The Stacks project

48.11 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 48.9. 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 48.11.1. 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 17.22.3. 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 26.7.1. $\square$

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 48.11.2. 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 48.11.1 and Derived Categories, Lemma 13.30.3. $\square$

Lemma 48.11.3. 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 48.11.4. 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 48.3.1 for $f$ and $\Phi $ is the equivalence of Derived Categories of Schemes, Lemma 36.5.4.

Proof. (The parenthetical remark follows from More on Morphisms, Lemma 37.60.9.) Since $f$ is pseudo-coherent, the $\mathcal{O}_ X$-module $f_*\mathcal{O}_ Y$ is pseudo-coherent, see More on Morphisms, Lemma 37.60.8. 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 36.10.8. 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 48.3.5 and 48.11.2. $\square$

Remark 48.11.5. 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 48.11.4. 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 48.7.

Comments (2)

Comment #1888 by Keenan Kidwell on

In the text block below Tag 48.11.1, "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."

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