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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}
    Adjointness follows from
    Modules, Lemma \ref{modules-lemma-adjoint-tensor-restrict}.
    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}
    \label{lemma-sheafhom-affine-adjoint}
    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
    Derived Categories, Lemma \ref{derived-lemma-derived-adjoint-functors}.
    \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
    \ref{lemma-sheafhom-affine-adjoint}.
    \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}

    Comments (2)

    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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