# The Stacks Project

## Tag 0AWZ

### 46.11. Right adjoint of pushforward for finite morphisms

If $i : Z \to X$ is a closed immersion of schemes, then there is a right adjoint $\mathop{\mathcal{H}\!\mathit{om}}\nolimits(\mathcal{O}_Z, -)$ to the functor $i_* : \textit{Mod}(\mathcal{O}_Z) \to \textit{Mod}(\mathcal{O}_X)$ whose derived extension $R\mathop{\mathcal{H}\!\mathit{om}}\nolimits(\mathcal{O}_Z, -)$ is the right adjoint to $Ri_* : D(\mathcal{O}_Z) \to D(\mathcal{O}_X)$. See Section 46.9. In the case of a finite morphism $f : Y \to X$ this strategy cannot work, as the functor $f_* : \textit{Mod}(\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$ is not exact in general and hence does not have a right adjoint. A replacement is to consider the exact functor $\textit{Mod}(f_*\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$ and consider the corresponding right adjoint and its derived extension.

Let $f : Y \to X$ be an affine morphism of schemes. For a sheaf of $\mathcal{O}_X$-modules $\mathcal{F}$ the sheaf $$\mathop{\mathcal{H}\!\mathit{om}}\nolimits_{\mathcal{O}_X}(f_*\mathcal{O}_Y, \mathcal{F})$$ is a sheaf of $f_*\mathcal{O}_Y$-modules. We obtain a functor $\textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(f_*\mathcal{O}_Y)$ which we will denote $\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)$.

Lemma 46.11.1. With notation as above. The functor $\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)$ is a right adjoint to the restriction functor $\textit{Mod}(f_*\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$. For an affine open $U \subset X$ we have $$\Gamma(U, \mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, \mathcal{F})) = \mathop{\mathrm{Hom}}\nolimits_A(B, \mathcal{F}(U))$$ where $A = \mathcal{O}_X(U)$ and $B = \mathcal{O}_Y(f^{-1}(U))$.

Proof. Adjointness follows from Modules, Lemma 17.20.6. As $f$ is affine we see that $f_*\mathcal{O}_Y$ is the quasi-coherent sheaf corresponding to $B$ viewed as an $A$-module. Hence the description of sections over $U$ follows from Schemes, Lemma 25.7.1. $\square$

The functor $\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)$ is left exact. Let $$R\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -) : D(\mathcal{O}_X) \longrightarrow D(f_*\mathcal{O}_Y)$$ be its derived extension.

Lemma 46.11.2. With notation as above. The functor $R\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)$ is the right adjoint of the functor $D(f_*\mathcal{O}_Y) \to D(\mathcal{O}_X)$.

Proof. Follows from Lemma 46.11.1 and Derived Categories, Lemma 13.28.5. $\square$

Lemma 46.11.3. With notation as above. The composition $$D(\mathcal{O}_X) \xrightarrow{R\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)} D(f_*\mathcal{O}_Y) \to D(\mathcal{O}_X)$$ is the functor $K \mapsto R\mathop{\mathcal{H}\!\mathit{om}}\nolimits_{\mathcal{O}_X}(f_*\mathcal{O}_Y, K)$.

Proof. This is immediate from the construction. $\square$

Lemma 46.11.4. Let $f : Y \to X$ be a finite pseudo-coherent morphism of schemes (a finite morphism of Noetherian schemes is pseudo-coherent). The functor $R\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)$ maps $D_\mathit{QCoh}^+(\mathcal{O}_X)$ into $D_\mathit{QCoh}^+(f_*\mathcal{O}_Y)$. If $X$ is quasi-compact and quasi-separated, then the diagram $$\xymatrix{ D_\mathit{QCoh}^+(\mathcal{O}_X) \ar[rr]_a \ar[rd]_{R\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)} & & D_\mathit{QCoh}^+(\mathcal{O}_Y) \ar[ld]^\Phi \\ & D_\mathit{QCoh}^+(f_*\mathcal{O}_Y) }$$ is commutative, where $a$ is the right adjoint of Lemma 46.3.1 for $f$ and $\Phi$ is the equivalence of Derived Categories of Schemes, Lemma 35.5.3.

