The Stacks project

Proof. Observe that the horizontal arrows in the diagram are equivalences of categories by Proposition 75.11.5. Hence we can identify these categories (and similarly for other quasi-compact algebraic spaces with affine diagonal) and then the statement of the lemma is that the canonical map $\Phi (K) \to Rf_*(K)$ is an isomorphism for all $K$ in $D(\mathit{QCoh}(\mathcal{O}_ X))$. Note that if $K_1 \to K_2 \to K_3 \to K_1[1]$ is a distinguished triangle in $D(\mathit{QCoh}(\mathcal{O}_ X))$ and the statement is true for two-out-of-three, then it is true for the third.

Let $\mathcal{B} \subset \mathop{\mathrm{Ob}}\nolimits (X_{spaces, {\acute{e}tale}})$ be the set of objects which are quasi-compact and have affine diagonal. For $U \in \mathcal{B}$ and any morphism $g : U \to Z$ where $Z$ is a quasi-compact algebraic space over $S$ with affine diagonal, denote

the derived extension of $g_*$. Let $P(U) =$ “for any $K$ in $D(\mathit{QCoh}(\mathcal{O}_ U))$ and any $g : U \to Z$ as above the map $\Phi _ g(K) \to Rg_*K$ is an isomorphism”. By Remark 75.9.5 conditions (1), (2), and (3)(a) of Lemma 75.9.4 hold and we are left with proving (3)(b) and (4).

Checking condition (3)(b). Let $U$ be an affine scheme étale over $X$. Let $g : U \to Z$ be as above. Since the diagonal of $Z$ is affine the morphism $g : U \to Z$ is affine (Morphisms of Spaces, Lemma 67.20.11). Hence $P(U)$ holds by Lemma 75.11.1.

Checking condition (4). Let $(U \subset W, V \to W)$ be an elementary distinguished square in $X_{spaces, {\acute{e}tale}}$ with $U, W, V$ in $\mathcal{B}$ and $V$ affine. Assume that $P$ holds for $U$, $V$, and $U \times _ W V$. We have to show that $P$ holds for $W$. Let $g : W \to Z$ be a morphism to a quasi-compact algebraic space with affine diagonal. Let $K$ be an object of $D(\mathit{QCoh}(\mathcal{O}_ W))$. Consider the distinguished triangle

in $D(\mathcal{O}_ W)$. By the two-out-of-three property mentioned above, it suffices to show that $\Phi _ g(Rj_{U, *}K|_ U) \to Rg_*(Rj_{U, *}K|_ U)$ is an isomorphism and similarly for $V$ and $U \times _ W V$. This is discussed in the next paragraph.

Let $j : U \to W$ be a morphism $X_{spaces, {\acute{e}tale}}$ with $U, W$ in $\mathcal{B}$ and $P$ holds for $U$. Let $g : W \to Z$ be a morphism to a quasi-compact algebraic space with affine diagonal. To finish the proof we have to show that $\Phi _ g(Rj_*K) \to Rg_*(Rj_*K)$ is an isomorphism for any $K$ in $D(\mathit{QCoh}(\mathcal{O}_ U))$. Let $\mathcal{I}^\bullet $ be a K-injective complex in $\mathit{QCoh}(\mathcal{O}_ U)$ representing $K$. From $P(U)$ applied to $j$ we see that $j_*\mathcal{I}^\bullet $ represents $Rj_*K$. Since $j_* : \mathit{QCoh}(\mathcal{O}_ U) \to \mathit{QCoh}(\mathcal{O}_ X)$ has an exact left adjoint $j^* : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ U)$ we see that $j_*\mathcal{I}^\bullet $ is a K-injective complex in $\mathit{QCoh}(\mathcal{O}_ W)$, see Derived Categories, Lemma 13.31.9. Hence $\Phi _ g(Rj_*K)$ is represented by $g_*j_*\mathcal{I}^\bullet = (g \circ j)_*\mathcal{I}^\bullet $. By $P(U)$ applied to $g \circ j$ we see that this represents $R_{g \circ j, *}(K) = Rg_*(Rj_*K)$. This finishes the proof. $\square$