75.25 Cohomology and base change, VI
A final section on cohomology and base change continuing the discussion of Sections 75.20, 75.21, and 75.22. An easy to grok special case is given in Remark 75.25.2.
Lemma 75.25.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. 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 $Y$, with support proper over $Y$. Then
\[ K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet ) \]
is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.
Proof.
The statement on base change is Lemma 75.21.4. Thus it suffices to show that $K$ is a perfect object. If $Y$ is Noetherian, then this follows from Lemma 75.22.2. 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 $Y$, hence we may assume $Y$ is affine. Say $Y = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits of Spaces, Lemma 70.7.1 there exists an $i$ and an algebraic space $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemma 70.7.2 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 of Spaces, Lemma 70.6.12. After increasing $i$ we may assume the support of $\mathcal{G}_ i^ n$ is proper over $R_ i$, see Limits of Spaces, Lemma 70.12.3. Finally, by Lemma 75.24.3 we may, after increasing $i$, assume there exists a perfect object $E_ i$ of $D(\mathcal{O}_{X_ i})$ whose pullback to $X$ is $E$. By Lemma 75.22.2 we have that $K_ i = Rf_{i, *}(E_ i \otimes _{\mathcal{O}_{X_ i}}^\mathbf {L} \mathcal{G}_ i^\bullet )$ is perfect on $\mathop{\mathrm{Spec}}(R_ i)$ where $f_ i : X_ i \to \mathop{\mathrm{Spec}}(R_ i)$ is the structure morphism. By the base change result (Lemma 75.21.4) the pullback of $K_ i$ to $Y = \mathop{\mathrm{Spec}}(R)$ is $K$ and we conclude.
$\square$
Lemma 75.25.3. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. 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 $Y$, with support proper over $Y$. Then
\[ K = Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{G}^\bullet ) \]
is a pseudo-coherent object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.
Proof.
The statement on base change is Lemma 75.21.4. Thus it suffices to show that $K$ is a pseudo-coherent object. This will follow from Lemma 75.25.1 by approximation by perfect complexes. We encourage the reader to skip the rest of the proof.
The question is étale local on $Y$, hence we may assume $Y$ is affine. Then $X$ is quasi-compact and quasi-separated. Moreover, there exists an integer $N$ such that total direct image $Rf_* : D_\mathit{QCoh}(\mathcal{O}_ X) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ has cohomological dimension $N$ as explained in Lemma 75.6.1. 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 75.14.7. 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 $Rf_*(C \otimes ^\mathbf {L} \mathcal{G})$ are zero in degrees $\geq m - 1$. Hence
\[ Rf_*(P \otimes ^\mathbf {L} \mathcal{G}) \to Rf_*(E \otimes ^\mathbf {L} \mathcal{G}) \]
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 75.25.1 the source is a perfect complex. We conclude that $K$ is $m$-pseudo-coherent as desired.
$\square$
Lemma 75.25.4. Let $S$ be a scheme. Let $f : X \to Y$ be a proper morphism of finite presentation of algebraic spaces over $S$.
Let $E \in D(\mathcal{O}_ X)$ be perfect and $f$ flat. Then $Rf_*E$ is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.
Let $\mathcal{G}$ be an $\mathcal{O}_ X$-module of finite presentation, flat over $S$. Then $Rf_*\mathcal{G}$ is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.
Proof.
Special cases of Lemma 75.25.1 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 75.25.5. Let $S$ be a scheme. Let $f : X \to Y$ be a flat proper morphism of finite presentation of algebraic spaces over $S$. Let $E \in D(\mathcal{O}_ X)$ be pseudo-coherent. Then $Rf_*E$ is a pseudo-coherent object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.
More generally, if $f : X \to Y$ is proper and $E$ on $X$ is pseudo-coherent relative to $Y$ (More on Morphisms of Spaces, Definition 76.45.3), then $Rf_*E$ is pseudo-coherent (but formation does not commute with base change in this generality). The case of this for schemes is proved in [Kiehl].
