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48.4 Right adjoint of pushforward and restriction to opens

In this section we study the question to what extend the right adjoint of pushforward commutes with restriction to open subschemes. This is a base change question, so let's first discuss this more generally.

We often want to know whether the right adjoints to pushforward commutes with base change. Thus we consider a cartesian square
\begin{equation} \label{duality-equation-base-change} \vcenter { \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } } \end{equation}

of quasi-compact and quasi-separated schemes. Denote

\[ a : D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ X) \quad \text{and}\quad a' : D_\mathit{QCoh}(\mathcal{O}_{Y'}) \to D_\mathit{QCoh}(\mathcal{O}_{X'}) \]

the right adjoints to $Rf_*$ and $Rf'_*$ (Lemma 48.3.1). Consider the base change map of Cohomology, Remark 20.28.3. It induces a transformation of functors

\[ Lg^* \circ Rf_* \longrightarrow Rf'_* \circ L(g')^* \]

on derived categories of sheaves with quasi-coherent cohomology. Hence a transformation between the right adjoints in the opposite direction

\[ a \circ Rg_* \longleftarrow Rg'_* \circ a' \]

Lemma 48.4.1. In diagram ( assume that $g$ is flat or more generally that $f$ and $g$ are Tor independent. Then $a \circ Rg_* \leftarrow Rg'_* \circ a'$ is an isomorphism.

Proof. In this case the base change map $Lg^* \circ Rf_* K \longrightarrow Rf'_* \circ L(g')^*K$ is an isomorphism for every $K$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ by Derived Categories of Schemes, Lemma 36.22.5. Thus the corresponding transformation between adjoint functors is an isomorphism as well. $\square$

Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $V \subset Y$ be a quasi-compact open subscheme and set $U = f^{-1}(V)$. This gives a cartesian square

\[ \xymatrix{ U \ar[r]_{j'} \ar[d]_{f|_ U} & X \ar[d]^ f \\ V \ar[r]^ j & Y } \]

as in ( By Lemma 48.4.1 the map $\xi : a \circ Rj_* \leftarrow Rj'_* \circ a'$ is an isomorphism where $a$ and $a'$ are the right adjoints to $Rf_*$ and $R(f|_ U)_*$. We obtain a transformation of functors $D_\mathit{QCoh}(\mathcal{O}_ Y) \to D_\mathit{QCoh}(\mathcal{O}_ U)$
\begin{equation} \label{duality-equation-sheafy} (j')^* \circ a \to (j')^* \circ a \circ Rj_* \circ j^* \xrightarrow {\xi ^{-1}} (j')^* \circ Rj'_* \circ a' \circ j^* \to a' \circ j^* \end{equation}

where the first arrow comes from $\text{id} \to Rj_* \circ j^*$ and the final arrow from the isomorphism $(j')^* \circ Rj'_* \to \text{id}$. In particular, we see that ( is an isomorphism when evaluated on $K$ if and only if $a(K)|_ U \to a(Rj_*(K|_ V))|_ U$ is an isomorphism.

Example 48.4.2. There is a finite morphism $f : X \to Y$ of Noetherian schemes such that ( is not an isomorphism when evaluated on some $K \in D_{\textit{Coh}}(\mathcal{O}_ Y)$. Namely, let $X = \mathop{\mathrm{Spec}}(B) \to Y = \mathop{\mathrm{Spec}}(A)$ with $A = k[x, \epsilon ]$ where $k$ is a field and $\epsilon ^2 = 0$ and $B = k[x] = A/(\epsilon )$. For $n \in \mathbf{N}$ set $M_ n = A/(\epsilon , x^ n)$. Observe that

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_ A(B, M_ n) = M_ n,\quad i \geq 0 \]

because $B$ has the free periodic resolution $\ldots \to A \to A \to A$ with maps given by multiplication by $\epsilon $. Consider the object $K = \bigoplus M_ n[n] = \prod M_ n[n]$ of $D_{\textit{Coh}}(A)$ (equality in $D(A)$ by Derived Categories, Lemmas 13.33.5 and 13.34.2). Then we see that $a(K)$ corresponds to $R\mathop{\mathrm{Hom}}\nolimits (B, K)$ by Example 48.3.2 and

\[ H^0(R\mathop{\mathrm{Hom}}\nolimits (B, K)) = \mathop{\mathrm{Ext}}\nolimits ^0_ A(B, K) = \prod \nolimits _{n \geq 1} \mathop{\mathrm{Ext}}\nolimits ^ n_ A(B. M_ n) = \prod \nolimits _{n \geq 1} M_ n \]

by the above. But this module has elements which are not annihilated by any power of $x$, whereas the complex $K$ does have every element of its cohomology annihilated by a power of $x$. In other words, for the map ( with $V = D(x)$ and $U = D(x)$ and the complex $K$ cannot be an isomorphism because $(j')^*(a(K))$ is nonzero and $a'(j^*K)$ is zero.

Lemma 48.4.3. Let $f : X \to Y$ be a morphism of quasi-compact and quasi-separated schemes. Let $a$ be the right adjoint to $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$. Let $V \subset Y$ be quasi-compact open with inverse image $U \subset X$.

  1. For every $Q \in D_\mathit{QCoh}^+(\mathcal{O}_ Y)$ supported on $Y \setminus V$ the image $a(Q)$ is supported on $X \setminus U$ if and only if ( is an isomorphism on all $K$ in $D_\mathit{QCoh}^+(\mathcal{O}_ Y)$.