Proof. (The parenthetical remark follows from More on Morphisms, Lemma 36.49.9.) Since $f$ is pseudo-coherent, the $\mathcal{O}_X$-module $f_*\mathcal{O}_Y$ is pseudo-coherent, see More on Morphisms, Lemma 36.49.8. Thus $R\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)$ maps $D_\mathit{QCoh}^+(\mathcal{O}_X)$ into $D_\mathit{QCoh}^+(f_*\mathcal{O}_Y)$, see Derived Categories of Schemes, Lemma 35.9.8. Then $\Phi \circ a$ and $R\mathop{\mathcal{H}\!\mathit{om}}\nolimits(f_*\mathcal{O}_Y, -)$ agree on $D_\mathit{QCoh}^+(\mathcal{O}_X)$ because these functors are both right adjoint to the restriction functor $D_\mathit{QCoh}^+(f_*\mathcal{O}_Y) \to D_\mathit{QCoh}^+(\mathcal{O}_X)$. To see this use Lemmas 46.3.5 and 46.11.2. $\square$

Remark 46.11.5. If $f : Y \to X$ is a finite morphism of Noetherian schemes, then the diagram $$\xymatrix{ Rf_*a(K) \ar[r]_-{\text{Tr}_{f, K}} \ar@{=}[d] & K \ar@{=}[d] \\ R\mathop{\mathcal{H}\!\mathit{om}}\nolimits_{\mathcal{O}_X}(f_*\mathcal{O}_Y, K) \ar[r] & K }$$ is commutative for $K \in D_\mathit{QCoh}^+(\mathcal{O}_X)$. This follows from Lemma 46.11.4. The lower horizontal arrow is induced by the map $\mathcal{O}_X \to f_*\mathcal{O}_Y$ and the upper horizontal arrow is the trace map discussed in Section 46.7.

The code snippet corresponding to this tag is a part of the file duality.tex and is located in lines 2319–2465 (see updates for more information).

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

\noindent
If $i : Z \to X$ is a closed immersion of schemes, then there is
a right adjoint $\SheafHom(\mathcal{O}_Z, -)$ to the functor
$i_* : \textit{Mod}(\mathcal{O}_Z) \to \textit{Mod}(\mathcal{O}_X)$
whose derived extension $R\SheafHom(\mathcal{O}_Z, -)$
is the right adjoint to $Ri_* : D(\mathcal{O}_Z) \to D(\mathcal{O}_X)$. See
Section \ref{section-sections-with-exact-support}.
In the case of a finite morphism $f : Y \to X$ this strategy
cannot work, as the functor
$f_* : \textit{Mod}(\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$
is not exact in general and hence does not have a right adjoint.
A replacement is to consider the exact functor
$\textit{Mod}(f_*\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$
and consider the corresponding right adjoint and its derived
extension.

\medskip\noindent
Let $f : Y \to X$ be an affine morphism of schemes. For a sheaf
of $\mathcal{O}_X$-modules $\mathcal{F}$ the sheaf
$$\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, \mathcal{F})$$
is a sheaf of $f_*\mathcal{O}_Y$-modules. We obtain a functor
$\textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(f_*\mathcal{O}_Y)$
which we will denote $\SheafHom(f_*\mathcal{O}_Y, -)$.

\begin{lemma}
\label{lemma-compute-sheafhom-affine}
With notation as above. The functor $\SheafHom(f_*\mathcal{O}_Y, -)$ is a
right adjoint to the restriction functor
$\textit{Mod}(f_*\mathcal{O}_Y) \to \textit{Mod}(\mathcal{O}_X)$.
For an affine open $U \subset X$ we have
$$\Gamma(U, \SheafHom(f_*\mathcal{O}_Y, \mathcal{F})) = \Hom_A(B, \mathcal{F}(U))$$
where $A = \mathcal{O}_X(U)$ and $B = \mathcal{O}_Y(f^{-1}(U))$.
\end{lemma}

\begin{proof}
As $f$ is affine we see that $f_*\mathcal{O}_Y$ is
the quasi-coherent sheaf corresponding to $B$ viewed
as an $A$-module. Hence the description of sections over $U$ follows from
Schemes, Lemma \ref{schemes-lemma-compare-constructions}.
\end{proof}

\noindent
The functor $\SheafHom(f_*\mathcal{O}_Y, -)$ is left exact. Let
$$R\SheafHom(f_*\mathcal{O}_Y, -) : D(\mathcal{O}_X) \longrightarrow D(f_*\mathcal{O}_Y)$$
be its derived extension.

