30.7 Cohomology and base change, II
Let $f : X \to S$ be a morphism of schemes and let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. If $f$ is quasi-compact and quasi-separated we would like to represent $Rf_*\mathcal{F}$ by a complex of quasi-coherent sheaves on $S$. This follows from the fact that the sheaves $R^ if_*\mathcal{F}$ are quasi-coherent if $S$ is quasi-compact and has affine diagonal, using that $D_\mathit{QCoh}(S)$ is equivalent to $D(\mathit{QCoh}(\mathcal{O}_ S))$, see Derived Categories of Schemes, Proposition 36.7.5.
In this section we will use a different approach which produces an explicit complex having a good base change property. The construction is particularly easy if $f$ and $S$ are separated, or more generally have affine diagonal. Since this is the case which by far the most often used we treat it separately.
Lemma 30.7.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $X$ is quasi-compact and $X$ and $S$ have affine diagonal (e.g., if $X$ and $S$ are separated). In this case we can compute $Rf_*\mathcal{F}$ as follows:
Choose a finite affine open covering $\mathcal{U} : X = \bigcup _{i = 1, \ldots , n} U_ i$.
For $i_0, \ldots , i_ p \in \{ 1, \ldots , n\} $ denote $f_{i_0 \ldots i_ p} : U_{i_0 \ldots i_ p} \to S$ the restriction of $f$ to the intersection $U_{i_0 \ldots i_ p} = U_{i_0} \cap \ldots \cap U_{i_ p}$.
Set $\mathcal{F}_{i_0 \ldots i_ p}$ equal to the restriction of $\mathcal{F}$ to $U_{i_0 \ldots i_ p}$.
Set
\[ \check{\mathcal{C}}^ p(\mathcal{U}, f, \mathcal{F}) = \bigoplus \nolimits _{i_0 \ldots i_ p} f_{i_0 \ldots i_ p *} \mathcal{F}_{i_0 \ldots i_ p} \]
and define differentials $d : \check{\mathcal{C}}^ p(\mathcal{U}, f, \mathcal{F}) \to \check{\mathcal{C}}^{p + 1}(\mathcal{U}, f, \mathcal{F})$ as in Cohomology, Equation (20.9.0.1).
Then the complex $\check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F})$ is a complex of quasi-coherent sheaves on $S$ which comes equipped with an isomorphism
\[ \check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf_*\mathcal{F} \]
in $D^{+}(S)$. This isomorphism is functorial in the quasi-coherent sheaf $\mathcal{F}$.
Proof.
Consider the resolution $\mathcal{F} \to {\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F})$ of Cohomology, Lemma 20.24.1. We have an equality of complexes $\check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) = f_*{\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F})$ of quasi-coherent $\mathcal{O}_ S$-modules. The morphisms $j_{i_0 \ldots i_ p} : U_{i_0 \ldots i_ p} \to X$ and the morphisms $f_{i_0 \ldots i_ p} : U_{i_0 \ldots i_ p} \to S$ are affine by Morphisms, Lemma 29.11.11 and Lemma 30.2.5. Hence $R^ qj_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}$ as well as $R^ qf_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}$ are zero for $q > 0$ (Lemma 30.2.3). Using $f \circ j_{i_0 \ldots i_ p} = f_{i_0 \ldots i_ p}$ and the spectral sequence of Cohomology, Lemma 20.13.8 we conclude that $R^ qf_*(j_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}) = 0$ for $q > 0$. Since the terms of the complex ${\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F})$ are finite direct sums of the sheaves $j_{i_0 \ldots i_ p *}\mathcal{F}_{i_0 \ldots i_ p}$ we conclude using Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) that
\[ Rf_* \mathcal{F} = f_*{\mathfrak C}^\bullet (\mathcal{U}, \mathcal{F}) = \check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) \]
as desired.
$\square$
Next, we are going to consider what happens if we do a base change.
Lemma 30.7.2. With notation as in diagram (30.5.0.1). Assume $f : X \to S$ and $\mathcal{F}$ satisfy the hypotheses of Lemma 30.7.1. Choose a finite affine open covering $\mathcal{U} : X = \bigcup U_ i$ of $X$. There is a canonical isomorphism
\[ g^*\check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) \longrightarrow Rf'_*\mathcal{F}' \]
in $D^{+}(S')$. Moreover, if $S' \to S$ is affine, then in fact
\[ g^*\check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) = \check{\mathcal{C}}^\bullet (\mathcal{U}', f', \mathcal{F}') \]
with $\mathcal{U}' : X' = \bigcup U_ i'$ where $U_ i' = (g')^{-1}(U_ i) = U_{i, S'}$ is also affine.
