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Move two lemmas and generalize one
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Relative dualizing complex for P(E)
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aisejohan committed Aug 27, 2015
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240 changes: 139 additions & 101 deletions dualizing.tex
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Expand Up @@ -4541,7 +4541,7 @@ \section{Right adjoint of pushforward}



\section{Base change for right adjoints to pushforward}
\section{Right adjoint of pushforward and base change}
\label{section-base-change-map}

\noindent
Expand Down Expand Up @@ -5016,7 +5016,7 @@ \section{Base change for right adjoints to pushforward}



\section{Trace maps for right adjoints to pushforward}
\section{Right adjoint of pushforward and trace maps}
\label{section-trace}

\noindent
Expand Down Expand Up @@ -5238,7 +5238,7 @@ \section{Trace maps for right adjoints to pushforward}



\section{Right adjoint to pushforward and pullback}
\section{Right adjoint of pushforward and pullback}
\label{section-compare-with-pullback}

\noindent
Expand Down Expand Up @@ -5428,7 +5428,7 @@ \section{Right adjoint to pushforward and pullback}



\section{Right adjoint for pushforward along a closed immersion}
\section{Right adjoint of pushforward for closed immersions}
\label{section-sections-with-exact-support}

\noindent
Expand Down Expand Up @@ -5671,7 +5671,7 @@ \section{Right adjoint for pushforward along a closed immersion}



\section{Right adjoint for pushforward along a finite morphism}
\section{Right adjoint of pushforward for finite morphisms}
\label{section-duality-finite}

\noindent
Expand Down Expand Up @@ -5784,7 +5784,7 @@ \section{Right adjoint for pushforward along a finite morphism}



\section{Right adjoint for pushforward along perfect proper morphisms}
\section{Right adjoint of pushforward for perfect proper morphisms}
\label{section-flat-and-proper}

\noindent
Expand Down Expand Up @@ -5908,6 +5908,13 @@ \section{Right adjoint for pushforward along perfect proper morphisms}
Lemma \ref{lemma-compare-with-pullback-perfect}.
\end{proof}

\noindent
The following lemma shows that the base change map
(\ref{equation-base-change-map}) is an isomorphism
for flat proper morphisms. We will see in
Example \ref{example-base-change-wrong}
that this does not remain true for perfect proper morphisms.

\begin{lemma}
\label{lemma-proper-flat-base-change}
Let $f : X \to Y$ be a flat proper morphism of Noetherian schemes.
Expand All @@ -5923,100 +5930,6 @@ \section{Right adjoint for pushforward along perfect proper morphisms}
case the statement follows from Lemma \ref{lemma-more-base-change}.
\end{proof}

\begin{lemma}
\label{lemma-upper-shriek-P1}
Let $f : X = \mathbf{P}^1_Y \to Y$ be the projection where $Y$ is
a Noetherian scheme. Let $a$ be the right adjoint for
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of
Lemma \ref{lemma-twisted-inverse-image}.
Then $a(\mathcal{O}_Y)$ is isomorphic to $\mathcal{O}_X(-2)[1]$.
\end{lemma}

\begin{proof}
Recall that there is an identification
$Rf_*(\mathcal{O}_X(-2)[1]) = \mathcal{O}_Y$, see
Cohomology of Schemes, Lemma
\ref{coherent-lemma-cohomology-projective-space-over-base} or
\ref{coherent-lemma-cohomology-projective-bundle}.
This determines a map $\mathcal{O}_X(-2)[1] \to a(\mathcal{O}_Y)$.
By Lemma \ref{lemma-proper-noetherian} construction of the $a$
is local on the base. In particular, to check that
$\mathcal{O}_X(-2)[1] \to a(\mathcal{O}_Y)$ is an isomorphism, we
may work locally on $Y$. In other words, we may assume $Y$ is affine.
In the affine case the sheaves $\mathcal{O}_X$ and $\mathcal{O}_X(-1)$
generate $D_\QCoh(X)$, see
Derived Categories of Schemes, Lemma \ref{perfect-lemma-generator-P1}.
Hence it suffices to show that the maps
\begin{align*}
H^{-n + 1}(X, \mathcal{O}(-2))
& =
\Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X[n], \mathcal{O}_X(-2)[1]) \\
& \to
\Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X[n], a(\mathcal{O}_Y)) \\
& =
\Hom_{D(\mathcal{O}_Y)}(Rf_*(\mathcal{O}_X)[n], \mathcal{O}_Y) \\
& =
H^{-n}(Y, \mathcal{O}_Y)
\end{align*}
and
\begin{align*}
H^{-n + 1}(X, \mathcal{O}_X(-1))
& =
\Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X(-1)[n], \mathcal{O}_X(-2)[1]) \\
& \to
\Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X(-1)[n], a(\mathcal{O}_Y)) \\
& =
\Hom_{D(\mathcal{O}_Y)}(Rf_*(\mathcal{O}_X(-1))[n], \mathcal{O}_Y) \\
& = 0
\end{align*}
(where we used Cohomology of Schemes, Lemma
\ref{coherent-lemma-cohomology-projective-space-over-ring})
are isomorphisms for all $n \in \mathbf{Z}$. This is clear from
the explicit computation of cohomology in
Cohomology of Schemes, Lemma
\ref{coherent-lemma-cohomology-projective-space-over-ring}.
\end{proof}

