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Tag 0AVV

35.5. Affine morphisms

In this section we collect some information about pushforward along an affine morphism of schemes.

Lemma 35.5.1. Let $f : X \to S$ be an affine morphism of schemes. Then $Rf_* : D_\mathit{QCoh}(\mathcal{O}_X) \to D_\mathit{QCoh}(\mathcal{O}_S)$ reflects isomorphisms.

Proof. The statement means that a morphism $\alpha : E \to F$ of $D_\mathit{QCoh}(\mathcal{O}_X)$ is an isomorphism if $Rf_*\alpha$ is an isomorphism. We may check this on cohomology sheaves. In particular, the question is local on $S$. Hence we may assume $S$ and therefore $X$ is affine. In this case the statement is clear from the description of the derived categories $D_\mathit{QCoh}(\mathcal{O}_X)$ and $D_\mathit{QCoh}(\mathcal{O}_S)$ given in Lemma 35.3.5. Some details omitted. $\square$

Lemma 35.5.2. Let $f : X \to S$ be an affine morphism of schemes. For $E$ in $D_\mathit{QCoh}(\mathcal{O}_S)$ we have $Rf_* Lf^* E = E \otimes^\mathbf{L}_{\mathcal{O}_S} f_*\mathcal{O}_X$.

Proof. Since $f$ is affine the map $f_*\mathcal{O}_X \to Rf_*\mathcal{O}_X$ is an isomorphism (Cohomology of Schemes, Lemma 29.2.3). There is a canonical map $E \otimes^\mathbf{L} f_*\mathcal{O}_X = E \otimes^\mathbf{L} Rf_*\mathcal{O}_X \to Rf_* Lf^* E$ adjoint to the map $$Lf^*(E \otimes^\mathbf{L} Rf_*\mathcal{O}_X) = Lf^*E \otimes^\mathbf{L} Lf^*Rf_*\mathcal{O}_X \longrightarrow Lf^* E \otimes^\mathbf{L} \mathcal{O}_X = Lf^* E$$ coming from $1 : Lf^*E \to Lf^*E$ and the canonical map $Lf^*Rf_*\mathcal{O}_X \to \mathcal{O}_X$. To check the map so constructed is an isomorphism we may work locally on $S$. Hence we may assume $S$ and therefore $X$ is affine. In this case the statement is clear from the description of the derived categories $D_\mathit{QCoh}(\mathcal{O}_X)$ and $D_\mathit{QCoh}(\mathcal{O}_S)$ and the functor $Lf^*$ given in Lemmas 35.3.5 and 35.3.8. Some details omitted. $\square$

Let $Y$ be a scheme. Let $\mathcal{A}$ be a sheaf of $\mathcal{O}_Y$-algebras. We will denote $D_\mathit{QCoh}(\mathcal{A})$ the inverse image of $D_\mathit{QCoh}(\mathcal{O}_X)$ under the restriction functor $D(\mathcal{A}) \to D(\mathcal{O}_X)$. In other words, $K \in D(\mathcal{A})$ is in $D_\mathit{QCoh}(\mathcal{A})$ if and only if its cohomology sheaves are quasi-coherent as $\mathcal{O}_X$-modules. If $\mathcal{A}$ is quasi-coherent itself this is the same as asking the cohomology sheaves to be quasi-coherent as $\mathcal{A}$-modules, see Morphisms, Lemma 28.11.6.

Lemma 35.5.3. Let $f : X \to Y$ be an affine morphism of schemes. Then $f_*$ induces an equivalence $$\Phi : D_\mathit{QCoh}(\mathcal{O}_X) \longrightarrow D_\mathit{QCoh}(f_*\mathcal{O}_X)$$ whose composition with $D_\mathit{QCoh}(f_*\mathcal{O}_X) \to D_\mathit{QCoh}(\mathcal{O}_Y)$ is $Rf_* : D_\mathit{QCoh}(\mathcal{O}_X) \to D_\mathit{QCoh}(\mathcal{O}_Y)$.