Proof.
In fact we may define $U_ i' = (g')^{-1}(U_ i) = U_{i, S'}$ no matter whether $S'$ is affine over $S$ or not. Let $\mathcal{U}' : X' = \bigcup U_ i'$ be the induced covering of $X'$. In this case we claim that
\[ g^*\check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) = \check{\mathcal{C}}^\bullet (\mathcal{U}', f', \mathcal{F}') \]
with $\check{\mathcal{C}}^\bullet (\mathcal{U}', f', \mathcal{F}')$ defined in exactly the same manner as in Lemma 30.7.1. This is clear from the case of affine morphisms (Lemma 30.5.1) by working locally on $S'$. Moreover, exactly as in the proof of Lemma 30.7.1 one sees that there is an isomorphism
\[ \check{\mathcal{C}}^\bullet (\mathcal{U}', f', \mathcal{F}') \longrightarrow Rf'_*\mathcal{F}' \]
in $D^{+}(S')$ since the morphisms $U_ i' \to X'$ and $U_ i' \to S'$ are still affine (being base changes of affine morphisms). Details omitted.
$\square$
The lemma above says that the complex
\[ \mathcal{K}^\bullet = \check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F}) \]
is a bounded below complex of quasi-coherent sheaves on $S$ which universally computes the higher direct images of $f : X \to S$. This is something about this particular complex and it is not preserved by replacing $\check{\mathcal{C}}^\bullet (\mathcal{U}, f, \mathcal{F})$ by a quasi-isomorphic complex in general! In other words, this is not a statement that makes sense in the derived category. The reason is that the pullback $g^*\mathcal{K}^\bullet $ is not equal to the derived pullback $Lg^*\mathcal{K}^\bullet $ of $\mathcal{K}^\bullet $ in general!
Here is a more general case where we can prove this statement. We remark that the condition of $S$ being separated is harmless in most applications, since this is usually used to prove some local property of the total derived image. The proof is significantly more involved and uses hypercoverings; it is a nice example of how you can use them sometimes.
Lemma 30.7.3. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Assume that $f$ is quasi-compact and quasi-separated and that $S$ is quasi-compact and separated. There exists a bounded below complex $\mathcal{K}^\bullet $ of quasi-coherent $\mathcal{O}_ S$-modules with the following property: For every morphism $g : S' \to S$ the complex $g^*\mathcal{K}^\bullet $ is a representative for $Rf'_*\mathcal{F}'$ with notation as in diagram (30.5.0.1).
Proof.
(If $f$ is separated as well, please see Lemma 30.7.2.) The assumptions imply in particular that $X$ is quasi-compact and quasi-separated as a scheme. Let $\mathcal{B}$ be the set of affine opens of $X$. By Hypercoverings, Lemma 25.11.4 we can find a hypercovering $K = (I, \{ U_ i\} )$ such that each $I_ n$ is finite and each $U_ i$ is an affine open of $X$. By Hypercoverings, Lemma 25.5.3 there is a spectral sequence with $E_2$-page
\[ E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F})) \]
converging to $H^{p + q}(X, \mathcal{F})$. Note that $\check{H}^ p(K, \underline{H}^ q(\mathcal{F}))$ is the $p$th cohomology group of the complex
\[ \prod \nolimits _{i \in I_0} H^ q(U_ i, \mathcal{F}) \to \prod \nolimits _{i \in I_1} H^ q(U_ i, \mathcal{F}) \to \prod \nolimits _{i \in I_2} H^ q(U_ i, \mathcal{F}) \to \ldots \]
Since each $U_ i$ is affine we see that this is zero unless $q = 0$ in which case we obtain
\[ \prod \nolimits _{i \in I_0} \mathcal{F}(U_ i) \to \prod \nolimits _{i \in I_1} \mathcal{F}(U_ i) \to \prod \nolimits _{i \in I_2} \mathcal{F}(U_ i) \to \ldots \]
Thus we conclude that $R\Gamma (X, \mathcal{F})$ is computed by this complex.