\begin{example}
\label{example-base-change-wrong}
The base change map (\ref{equation-base-change-map}) is not an
isomorphism if $f$ is perfect proper and $g$ is perfect.
Let $k$ be a field. Let $Y = \mathbf{A}^2_k$ and let $f : X \to Y$
be the blow up of $Y$ in the origin. Denote $E \subset X$ the
exceptional divisor. Then we can factor $f$ as
$$
X \xrightarrow{i} \mathbf{P}^1_Y \xrightarrow{p} Y
$$
This gives a factorization $a = c \circ b$ where
$b$ and $c$ are the right adjoints of
Lemma \ref{lemma-twisted-inverse-image}
for $p$ and $i$. Denote $\mathcal{O}(n)$ the
Serre twist of the structure sheaf on $\mathbf{P}^1_Y$ and
denote $\mathcal{O}_X(n)$ its restriction to $X$.
Note that $X \subset \mathbf{P}^1_Y$ is cut out by
a degree one equation, hence $\mathcal{O}(X) = \mathcal{O}(1)$.
By Lemma \ref{lemma-upper-shriek-P1} we have
$b(\mathcal{O}_Y) = \mathcal{O}(-2)[1]$.
By Lemma \ref{lemma-twisted-inverse-image-closed}
we have
$$
a(\mathcal{O}_Y) = c(b(\mathcal{O}_Y)) =
c(\mathcal{O}(-2)[1]) =
R\SheafHom(\mathcal{O}_X, \mathcal{O}(-2)[1]) =
\mathcal{O}_X(-1)
$$
Last equality by Lemma \ref{lemma-sheaf-with-exact-support-effective-Cartier}.
Let $Y' = \Spec(k)$ be the origin in $Y$. The restriction of
$a(\mathcal{O}_Y)$ to $X' = E = \mathbf{P}^1_k$
is an invertible sheaf of degree $-1$ placed in cohomological
degree $0$. But on the other hand,
$a'(\mathcal{O}_{\Spec(k)}) = \mathcal{O}_E(-2)[1]$
which is an invertible sheaf of degree $-2$ placed in
cohomological degree $-1$, so different. In this example
(4) is the only hypothesis of Lemma \ref{lemma-more-base-change}
which is violated.
\end{example}

\begin{remark}
\label{remark-relative-dualizing-complex}
Let $f : X \to Y$ be a flat proper morphism of Noetherian schemes.
Expand Down Expand Up @@ -6060,7 +5973,7 @@ \section{Right adjoint for pushforward along perfect proper morphisms}



\section{Right adjoint to pushforward for effective Cartier divisors}
\section{Right adjoint of pushforward for effective Cartier divisors}
\label{section-dualizing-Cartier}

\noindent
Expand Down Expand Up @@ -6206,6 +6119,131 @@ \section{Right adjoint to pushforward for effective Cartier divisors}



\section{Right adjoint of pushforward in examples}
\label{section-examples}

\noindent
In this section we compute the right adjoint to pushforward in
some examples.

\begin{lemma}
\label{lemma-upper-shriek-P1}
Let $Y$ be a Noetherian scheme. Let $\mathcal{E}$ be a finite locally
free $\mathcal{O}_Y$-module of rank $n + 1$ with determinant
$\mathcal{L} = \wedge^{n + 1}(\mathcal{E})$.
Let $f : X = \mathbf{P}(\mathcal{E}) \to Y$ be the projection.
Let $a$ be the right adjoint for
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$ of
Lemma \ref{lemma-twisted-inverse-image}.
Then there is a canonical isomorphism
$$
c : f^*\mathcal{L}(-n - 1)[n] \longrightarrow a(\mathcal{O}_Y)
$$
In particular, if $\mathcal{E} = \mathcal{O}_Y^{\oplus n + 1}$, then
$X = \mathbf{P}^n_Y$ and we obtain
$a(\mathcal{O}_Y) = \mathcal{O}_X(-n - 1)[n]$.
\end{lemma}

