The Stacks project

Proof. Let $E$ be an object of $D_\mathit{QCoh}(\mathcal{O}_ X)$ and let $A$ be an object of $D(\mathit{QCoh}(\mathcal{O}_ X))$. We have to show that the adjunction maps

are isomorphisms. Consider the hypothesis $H_ n$: the adjunction maps above are isomorphisms whenever $E$ and $i_ X(A)$ are supported (Definition 36.6.1) on a closed subset of $X$ which is contained in the union of $n$ affine opens of $X$. We will prove $H_ n$ by induction on $n$.

Base case: $n = 0$. In this case $E = 0$, hence the map $E \to i_ X(RQ_ X(E))$ is an isomorphism. Similarly $i_ X(A) = 0$. Thus the cohomology sheaves of $i_ X(A)$ are zero. Since the inclusion functor $\mathit{QCoh}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ X)$ is fully faithful and exact, we conclude that the cohomology objects of $A$ are zero, i.e., $A = 0$ and $RQ_ X(i_ X(A)) \to A$ is an isomorphism as well.

Induction step. Suppose that $E$ and $i_ X(A)$ are supported on a closed subset $T$ of $X$ contained in $U_1 \cup \ldots \cup U_ n$ with $U_ i \subset X$ affine open. Set $U = U_ n$. Consider the distinguished triangles

where $\Phi$ is as in the statement of the lemma. Note that $E \to Rj_*(E|_ U)$ is a quasi-isomorphism over $U = U_ n$. Since $i_ X \circ \Phi = Rj_* \circ i_ U$ by assumption and since $i_ X(A)|_ U = i_ U(A|_ U)$ we see that $i_ X(A) \to i_ X(\Phi (A|_ U))$ is a quasi-isomorphism over $U$. Hence $i_ X(A')$ and $E'$ are supported on the closed subset $T \setminus U$ of $X$ which is contained in $U_1 \cup \ldots \cup U_{n - 1}$. By induction hypothesis the statement is true for $A'$ and $E'$. By Derived Categories, Lemma 13.4.3 it suffices to prove the maps

are isomorphisms. By assumption and by Lemma 36.7.2 (the inclusion morphism $j : U \to X$ is flat, quasi-compact, and quasi-separated) we have

and

Finally, the maps

are isomorphisms by Lemma 36.7.3. The result follows. $\square$