  2. For every $Q \in D_\mathit{QCoh}(\mathcal{O}_ Y)$ supported on $Y \setminus V$ the image $a(Q)$ is supported on $X \setminus U$ if and only if ( is an isomorphism on all $K$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$.

  3. If $a$ commutes with direct sums, then the equivalent conditions of (1) imply the equivalent conditions of (2).

Proof. Proof of (1). Let $K \in D_\mathit{QCoh}^+(\mathcal{O}_ Y)$. Choose a distinguished triangle

\[ K \to Rj_*K|_ V \to Q \to K[1] \]

Observe that $Q$ is in $D_\mathit{QCoh}^+(\mathcal{O}_ Y)$ (Derived Categories of Schemes, Lemma 36.4.1) and is supported on $Y \setminus V$ (Derived Categories of Schemes, Definition 36.6.1). Applying $a$ we obtain a distinguished triangle

\[ a(K) \to a(Rj_*K|_ V) \to a(Q) \to a(K)[1] \]

on $X$. If $a(Q)$ is supported on $X \setminus U$, then restricting to $U$ the map $a(K)|_ U \to a(Rj_*K|_ V)|_ U$ is an isomorphism, i.e., ( is an isomorphism on $K$. The converse is immediate.

The proof of (2) is exactly the same as the proof of (1).

Proof of (3). Assume the equivalent conditions of (1) hold. Set $T = Y \setminus V$. We will use the notation $D_{\mathit{QCoh}, T}(\mathcal{O}_ Y)$ and $D_{\mathit{QCoh}, f^{-1}(T)}(\mathcal{O}_ X)$ to denote complexes whose cohomology sheaves are supported on $T$ and $f^{-1}(T)$. Since $a$ commutes with direct sums, the strictly full, saturated, triangulated subcategory $\mathcal{D}$ with objects

\[ \{ Q \in D_{\mathit{QCoh}, T}(\mathcal{O}_ Y) \mid a(Q) \in D_{\mathit{QCoh}, f^{-1}(T)}(\mathcal{O}_ X)\} \]

is preserved by direct sums and hence derived colimits. On the other hand, the category $D_{\mathit{QCoh}, T}(\mathcal{O}_ Y)$ is generated by a perfect object $E$ (see Derived Categories of Schemes, Lemma 36.15.4). By assumption we see that $E \in \mathcal{D}$. By Derived Categories, Lemma 13.37.3 every object $Q$ of $D_{\mathit{QCoh}, T}(\mathcal{O}_ Y)$ is a derived colimit of a system $Q_1 \to Q_2 \to Q_3 \to \ldots $ such that the cones of the transition maps are direct sums of shifts of $E$. Arguing by induction we see that $Q_ n \in \mathcal{D}$ for all $n$ and finally that $Q$ is in $\mathcal{D}$. Thus the equivalent conditions of (2) hold. $\square$

Lemma 48.4.4. Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a proper morphism. If1

  1. $f$ is flat and of finite presentation, or

  2. $Y$ is Noetherian

then the equivalent conditions of Lemma 48.4.3 part (1) hold for all quasi-compact opens $V$ of $Y$.

Proof. Let $Q \in D^+_\mathit{QCoh}(\mathcal{O}_ Y)$ be supported on $Y \setminus V$. To get a contradiction, assume that $a(Q)$ is not supported on $X \setminus U$. Then we can find a perfect complex $P_ U$ on $U$ and a nonzero map $P_ U \to a(Q)|_ U$ (follows from Derived Categories of Schemes, Theorem 36.15.3). Then using Derived Categories of Schemes, Lemma 36.13.11 we may assume there is a perfect complex $P$ on $X$ and a map $P \to a(Q)$ whose restriction to $U$ is nonzero. By definition of $a$ this map is adjoint to a map $Rf_*P \to Q$.

The complex $Rf_*P$ is pseudo-coherent. In case (1) this follows from Derived Categories of Schemes, Lemma 36.30.5. In case (2) this follows from Derived Categories of Schemes, Lemmas 36.11.3 and 36.10.3. Thus we may apply Derived Categories of Schemes, Lemma 36.17.5 and get a map $I \to \mathcal{O}_ Y$ of perfect complexes whose restriction to $V$ is an isomorphism such that the composition $I \otimes ^\mathbf {L}_{\mathcal{O}_ Y} Rf_*P \to Rf_*P \to Q$ is zero. By Derived Categories of Schemes, Lemma 36.22.1 we have $I \otimes ^\mathbf {L}_{\mathcal{O}_ Y} Rf_*P = Rf_*(Lf^*I \otimes ^\mathbf {L}_{\mathcal{O}_ X} P)$. We conclude that the composition

\[ Lf^*I \otimes ^\mathbf {L}_{\mathcal{O}_ X} P \to P \to a(Q) \]

is zero. However, the restriction to $U$ is the map $P|_ U \to a(Q)|_ U$ which we assumed to be nonzero. This contradiction finishes the proof. $\square$

[1] This proof works for those morphisms of quasi-compact and quasi-separated schemes such that $Rf_*P$ is pseudo-coherent for all $P$ perfect on $X$. It follows easily from a theorem of Kiehl [Kiehl] that this holds if $f$ is proper and pseudo-coherent. This is the correct generality for this lemma and some of the other results in this chapter.

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