\begin{lemma}
With notation as above. The functor $R\SheafHom(f_*\mathcal{O}_Y, -)$
is the right adjoint of the functor $D(f_*\mathcal{O}_Y) \to D(\mathcal{O}_X)$.
\end{lemma}

\begin{proof}
Follows from Lemma \ref{lemma-compute-sheafhom-affine}
and
\end{proof}

\begin{lemma}
\label{lemma-sheafhom-affine-ext}
With notation as above. The composition
$$D(\mathcal{O}_X) \xrightarrow{R\SheafHom(f_*\mathcal{O}_Y, -)} D(f_*\mathcal{O}_Y) \to D(\mathcal{O}_X)$$
is the functor $K \mapsto R\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, K)$.
\end{lemma}

\begin{proof}
This is immediate from the construction.
\end{proof}

\begin{lemma}
\label{lemma-finite-twisted}
Let $f : Y \to X$ be a finite pseudo-coherent morphism of schemes
(a finite morphism of Noetherian schemes is pseudo-coherent).
The functor $R\SheafHom(f_*\mathcal{O}_Y, -)$ maps
$D_\QCoh^+(\mathcal{O}_X)$ into $D_\QCoh^+(f_*\mathcal{O}_Y)$.
If $X$ is quasi-compact and quasi-separated, then the diagram
$$\xymatrix{ D_\QCoh^+(\mathcal{O}_X) \ar[rr]_a \ar[rd]_{R\SheafHom(f_*\mathcal{O}_Y, -)} & & D_\QCoh^+(\mathcal{O}_Y) \ar[ld]^\Phi \\ & D_\QCoh^+(f_*\mathcal{O}_Y) }$$
is commutative, where $a$ is the right adjoint of
Lemma \ref{lemma-twisted-inverse-image} for $f$ and $\Phi$ is the equivalence
of Derived Categories of Schemes, Lemma
\ref{perfect-lemma-affine-morphism-equivalence}.
\end{lemma}

\begin{proof}
(The parenthetical remark follows from More on Morphisms, Lemma
\ref{more-morphisms-lemma-Noetherian-pseudo-coherent}.)
Since $f$ is pseudo-coherent, the $\mathcal{O}_X$-module $f_*\mathcal{O}_Y$
is pseudo-coherent, see More on Morphisms, Lemma
\ref{more-morphisms-lemma-finite-pseudo-coherent}.
Thus $R\SheafHom(f_*\mathcal{O}_Y, -)$ maps
$D_\QCoh^+(\mathcal{O}_X)$ into
$D_\QCoh^+(f_*\mathcal{O}_Y)$, see
Derived Categories of Schemes, Lemma
\ref{perfect-lemma-quasi-coherence-internal-hom}.
Then $\Phi \circ a$ and $R\SheafHom(f_*\mathcal{O}_Y, -)$
agree on $D_\QCoh^+(\mathcal{O}_X)$ because these functors are
both right adjoint to the restriction functor
$D_\QCoh^+(f_*\mathcal{O}_Y) \to D_\QCoh^+(\mathcal{O}_X)$. To see this
use Lemmas \ref{lemma-twisted-inverse-image-bounded-below} and
\end{proof}

\begin{remark}
\label{remark-trace-map-finite}
If $f : Y \to X$ is a finite morphism of Noetherian schemes, then the diagram
$$\xymatrix{ Rf_*a(K) \ar[r]_-{\text{Tr}_{f, K}} \ar@{=}[d] & K \ar@{=}[d] \\ R\SheafHom_{\mathcal{O}_X}(f_*\mathcal{O}_Y, K) \ar[r] & K }$$
is commutative for $K \in D_\QCoh^+(\mathcal{O}_X)$. This follows
from Lemma \ref{lemma-finite-twisted}. The lower horizontal
arrow is induced by the map $\mathcal{O}_X \to f_*\mathcal{O}_Y$ and the
upper horizontal arrow is the trace map discussed in
Section \ref{section-trace}.
\end{remark}

Comment #1888 by Keenan Kidwell on April 1, 2016 a 11:11 pm UTC

In the text block below Tag 0BUZ, "let us denote be its..." should have "us denote" removed or perhaps "be its" changed to just "its." I vote for the first one. Or possibly "let us denote by...its."

Comment #1906 by Johan (site) on April 2, 2016 a 12:33 am UTC

Thanks, fixed here.

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