The Stacks project

Proof. The statement on base change is Lemma 36.26.4. Thus it suffices to show that $K$ is a perfect object. If $S$ is Noetherian, then this follows from Lemma 36.27.2. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on $S$, hence we may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits, Lemma 32.10.1 there exists an $i$ and a scheme $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits, Lemma 32.10.2 we may assume after increasing $i$, that there exists a bounded complex of finitely presented $\mathcal{O}_{X_ i}$-modules $\mathcal{G}_ i^\bullet $ whose pullback to $X$ is $\mathcal{G}^\bullet $. After increasing $i$ we may assume $\mathcal{G}_ i^ n$ is flat over $R_ i$, see Limits, Lemma 32.10.4. After increasing $i$ we may assume the support of $\mathcal{G}_ i^ n$ is proper over $R_ i$, see Limits, Lemma 32.13.5 and Cohomology of Schemes, Lemma 30.26.7. Finally, by Lemma 36.29.3 we may, after increasing $i$, assume there exists a perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. Applying Lemma 36.27.2 to $X_ i \to \mathop{\mathrm{Spec}}(R_ i)$, $E_ i$, $\mathcal{G}_ i^\bullet $ and using the base change property already shown we obtain the result. $\square$

Proof. The statement on base change is Lemma 36.26.4. Thus it suffices to show that $K$ is a pseudo-coherent object. This will follow from Lemma 36.30.1 by approximation by perfect complexes. We encourage the reader to skip the rest of the proof.

The question is local on $S$, hence we may assume $S$ is affine. Then $X$ is quasi-compact and quasi-separated. Moreover, there exists an integer $N$ such that total direct image $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ S)$ has cohomological dimension $N$ as explained in Lemma 36.4.1. Choose an integer $b$ such that $\mathcal{G}^ i = 0$ for $i > b$. It suffices to show that $K$ is $m$-pseudo-coherent for every $m$. Choose an approximation $P \to E$ by a perfect complex $P$ of $(X, E, m - N - 1 - b)$. This is possible by Theorem 36.14.6. Choose a distinguished triangle

in $D_\mathit{QCoh}(\mathcal{O}_ X)$. The cohomology sheaves of $C$ are zero in degrees $\geq m - N - 1 - b$. Hence the cohomology sheaves of $C \otimes ^\mathbf {L} \mathcal{G}^\bullet $ are zero in degrees $\geq m - N - 1$. Thus the cohomology sheaves of $Rf_*(C \otimes ^\mathbf {L} \mathcal{G}^\bullet )$ are zero in degrees $\geq m - 1$. Hence

is an isomorphism on cohomology sheaves in degrees $\geq m$. Next, suppose that $H^ i(P) = 0$ for $i > a$. Then $ P \otimes ^\mathbf {L} \sigma _{\geq m - N - 1 - a}\mathcal{G}^\bullet \longrightarrow P \otimes ^\mathbf {L} \mathcal{G}^\bullet $ is an isomorphism on cohomology sheaves in degrees $\geq m - N - 1$. Thus again we find that

is an isomorphism on cohomology sheaves in degrees $\geq m$. By Lemma 36.30.1 the source is a perfect complex. We conclude that $K$ is $m$-pseudo-coherent as desired. $\square$

Proof. Special cases of Lemma 36.30.1 applied with (1) $\mathcal{G}^\bullet $ equal to $\mathcal{O}_ X$ in degree $0$ and (2) $E = \mathcal{O}_ X$ and $\mathcal{G}^\bullet $ consisting of $\mathcal{G}$ sitting in degree $0$. $\square$

Proof. Special case of Lemma 36.30.3 applied with $\mathcal{G}^\bullet $ equal to $\mathcal{O}_ X$ in degree $0$. $\square$

Proof. The statement means that for any object $E$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the induced map

is an isomorphism. Since $D_\mathit{QCoh}(\mathcal{O}_ X)$ has a perfect generator (Theorem 36.15.3) it suffices to check this for perfect $E$. By Lemma 36.3.2 we have $\mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n = R\mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n$. The exact functor $R\mathop{\mathrm{Hom}}\nolimits _ X(E, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(R)$ of Cohomology, Section 20.44 commutes with products and hence with derived limits, whence

Let $E^\vee $ be the dual perfect complex, see Cohomology, Lemma 20.50.5. We have

by Lemma 36.22.1. From Lemma 36.30.4 we see $R\Gamma (X, E^\vee )$ is a perfect complex of $R$-modules. In particular it is a pseudo-coherent complex and by More on Algebra, Lemma 15.102.3 we obtain

as desired. $\square$

Proof. The statement on base change is Lemma 36.26.5. Thus it suffices to show that $K$ is a perfect object. If $S$ is Noetherian, then this follows from Lemma 36.27.3. We will reduce to this case by Noetherian approximation. We encourage the reader to skip the rest of this proof.

The question is local on $S$, hence we may assume $S$ is affine. Say $S = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits, Lemma 32.10.1 there exists an $i$ and a scheme $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits, Lemma 32.10.2 we may assume after increasing $i$, that there exists a bounded complex of finitely presented $\mathcal{O}_{X_ i}$-modules $\mathcal{G}_ i^\bullet $ whose pullback to $X$ is $\mathcal{G}^\bullet $. After increasing $i$ we may assume $\mathcal{G}_ i^ n$ is flat over $R_ i$, see Limits, Lemma 32.10.4. After increasing $i$ we may assume the support of $\mathcal{G}_ i^ n$ is proper over $R_ i$, see Limits, Lemma 32.13.5 and Cohomology of Schemes, Lemma 30.26.7. Finally, by Lemma 36.29.3 we may, after increasing $i$, assume there exists a perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. Applying Lemma 36.27.3 to $X_ i \to \mathop{\mathrm{Spec}}(R_ i)$, $E_ i$, $\mathcal{G}_ i^\bullet $ and using the base change property already shown we obtain the result. $\square$