Proof.
Special case of Lemma 75.25.3 applied with $\mathcal{G} = \mathcal{O}_ X$.
$\square$
Lemma 75.25.6. Let $R$ be a ring. Let $X$ be an algebraic space 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 75.15.4) it suffices to check this for perfect $E$. By Lemma 75.5.4 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 on Sites, Section 21.36 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 on Sites, Lemma 21.48.4. 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 75.20.1. From Lemma 75.25.4 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
\[ R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, E^\vee ) \otimes _ R^\mathbf {L} M_ n = R\Gamma (X, E^\vee ) \otimes _ R^\mathbf {L} \mathop{\mathrm{lim}}\nolimits M_ n \]
as desired.
$\square$
Lemma 75.25.7. Let $A$ be a ring. Let $X$ be an algebraic space over $A$ which is quasi-compact and quasi-separated. Let $K \in D^-_\mathit{QCoh}(\mathcal{O}_ X)$. If $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is pseudo-coherent in $D(A)$ for every perfect $E$ in $D(\mathcal{O}_ X)$, then $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is pseudo-coherent in $D(A)$ for every pseudo-coherent $E$ in $D(\mathcal{O}_ X)$.
Proof.
There exists an integer $N$ such that $R\Gamma (X, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A)$ has cohomological dimension $N$ as explained in Lemma 75.6.1. Let $b \in \mathbf{Z}$ be such that $H^ i(K) = 0$ for $i > b$. Let $E$ be pseudo-coherent on $X$. It suffices to show that $R\Gamma (X, E \otimes ^\mathbf {L} K)$ 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 75.14.7. 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 $C \otimes ^\mathbf {L} K$ are zero in degrees $\geq m - N - 1$. Thus the cohomology of $R\Gamma (X, C \otimes ^\mathbf {L} K)$ are zero in degrees $\geq m - 1$. Hence
\[ R\Gamma (X, P \otimes ^\mathbf {L} K) \to R\Gamma (X, E \otimes ^\mathbf {L} K) \]
is an isomorphism on cohomology in degrees $\geq m$. By assumption the source is pseudo-coherent. We conclude that $R\Gamma (X, E \otimes ^\mathbf {L} K)$ is $m$-pseudo-coherent as desired.
$\square$
Lemma 75.25.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of finite presentation between algebraic spaces over $S$. 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 $Y$, with support proper over $Y$. Then
\[ K = Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (E, \mathcal{G}^\bullet ) \]
is a perfect object of $D(\mathcal{O}_ Y)$ and its formation commutes with arbitrary base change.
Proof.
The statement on base change is Lemma 75.21.5. Thus it suffices to show that $K$ is a perfect object. If $Y$ is Noetherian, then this follows from Lemma 75.22.3. 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 $Y$, hence we may assume $Y$ is affine. Say $Y = \mathop{\mathrm{Spec}}(R)$. We write $R = \mathop{\mathrm{colim}}\nolimits R_ i$ as a filtered colimit of Noetherian rings $R_ i$. By Limits of Spaces, Lemma 70.7.1 there exists an $i$ and an algebraic space $X_ i$ of finite presentation over $R_ i$ whose base change to $R$ is $X$. By Limits of Spaces, Lemma 70.7.2 we may assume after increasing $i$, that there exists a bounded complex of finitely presented $\mathcal{O}_{X_ i}$-module $\mathcal{G}_ i^\bullet $ whose pullback to $X$ is $\mathcal{G}$. After increasing $i$ we may assume $\mathcal{G}_ i^ n$ is flat over $R_ i$, see Limits of Spaces, Lemma 70.6.12. After increasing $i$ we may assume the support of $\mathcal{G}_ i^ n$ is proper over $R_ i$, see Limits of Spaces, Lemma 70.12.3. Finally, by Lemma 75.13.5 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 75.23.2 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.
$\square$
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