Proof. Recall that $Rf_*$ is computed on an object $K \in D_\mathit{QCoh}(\mathcal{O}_X)$ by choosing a K-injective complex $\mathcal{I}^\bullet$ of $\mathcal{O}_X$-modules representing $K$ and taking $f_*\mathcal{I}^\bullet$. Thus we let $\Phi(K)$ be the complex $f_*\mathcal{I}^\bullet$ viewed as a complex of $f_*\mathcal{O}_X$-modules. Denote $g : (X, \mathcal{O}_X) \to (Y, f_*\mathcal{O}_X)$ the obvious morphism of ringed spaces. Then $g$ is a flat morphism of ringed spaces (see below for a description of the stalks) and $\Phi$ is the restriction of $Rg_*$ to $D_\mathit{QCoh}(\mathcal{O}_X)$. We claim that $Lg^*$ is a quasi-inverse. First, observe that $Lg^*$ sends $D_\mathit{QCoh}(f_*\mathcal{O}_X)$ into $D_\mathit{QCoh}(\mathcal{O}_X)$ because $g^*$ transforms quasi-coherent modules into quasi-coherent modules (Modules, Lemma 17.10.4). To finish the proof it suffices to show that the adjunction mappings $$Lg^*\Phi(K) = Lg^*Rg_*K \to K \quad\text{and}\quad M \to Rg_*Lg^*M = \Phi(Lg^*M)$$ are isomorphisms for $K \in D_\mathit{QCoh}(\mathcal{O}_X)$ and $M \in D_\mathit{QCoh}(f_*\mathcal{O}_X)$. This is a local question, hence we may assume $Y$ and therefore $X$ are affine.

Assume $Y = \mathop{\mathrm{Spec}}(B)$ and $X = \mathop{\mathrm{Spec}}(A)$. Let $\mathfrak p = x \in \mathop{\mathrm{Spec}}(A) = X$ be a point mapping to $\mathfrak q = y \in \mathop{\mathrm{Spec}}(B) = Y$. Then $(f_*\mathcal{O}_X)_y = A_\mathfrak q$ and $\mathcal{O}_{X, x} = A_\mathfrak p$ hence $g$ is flat. Hence $g^*$ is exact and $H^i(Lg^*M) = g^*H^i(M)$ for any $M$ in $D(f_*\mathcal{O}_X)$. For $K \in D_\mathit{QCoh}(\mathcal{O}_X)$ we see that $$H^i(\Phi(K)) = H^i(Rf_*K) = f_*H^i(K)$$ by the vanishing of higher direct images (Cohomology of Schemes, Lemma 29.2.3) and Lemma 35.3.4. Thus it suffice to show that $$g^*g_*\mathcal{F} \to \mathcal{F} \quad\text{and}\quad \mathcal{G} \to g_*g^*\mathcal{F}$$ are isomorphisms where $\mathcal{F}$ is a quasi-coherent $\mathcal{O}_X$-module and $\mathcal{G}$ is a quasi-coherent $f_*\mathcal{O}_X$-module. This follows from Morphisms, Lemma 28.11.6. $\square$

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\section{Affine morphisms}
\label{section-affine-morphisms}

\noindent
In this section we collect some information about pushforward
along an affine morphism of schemes.

\begin{lemma}
\label{lemma-affine-morphism}
Let $f : X \to S$ be an affine morphism of schemes.
Then
$Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_S)$
reflects isomorphisms.
\end{lemma}

\begin{proof}
The statement means that a morphism $\alpha : E \to F$ of
$D_\QCoh(\mathcal{O}_X)$ is an isomorphism if
$Rf_*\alpha$ is an isomorphism. We may check this on cohomology sheaves.
In particular, the question is local on $S$. Hence we may assume $S$
and therefore $X$ is affine. In this case the statement is clear from
the description of the derived categories
$D_\QCoh(\mathcal{O}_X)$ and
$D_\QCoh(\mathcal{O}_S)$ given in
Lemma \ref{lemma-affine-compare-bounded}.
Some details omitted.
\end{proof}

\begin{lemma}
\label{lemma-affine-morphism-pull-push}
Let $f : X \to S$ be an affine morphism of schemes.
For $E$ in $D_\QCoh(\mathcal{O}_S)$ we have
$Rf_* Lf^* E = E \otimes^\mathbf{L}_{\mathcal{O}_S} f_*\mathcal{O}_X$.
\end{lemma}

\begin{proof}
Since $f$ is affine the map $f_*\mathcal{O}_X \to Rf_*\mathcal{O}_X$
is an isomorphism
(Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing}).
There is a canonical map $E \otimes^\mathbf{L} f_*\mathcal{O}_X = E \otimes^\mathbf{L} Rf_*\mathcal{O}_X \to Rf_* Lf^* E$
adjoint to the map
$$Lf^*(E \otimes^\mathbf{L} Rf_*\mathcal{O}_X) = Lf^*E \otimes^\mathbf{L} Lf^*Rf_*\mathcal{O}_X \longrightarrow Lf^* E \otimes^\mathbf{L} \mathcal{O}_X = Lf^* E$$
coming from $1 : Lf^*E \to Lf^*E$ and the canonical map
$Lf^*Rf_*\mathcal{O}_X \to \mathcal{O}_X$. To check the map so constructed
is an isomorphism we may work locally on $S$. Hence we may assume
$S$ and therefore $X$ is affine. In this case the statement is clear from
the description of the derived categories
$D_\QCoh(\mathcal{O}_X)$ and
$D_\QCoh(\mathcal{O}_S)$ and the functor $Lf^*$ given in
Lemmas \ref{lemma-affine-compare-bounded} and
\ref{lemma-quasi-coherence-pullback}.
Some details omitted.
\end{proof}

