The Stacks project

Proof. We have

the first equality by Cohomology, Lemma 20.27.4, the second as $\mathcal{G}^ n$ is a flat $f^{-1}\mathcal{O}_ S$-module, and the third by definition of pullbacks. Hence we obtain

The first equality by the above, the second by Leray (Cohomology, Lemma 20.13.1), and the third equality by Lemma 36.22.1. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ of quasi-coherent $\mathcal{O}_ S$-modules, the isomorphisms fit into commutative diagrams

where the boundary maps come from the distinguished triangle

and the distinguished triangle in $D(\mathcal{O}_ X)$ associated to the short exact sequence

of complexes of $\mathcal{O}_ X$-modules. This sequence is exact because $\mathcal{G}^ n$ is flat over $S$. We omit the verification of the commutativity of the displayed diagram. $\square$

Proof. As in the proof of Lemma 36.27.3 let $E^\vee$ be the dual perfect complex and recall that $K = Rf_*(E^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet )$. Since we also have

by construction of $E^\vee$, the existence of the isomorphisms follows from Lemma 36.28.1 applied to $E^\vee$ and $\mathcal{G}^\bullet$. The statement on boundary maps means the following: Given a short exact sequence $0 \to \mathcal{F}_1 \to \mathcal{F}_2 \to \mathcal{F}_3 \to 0$ then the isomorphisms fit into commutative diagrams

where the boundary maps come from the distinguished triangle

and the distinguished triangle in $D(\mathcal{O}_ X)$ associated to the short exact sequence

of complexes. This sequence is exact because $\mathcal{G}$ is flat over $S$. We omit the verification of the commutativity of the displayed diagram. $\square$

Proof. Proof of (A). Suppose $\mathcal{G}^ i$ is nonzero only for $i \in [a, b]$. We may replace $X$ by a quasi-compact open neighbourhood of the union of the supports of $\mathcal{G}^ i$. Hence we may assume $X$ is Noetherian. In this case $X$ and $f$ are quasi-compact and quasi-separated. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, -m - 1 + a)$ (possible by Theorem 36.14.6). Then the induced map

is an isomorphism for $i \leq m$. Namely, the kernel, resp. cokernel of this map is a quotient, resp. submodule of

where $C$ is the cone of $P \to E$. Since $C$ has vanishing cohomology sheaves in degrees $\geq -m - 1 + a$ these $\mathop{\mathrm{Ext}}\nolimits$-groups are zero for $i \leq m + 1$ by Derived Categories, Lemma 13.27.3. This reduces us to the case that $E$ is a perfect complex which is Lemma 36.28.2. The statement on boundaries is explained in the proof of Lemma 36.28.2.

Proof of (B). As in the proof of (A) we may assume $X$ is Noetherian. Observe that $E$ is pseudo-coherent by Lemma 36.10.3. By Lemma 36.19.1 we can write $E = \text{hocolim} E_ n$ with $E_ n$ perfect and $E_ n \to E$ inducing an isomorphism on truncations $\tau _{\geq -n}$. Let $E_ n^\vee$ be the dual perfect complex (Cohomology, Lemma 20.50.5). We obtain an inverse system $\ldots \to E_3^\vee \to E_2^\vee \to E_1^\vee$ of perfect objects. This in turn gives rise to an inverse system

perfect on $S$, see Lemma 36.27.2. By Lemma 36.28.2 and its proof and by the arguments in the previous paragraph (with $P = E_ n$) for any quasi-coherent $\mathcal{F}$ on $S$ we have functorial canonical maps

which are isomorphisms for $i \leq n + a$. Let $L_ n = K_ n^\vee$ be the dual perfect complex. Then we see that $L_1 \to L_2 \to L_3 \to \ldots$ is a system of perfect objects in $D(\mathcal{O}_ S)$ such that for any quasi-coherent $\mathcal{F}$ on $S$ the maps

are isomorphisms for $i \leq n + a - 1$. This implies that $L_ n \to L_{n + 1}$ induces an isomorphism on truncations $\tau _{\geq -n - a + 2}$ (hint: take cone of $L_ n \to L_{n + 1}$ and look at its last nonvanishing cohomology sheaf). Thus $L = \text{hocolim} L_ n$ is pseudo-coherent, see Lemma 36.19.1. The mapping property of homotopy colimits gives that $\mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L, \mathcal{F}) = \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ S}(L_ n, \mathcal{F})$ for $i \leq n + a - 3$ which finishes the proof. $\square$