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48.3 Right adjoint of pushforward

References for this section and the following are [Neeman-Grothendieck], [LN], [Lipman-notes], and [Neeman-improvement].

Let $f : X \to Y$ be a morphism of schemes. In this section we consider the right adjoint to the functor $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. In the literature, if this functor exists, then it is sometimes denoted $f^{\times }$. This notation is not universally accepted and we refrain from using it. We will not use the notation $f^!$ for such a functor, as this would clash (for general morphisms $f$) with the notation in [RD].

reference

Lemma 48.3.1. Let $f : X \to Y$ be a morphism between quasi-separated and quasi-compact schemes. The functor $Rf_* : D_\mathit{QCoh}(X) \to D_\mathit{QCoh}(Y)$ has a right adjoint.

Proof. We will prove a right adjoint exists by verifying the hypotheses of Derived Categories, Proposition 13.37.2. First off, the category $D_\mathit{QCoh}(\mathcal{O}_ X)$ has direct sums, see Derived Categories of Schemes, Lemma 36.3.1. The category $D_\mathit{QCoh}(\mathcal{O}_ X)$ is compactly generated by Derived Categories of Schemes, Theorem 36.14.3. Since $X$ and $Y$ are quasi-compact and quasi-separated, so is $f$, see Schemes, Lemmas 26.21.13 and 26.21.14. Hence the functor $Rf_*$ commutes with direct sums, see Derived Categories of Schemes, Lemma 36.4.2. This finishes the proof. $\square$

Example 48.3.2. Let $A \to B$ be a ring map. Let $Y = \mathop{\mathrm{Spec}}(A)$ and $X = \mathop{\mathrm{Spec}}(B)$ and $f : X \to Y$ the morphism corresponding to $A \to B$. Then $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ corresponds to restriction $D(B) \to D(A)$ via the equivalences $D(B) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ and $D(A) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. Hence the right adjoint corresponds to the functor $K \longmapsto R\mathop{\mathrm{Hom}}\nolimits (B, K)$ of Dualizing Complexes, Section 47.13.

Example 48.3.3. If $f : X \to Y$ is a separated finite type morphism of Noetherian schemes, then the right adjoint of $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ does not map $D_{\textit{Coh}}(\mathcal{O}_ Y)$ into $D_{\textit{Coh}}(\mathcal{O}_ X)$. Namely, let $k$ be a field and consider the morphism $f : \mathbf{A}^1_ k \to \mathop{\mathrm{Spec}}(k)$. By Example 48.3.2 this corresponds to the question of whether $R\mathop{\mathrm{Hom}}\nolimits (B, -)$ maps $D_{\textit{Coh}}(A)$ into $D_{\textit{Coh}}(B)$ where $A = k$ and $B = k[x]$. This is not true because

\[ R\mathop{\mathrm{Hom}}\nolimits (k[x], k) = \left(\prod \nolimits _{n \geq 0} k\right)[0] \]

which is not a finite $k[x]$-module. Hence $a(\mathcal{O}_ Y)$ does not have coherent cohomology sheaves.

Example 48.3.4. If $f : X \to Y$ is a proper or even finite morphism of Noetherian schemes, then the right adjoint of $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ does not map $D_\mathit{QCoh}^-(\mathcal{O}_ Y)$ into $D_\mathit{QCoh}^-(\mathcal{O}_ X)$. Namely, let $k$ be a field, let $k[\epsilon ]$ be the dual numbers over $k$, let $X = \mathop{\mathrm{Spec}}(k)$, and let $Y = \mathop{\mathrm{Spec}}(k[\epsilon ])$. Then $\mathop{\mathrm{Ext}}\nolimits ^ i_{k[\epsilon ]}(k, k)$ is nonzero for all $i \geq 0$. Hence $a(\mathcal{O}_ Y)$ is not bounded above by Example 48.3.2.

Lemma 48.3.5. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ X)$ be the right adjoint to $Rf_*$ of Lemma 48.3.1. Then $a$ maps $D^+_\mathit{QCoh}(\mathcal{O}_ Y)$ into $D^+_\mathit{QCoh}(\mathcal{O}_ X)$. In fact, there exists an integer $N$ such that $H^ i(K) = 0$ for $i \leq c$ implies $H^ i(a(K)) = 0$ for $i \leq c - N$.

Proof. By Derived Categories of Schemes, Lemma 36.4.1 the functor $Rf_*$ has finite cohomological dimension. In other words, there exist an integer $N$ such that $H^ i(Rf_*L) = 0$ for $i \geq N + c$ if $H^ i(L) = 0$ for $i \geq c$. Say $K \in D^+_\mathit{QCoh}(\mathcal{O}_ Y)$ has $H^ i(K) = 0$ for $i \leq c$. Then

\[ \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ X)}(\tau _{\leq c - N}a(K), a(K)) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}_ Y)}(Rf_*\tau _{\leq c - N}a(K), K) = 0 \]

by what we said above. Clearly, this implies that $H^ i(a(K)) = 0$ for $i \leq c - N$. $\square$

Let $f : X \to Y$ be a morphism of quasi-separated and quasi-compact schemes. Let $a$ denote the right adjoint to $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. For every $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ and $L \in D_\mathit{QCoh}(\mathcal{O}_ X)$ we obtain a canonical map

48.3.5.1
\begin{equation} \label{duality-equation-sheafy-trace} Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K)) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, K) \end{equation}

Namely, this map is constructed as the composition

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K)) \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, Rf_*a(K)) \to R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, K) \]

where the first arrow is Cohomology, Remark 20.38.10 and the second arrow is the counit $Rf_*a(K) \to K$ of the adjunction.

Lemma 48.3.6. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a$ be the right adjoint to $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. Then (48.3.5.1)

\[ Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K)) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, K) \]

is an isomorphism for all $L \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$.

Proof. Let $M \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. Then we have the following

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ Y(M, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K))) & = \mathop{\mathrm{Hom}}\nolimits _ X(Lf^*M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K))) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(Lf^*M \otimes _{\mathcal{O}_ X}^\mathbf {L} L, a(K)) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*(Lf^*M \otimes _{\mathcal{O}_ X}^\mathbf {L} L), K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(M \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*L, K) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(M, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(Rf_*L, K)) \end{align*}

The first equality holds by Cohomology, Lemma 20.28.1. The second equality by Cohomology, Lemma 20.38.2. The third equality by construction of $a$. The fourth equality by Derived Categories of Schemes, Lemma 36.21.1 (this is the important step). The fifth by Cohomology, Lemma 20.38.2. Thus the result holds by the Yoneda lemma. $\square$

Lemma 48.3.7. Let $f : X \to Y$ be a morphism of quasi-separated and quasi-compact schemes. For all $L \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ (48.3.5.1) induces an isomorphism $R\mathop{\mathrm{Hom}}\nolimits _ X(L, a(K)) \to R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*L, K)$ of global derived homs.

Proof. By the construction in Cohomology, Section 20.40 we have

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(L, a(K)) = R\Gamma (X, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K))) = R\Gamma (Y, Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(L, a(K))) \]

and

\[ R\mathop{\mathrm{Hom}}\nolimits _ Y(Rf_*L, K) = R\Gamma (Y, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(Rf_*L, a(K))) \]

Thus the lemma is a consequence of Lemma 48.3.6. $\square$


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