\noindent
Let $Y$ be a scheme. Let $\mathcal{A}$ be a sheaf of $\mathcal{O}_Y$-algebras.
We will denote $D_\QCoh(\mathcal{A})$ the inverse image of
$D_\QCoh(\mathcal{O}_X)$ under the restriction functor
$D(\mathcal{A}) \to D(\mathcal{O}_X)$. In other words, $K \in D(\mathcal{A})$
is in $D_\QCoh(\mathcal{A})$ if and only if its cohomology sheaves are
quasi-coherent as $\mathcal{O}_X$-modules. If $\mathcal{A}$ is quasi-coherent
itself this is the same as asking the cohomology sheaves to be quasi-coherent
as $\mathcal{A}$-modules, see
Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-modules}.

\begin{lemma}
\label{lemma-affine-morphism-equivalence}
Let $f : X \to Y$ be an affine morphism of schemes. Then $f_*$ induces
an equivalence
$$\Phi : D_\QCoh(\mathcal{O}_X) \longrightarrow D_\QCoh(f_*\mathcal{O}_X)$$
whose composition with $D_\QCoh(f_*\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$
is $Rf_* : D_\QCoh(\mathcal{O}_X) \to D_\QCoh(\mathcal{O}_Y)$.
\end{lemma}

\begin{proof}
Recall that $Rf_*$ is computed on an object $K \in D_\QCoh(\mathcal{O}_X)$
by choosing a K-injective complex $\mathcal{I}^\bullet$ of
$\mathcal{O}_X$-modules representing $K$ and taking $f_*\mathcal{I}^\bullet$.
Thus we let $\Phi(K)$ be the complex $f_*\mathcal{I}^\bullet$
viewed as a complex of $f_*\mathcal{O}_X$-modules.
Denote $g : (X, \mathcal{O}_X) \to (Y, f_*\mathcal{O}_X)$ the
obvious morphism of ringed spaces. Then $g$ is a flat morphism of
ringed spaces (see below for a description of the stalks) and
$\Phi$ is the restriction of $Rg_*$ to $D_\QCoh(\mathcal{O}_X)$.
We claim that $Lg^*$ is a quasi-inverse. First, observe that
$Lg^*$ sends $D_\QCoh(f_*\mathcal{O}_X)$ into $D_\QCoh(\mathcal{O}_X)$
because $g^*$ transforms quasi-coherent modules into quasi-coherent
modules (Modules, Lemma \ref{modules-lemma-pullback-quasi-coherent}).
To finish the proof it suffices to show that
$$Lg^*\Phi(K) = Lg^*Rg_*K \to K \quad\text{and}\quad M \to Rg_*Lg^*M = \Phi(Lg^*M)$$
are isomorphisms for $K \in D_\QCoh(\mathcal{O}_X)$ and
$M \in D_\QCoh(f_*\mathcal{O}_X)$. This is a local question, hence
we may assume $Y$ and therefore $X$ are affine.

\medskip\noindent
Assume $Y = \Spec(B)$ and $X = \Spec(A)$. Let
$\mathfrak p = x \in \Spec(A) = X$ be a point mapping to
$\mathfrak q = y \in \Spec(B) = Y$. Then
$(f_*\mathcal{O}_X)_y = A_\mathfrak q$ and $\mathcal{O}_{X, x} = A_\mathfrak p$
hence $g$ is flat. Hence $g^*$ is exact and $H^i(Lg^*M) = g^*H^i(M)$
for any $M$ in $D(f_*\mathcal{O}_X)$.
For $K \in D_\QCoh(\mathcal{O}_X)$ we see that
$$H^i(\Phi(K)) = H^i(Rf_*K) = f_*H^i(K)$$
by the vanishing of higher direct images
(Cohomology of Schemes, Lemma \ref{coherent-lemma-relative-affine-vanishing})
and Lemma \ref{lemma-application-nice-K-injective}.
Thus it suffice to show that
$$g^*g_*\mathcal{F} \to \mathcal{F} \quad\text{and}\quad \mathcal{G} \to g_*g^*\mathcal{F}$$
are isomorphisms where $\mathcal{F}$ is
a quasi-coherent $\mathcal{O}_X$-module and $\mathcal{G}$ is
a quasi-coherent $f_*\mathcal{O}_X$-module. This follows from
Morphisms, Lemma \ref{morphisms-lemma-affine-equivalence-modules}.
\end{proof}

Comment #3235 by denis lieberman (site) on March 13, 2018 a 2:13 pm UTC

In the Proof of Lemma 35.5.3 should the obvious morphism of ringed spaces for sheaves be g:(X,O_x) -->(Y,f_O_y) and not -->(Y,f_O_x).

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