\begin{proof}
In (the proof of) Cohomology of Schemes, Lemma
\ref{coherent-lemma-cohomology-projective-bundle}
we constructed a canonical isomorphism
$$
\mathcal{L} \otimes_{\mathcal{O}_Y} R^nf_*(\mathcal{O}_X(-n - 1))
\longrightarrow
\mathcal{O}_Y
$$
Moreover, $Rf_*\mathcal{O}_X(-n - 1) = R^nf_*\mathcal{O}_X(-n - 1)[n]$,
i.e., the other higher direct images are zero. Using the projection
formula (Cohomology, Lemma \ref{cohomology-lemma-projection-formula}) to bring
$\mathcal{L}$ inside we find an isomorphism
$$
Rf_*(f^*\mathcal{L}(-n - 1)[n]) \longrightarrow \mathcal{O}_Y
$$
This isomorphism determines $c$ as in the statement of the lemma
because $a$ is the right adjoint of $Rf_*$.
By Lemma \ref{lemma-proper-noetherian} construction of the $a$
is local on the base. In particular, to check that
$c$ is an isomorphism, we may work locally on $Y$.
In other words, we may assume $Y$ is affine and
$\mathcal{E} = \mathcal{O}_Y^{\oplus n + 1}$.
In this case the sheaves $\mathcal{O}_X, \mathcal{O}_X(-1), \ldots,
\mathcal{O}_X(-n)$ generate $D_\QCoh(X)$, see
Derived Categories of Schemes, Lemma \ref{perfect-lemma-generator-P1}.
Hence it suffices to show that
$c : \mathcal{O}_X(-n - 1)[n] \to a(\mathcal{O}_Y)$
is transformed into an isomorphism under the functors
$$
F_{i, p}(-) = \Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X(i), (-)[p])
$$
for $i \in \{-n, \ldots, 0\}$ and $p \in \mathbf{Z}$.
For $F_{0, p}$ this holds by construction of the arrow $c$!
For $i \in \{-n, \ldots, -1\}$ we have
$$
\Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X(i), \mathcal{O}_X(-n - 1)[n + p]) =
H^p(X, \mathcal{O}_X(-n - 1 - i)) = 0
$$
by the computation of cohomology of projective space
(Cohomology of Schemes, Lemma
\ref{coherent-lemma-cohomology-projective-space-over-ring})
and we have
$$
\Hom_{D(\mathcal{O}_X)}(\mathcal{O}_X(i), a(\mathcal{O}_Y)[p]) =
\Hom_{D(\mathcal{O}_Y)}(Rf_*\mathcal{O}_X(i), \mathcal{O}_Y[p]) = 0
$$
because $Rf_*\mathcal{O}_X(i) = 0$ by the same lemma.
Hence the source and the target of $F_{i, p}(c)$ vanish
and $F_{i, p}(c)$ is necessarily an isomorphism.
This finishes the proof.
\end{proof}

\begin{example}
\label{example-base-change-wrong}
The base change map (\ref{equation-base-change-map}) is not an
isomorphism if $f$ is perfect proper and $g$ is perfect.
Let $k$ be a field. Let $Y = \mathbf{A}^2_k$ and let $f : X \to Y$
be the blow up of $Y$ in the origin. Denote $E \subset X$ the
exceptional divisor. Then we can factor $f$ as
$$
X \xrightarrow{i} \mathbf{P}^1_Y \xrightarrow{p} Y
$$
This gives a factorization $a = c \circ b$ where
$a$, $b$, and $c$ are the right adjoints of
Lemma \ref{lemma-twisted-inverse-image}
of $Rf_*$, $Rp_*$, and $Ri_*$. Denote $\mathcal{O}(n)$ the
Serre twist of the structure sheaf on $\mathbf{P}^1_Y$ and
denote $\mathcal{O}_X(n)$ its restriction to $X$.
Note that $X \subset \mathbf{P}^1_Y$ is cut out by
a degree one equation, hence $\mathcal{O}(X) = \mathcal{O}(1)$.
By Lemma \ref{lemma-upper-shriek-P1} we have
$b(\mathcal{O}_Y) = \mathcal{O}(-2)[1]$.
By Lemma \ref{lemma-twisted-inverse-image-closed}
we have
$$
a(\mathcal{O}_Y) = c(b(\mathcal{O}_Y)) =
c(\mathcal{O}(-2)[1]) =
R\SheafHom(\mathcal{O}_X, \mathcal{O}(-2)[1]) =
\mathcal{O}_X(-1)
$$
Last equality by Lemma \ref{lemma-sheaf-with-exact-support-effective-Cartier}.
Let $Y' = \Spec(k)$ be the origin in $Y$. The restriction of
$a(\mathcal{O}_Y)$ to $X' = E = \mathbf{P}^1_k$
is an invertible sheaf of degree $-1$ placed in cohomological
degree $0$. But on the other hand,
$a'(\mathcal{O}_{\Spec(k)}) = \mathcal{O}_E(-2)[1]$
which is an invertible sheaf of degree $-2$ placed in
cohomological degree $-1$, so different. In this example
(4) is the only hypothesis of Lemma \ref{lemma-more-base-change}
which is violated.
\end{example}








\section{Compactifications}
\label{section-compactify}

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2 changes: 1 addition & 1 deletion perfect.tex
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Expand Up @@ -3422,7 +3422,7 @@ \section{Derived categories as module categories}
\begin{example}
\label{example-Pn-module-category}
Let $A$ be a ring. Let $X = \mathbf{P}^n_A = \text{Proj}(S)$
where $S = A[x_0, \ldots, X_n]$. By Lemma \ref{lemma-generator-P1}
where $S = A[X_0, \ldots, X_n]$. By Lemma \ref{lemma-generator-P1}
we know that
$$
P =
Expand Down

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