The Stacks project

Lemma 75.21.2. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

be a cartesian diagram of algebraic spaces over $S$. Let $K \in D_\mathit{QCoh}(\mathcal{O}_ X)$ and let $L(g')^*K \to K'$ be a map in $D_\mathit{QCoh}(\mathcal{O}_{X'})$. If

  1. the equivalent conditions of Lemma 75.21.1 hold, and

  2. $f$ is quasi-compact and quasi-separated,

then the composition $Lg^*Rf_*K \to Rf'_*L(g')^*K \to Rf'_*K'$ is an isomorphism.

Proof. To check the map is an isomorphism we may work étale locally on $Y'$. Hence we may assume $g : Y' \to Y$ is a morphism of affine schemes. In this case, we will use the induction principle of Lemma 75.9.3 to prove that for a quasi-compact and quasi-separated algebraic space $U$ étale over $X$ the similarly constructed map $Lg^*R(U \to Y)_*K|_ U \to R(U' \to Y')_*K'|_{U'}$ is an isomorphism. Here $U' = X' \times _{g', X} U = Y' \times _{g, Y} U$.

If $U$ is a scheme (for example affine), then the result holds. Namely, then $Y, Y', U, U'$ are schemes, $K$ and $K'$ come from objects of the derived category of the underlying schemes by Lemma 75.4.2 and the condition of Derived Categories of Schemes, Lemma 36.26.1 holds for these complexes by Lemma 75.21.1. Thus (by the compatibilities explained in Remark 75.6.3) we can apply the result in the case of schemes which is Derived Categories of Schemes, Lemma 36.26.2.

The induction step. Let $(U \subset W, V \to W)$ be an elementary distinguished square with $W$ a quasi-compact and quasi-separated algebraic space étale over $X$, with $U$ quasi-compact, $V$ affine and the result holds for $U$, $V$, and $U \times _ W V$. To easy notation we replace $W$ by $X$ (this is permissible at this point). Denote $a : U \to Y$, $b : V \to Y$, and $c : U \times _ X V \to Y$ the obvious morphisms. Let $a' : U' \to Y'$, $b' : V' \to Y'$ and $c' : U' \times _{X'} V' \to Y'$ be the base changes of $a$, $b$, and $c$. Using the distinguished triangles from relative Mayer-Vietoris (Lemma 75.10.3) we obtain a commutative diagram

\[ \xymatrix{ Lg^*Rf_*K \ar[r] \ar[d] & Rf'_*K' \ar[d] \\ Lg^*Ra_*K|_ U \oplus Lg^*Rb_*K|_ V \ar[r] \ar[d] & Ra'_* K'|_{U'} \oplus Rb'_* K'|_{V'} \ar[d] \\ Lg^*Rc_*K|_{U \times _ X V} \ar[r] \ar[d] & Rc'_*K'|_{U' \times _{X'} V'} \ar[d] \\ Lg^*Rf_* K[1] \ar[r] & Rf'_* K'[1] } \]

Since the 2nd and 3rd horizontal arrows are isomorphisms so is the first (Derived Categories, Lemma 13.4.3) and the proof of the lemma is finished. $\square$

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