The Stacks project

Lemma 74.23.3. Let $S$ be a scheme. Let $f : X \to B$ be a morphism of algebraic spaces over $S$, $E \in D(\mathcal{O}_ X)$, and $\mathcal{F}^\bullet $ a complex of $\mathcal{O}_ X$-modules. Assume

  1. $B$ is Noetherian,

  2. $f$ is locally of finite type and quasi-separated,

  3. $E \in D^-_{\textit{Coh}}(\mathcal{O}_ X)$,

  4. $\mathcal{G}^\bullet $ is a bounded complex of coherent $\mathcal{O}_ X$-module flat over $B$ with support proper over $B$.

Then the following two statements are true

  1. for every $m \in \mathbf{Z}$ there exists a perfect object $K$ of $D(\mathcal{O}_ B)$ and functorial maps

    \[ \alpha ^ i_\mathcal {F} : \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \longrightarrow H^ i(B, K \otimes ^\mathbf {L}_{\mathcal{O}_ B} \mathcal{F}) \]

    for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps (see proof) such that $\alpha ^ i_\mathcal {F}$ is an isomorphism for $i \leq m$, and

  2. there exists a pseudo-coherent $L \in D(\mathcal{O}_ B)$ and functorial isomorphisms

    \[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ B}(L, \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

    for $\mathcal{F}$ quasi-coherent on $B$ compatible with boundary maps.

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 74.14.7). Then the induced map

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(P, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

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

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(C, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \quad \text{resp.}\quad \mathop{\mathrm{Ext}}\nolimits ^{i + 1}_{\mathcal{O}_ X}(C, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \]

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 74.23.2. The statement on boundaries is explained in the proof of Lemma 74.23.2.

Proof of (B). As in the proof of (A) we may assume $X$ is Noetherian. Observe that $E$ is pseudo-coherent by Lemma 74.13.7. By Lemma 74.18.1 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 on Sites, Lemma 21.48.4). 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

\[ \ldots \to K_3 \to K_2 \to K_1\quad \text{with}\quad K_ n = Rf_*(E_ n^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet ) \]

perfect on $Y$, see Lemma 74.22.2. By Lemma 74.23.2 and its proof and by the arguments in the previous paragraph (with $P = E_ n$) for any quasi-coherent $\mathcal{F}$ on $Y$ we have functorial canonical maps

\[ \xymatrix{ & \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(E, \mathcal{G}^\bullet \otimes _{\mathcal{O}_ X} f^*\mathcal{F}) \ar[ld] \ar[rd] \\ H^ i(Y, K_{n + 1} \otimes _{\mathcal{O}_ Y}^\mathbf {L} \mathcal{F}) \ar[rr] & & H^ i(Y, K_ n \otimes _{\mathcal{O}_ Y}^\mathbf {L} \mathcal{F}) } \]

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 $D(\mathcal{O}_ Y)$ such that for any quasi-coherent $\mathcal{F}$ on $Y$ the maps

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ Y}(L_{n + 1}, \mathcal{F}) \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ Y}(L_ n, \mathcal{F}) \]

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 74.18.1. The mapping property of homotopy colimits gives that $\mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ Y}(L, \mathcal{F}) = \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ Y}(L_ n, \mathcal{F})$ for $i \leq n + a - 3$ which finishes the proof. $\square$


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