then
the de Rham complex of log poles is defined for $Y \subset X$ over $S$.
If we ever need this result we will formulate a precise statement and
add a proof here.
\end{remark}

\begin{remark}
\label{remark-check-log-completion-2}
Let $S$ be a locally Noetherian scheme. Let $X$ be locally of finite
type over $S$. Let $Y \subset X$ be an effective Cartier divisor.
If for every $y \in Y$ we can find a diagram of schemes over $S$
$$
X \xleftarrow{\varphi} U \xrightarrow{\psi} V
$$
with $\varphi$ \'etale and $\psi|_{\varphi^{-1}(Y)} : \varphi^{-1}(Y) \to V$
\'etale, then the de Rham complex of log poles is defined for
$Y \subset X$ over $S$. A special case is when the pair $(X, Y)$
\'etale locally looks like $(V \times \mathbf{A}^1, V \times \{0\})$.
If we ever need this result we will formulate
a precise statement and add a proof here.
\end{remark}









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

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