The Stacks project

36.27 Cohomology and base change, VI

A final section on cohomology and base change continuing the discussion of Sections 36.21, 36.23, and 36.24. An easy to grok special case is given in Remark 36.27.2.

Lemma 36.27.1. Let $f : X \to S$ be a morphism of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. Let $\mathcal{G}^\bullet $ be a bounded complex of finitely presented $\mathcal{O}_ X$-modules, flat over $S$, with support proper over $S$. Then

\[ K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet ) \]

is a perfect object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma 36.23.4. Thus it suffices to show that $K$ is a perfect object. If $S$ is Noetherian, then this follows from Lemma 36.24.2. 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 36.26.3 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 36.24.2 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. $\square$

Remark 36.27.2. Let $R$ be a ring. Let $X$ be a scheme of finite presentation over $R$. Let $\mathcal{G}$ be a finitely presented $\mathcal{O}_ X$-module flat over $R$ with support proper over $R$. By Lemma 36.27.1 there exists a finite complex of finite projective $R$-modules $M^\bullet $ such that we have

\[ R\Gamma (X_{R'}, \mathcal{G}_{R'}) = M^\bullet \otimes _ R R' \]

functorially in the $R$-algebra $R'$.

Lemma 36.27.3. Let $f : X \to S$ be a morphism of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be a pseudo-coherent object. Let $\mathcal{G}^\bullet $ be a bounded above complex of finitely presented $\mathcal{O}_ X$-modules, flat over $S$, with support proper over $S$. Then

\[ K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet ) \]

is a pseudo-coherent object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma 36.23.4. Thus it suffices to show that $K$ is a pseudo-coherent object. This will follow from Lemma 36.27.1 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 36.13.6. Choose a distinguished triangle

\[ P \to E \to C \to P[1] \]

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

\[ Rf_*(P \otimes ^\mathbf {L} \mathcal{G}^\bullet ) \to Rf_*(E \otimes ^\mathbf {L} \mathcal{G}^\bullet ) \]

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

\[ Rf_*(P \otimes ^\mathbf {L} \sigma _{\geq m - N - 1 - a}\mathcal{G}^\bullet ) \to Rf_*(P \otimes ^\mathbf {L} \mathcal{G}^\bullet ) \]

is an isomorphism on cohomology sheaves in degrees $\geq m$. By Lemma 36.27.1 the source is a perfect complex. We conclude that $K$ is $m$-pseudo-coherent as desired. $\square$

Lemma 36.27.4. Let $S$ be a scheme. Let $f : X \to S$ be a proper morphism of finite presentation.

  1. Let $E \in D(\mathcal{O}_ X)$ be perfect and $f$ flat. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

  2. Let $\mathcal{G}$ be an $\mathcal{O}_ X$-module of finite presentation, flat over $S$. Then $Rf_*\mathcal{G}$ is a perfect object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

Proof. Special cases of Lemma 36.27.1 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 $0$. $\square$

Lemma 36.27.5. Let $S$ be a scheme. Let $f : X \to S$ be a flat proper morphism of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

More generally, if $f : X \to S$ is proper and $E$ on $X$ is pseudo-coherent relative to $S$ (More on Morphisms, Definition 37.51.2), then $Rf_*E$ is pseudo-coherent (but formation does not commute with base change in this generality). See [Kiehl].

Proof. Special case of Lemma 36.27.3 applied with $\mathcal{G}^\bullet $ equal to $\mathcal{O}_ X$ in degree $0$. $\square$

Lemma 36.27.6. Let $R$ be a ring. Let $X$ be a scheme and let $f : X \to \mathop{\mathrm{Spec}}(R)$ be proper, flat, and of finite presentation. Let $(M_ n)$ be an inverse system of $R$-modules with surjective transition maps. Then the canonical map

\[ \mathcal{O}_ X \otimes _ R (\mathop{\mathrm{lim}}\nolimits M_ n) \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n \]

induces an isomorphism from the source to $DQ_ X$ applied to the target.

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

\[ \mathop{\mathrm{Hom}}\nolimits (E, \mathcal{O}_ X \otimes _ R (\mathop{\mathrm{lim}}\nolimits M_ n)) \longrightarrow \mathop{\mathrm{Hom}}\nolimits (E, \mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n) \]

is an isomorphism. Since $D_\mathit{QCoh}(\mathcal{O}_ X)$ has a perfect generator (Theorem 36.14.3) 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 20.40 commutes with products and hence with derived limits, whence

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(E, \mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n) = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathrm{Hom}}\nolimits _ X(E, \mathcal{O}_ X \otimes _ R M_ n) \]

Let $E^\vee $ be the dual perfect complex, see Cohomology, Lemma 20.46.4. We have

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(E, \mathcal{O}_ X \otimes _ R M_ n) = R\Gamma (X, E^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*M_ n) = R\Gamma (X, E^\vee ) \otimes _ R^\mathbf {L} M_ n \]

by Lemma 36.21.1. From Lemma 36.27.4 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 15.93.5 we obtain

\[ R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, E^\vee ) \otimes _ R^\mathbf {L} M_ n = R\Gamma (X, E^\vee ) \otimes _ R^\mathbf {L} \mathop{\mathrm{lim}}\nolimits M_ n \]

as desired. $\square$

Lemma 36.27.7. Let $f : X \to S$ be a morphism of finite presentation. Let $E \in D(\mathcal{O}_ X)$ be a perfect object. Let $\mathcal{G}^\bullet $ be a bounded complex of finitely presented $\mathcal{O}_ X$-modules, flat over $S$, with support proper over $S$. Then

\[ K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet ) \]

is a perfect object of $D(\mathcal{O}_ S)$ and its formation commutes with arbitrary base change.

Proof. The statement on base change is Lemma 36.23.5. Thus it suffices to show that $K$ is a perfect object. If $S$ is Noetherian, then this follows from Lemma 36.24.3. 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 36.26.3 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 36.24.3 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. $\square$


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