For any $n$ and $i \in I_ n$ denote $f_ i : U_ i \to S$ the restriction of $f$ to $U_ i$. As $S$ is separated and $U_ i$ is affine this morphism is affine. Consider the complex of quasi-coherent sheaves
\[ \mathcal{K}^\bullet = ( \prod \nolimits _{i \in I_0} f_{i, *}\mathcal{F}|_{U_ i} \to \prod \nolimits _{i \in I_1} f_{i, *}\mathcal{F}|_{U_ i} \to \prod \nolimits _{i \in I_2} f_{i, *}\mathcal{F}|_{U_ i} \to \ldots ) \]
on $S$. As in Hypercoverings, Lemma 25.5.3 we obtain a map $\mathcal{K}^\bullet \to Rf_*\mathcal{F}$ in $D(\mathcal{O}_ S)$ by choosing an injective resolution of $\mathcal{F}$ (details omitted). Consider any affine scheme $V$ and a morphism $g : V \to S$. Then the base change $X_ V$ has a hypercovering $K_ V = (I, \{ U_{i, V}\} )$ obtained by base change. Moreover, $g^*f_{i, *}\mathcal{F} = f_{i, V, *}(g')^*\mathcal{F}|_{U_{i, V}}$. Thus the arguments above prove that $\Gamma (V, g^*\mathcal{K}^\bullet )$ computes $R\Gamma (X_ V, (g')^*\mathcal{F})$. This finishes the proof of the lemma as it suffices to prove the equality of complexes Zariski locally on $S'$.
$\square$
The following lemma is a variant to flat base change.
Lemma 30.7.4. Consider a cartesian diagram of schemes
\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ S' \ar[r]^ g & S } \]
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $\mathcal{G}$ be a quasi-coherent $\mathcal{O}_{S'}$-module flat over $S$. Assume $f$ is quasi-compact and quasi-separated. For any $i \geq 0$ there is an identification
\[ \mathcal{G} \otimes _{\mathcal{O}_{S'}} g^*R^ if_*\mathcal{F} = R^ if'_*\left((f')^*\mathcal{G} \otimes _{\mathcal{O}_{X'}} (g')^*\mathcal{F}\right) \]
Proof.
Let us construct a map from left to right. First, we have the base change map $Lg^*Rf_*\mathcal{F} \to Rf'_*L(g')^*\mathcal{F}$. There is also the adjunction map $\mathcal{G} \to Rf'_*L(f')^*\mathcal{G}$. Using the relative cup product We obtain
\begin{align*} \mathcal{G} \otimes _{\mathcal{O}_{S'}}^\mathbf {L} Lg^*Rf_*\mathcal{F} & \to Rf'_*L(f')^*\mathcal{G} \otimes _{\mathcal{O}_{S'}}^\mathbf {L} Rf'_*L(g')^*\mathcal{F} \\ & \to Rf'_*\left(L(f')^*\mathcal{G} \otimes _{\mathcal{O}_{X'}}^\mathbf {L} L(g')^*\mathcal{F}\right) \\ & \to Rf'_*\left((f')^*\mathcal{G} \otimes _{\mathcal{O}_{X'}} (g')^*\mathcal{F}\right) \end{align*}
where for the middle arrow we used the relative cup product, see Cohomology, Remark 20.28.7. The source of the composition is
\[ \mathcal{G} \otimes _{\mathcal{O}_{S'}}^\mathbf {L} Lg^*Rf_*\mathcal{F} = \mathcal{G} \otimes _{g^{-1}\mathcal{O}_ S}^\mathbf {L} g^{-1}Rf_*\mathcal{F} \]
by Cohomology, Lemma 20.27.4. Since $\mathcal{G}$ is flat as a sheaf of $g^{-1}\mathcal{O}_ S$-modules and since $g^{-1}$ is an exact functor, this is a complex whose $i$th cohomology sheaf is $\mathcal{G} \otimes _{g^{-1}\mathcal{O}_ S} g^{-1}R^ if_*\mathcal{F} = \mathcal{G} \otimes _{\mathcal{O}_{S'}} g^*R^ if_*\mathcal{F}$. In this way we obtain global maps from left to right in the equality of the lemma. To show this map is an isomorphism we may work locally on $S'$. Thus we may and do assume that $S$ and $S'$ are affine schemes.
Proof in case $S$ and $S'$ are affine. Say $S = \mathop{\mathrm{Spec}}(A)$ and $S' = \mathop{\mathrm{Spec}}(B)$ and say $\mathcal{G}$ corresponds to the $B$-module $N$ which is assumed to be $A$-flat. Since $S$ is affine, $X$ is quasi-compact and quasi-separated. We will use a hypercovering argument to finish the proof; if $X$ is separated or has affine diagonal, then you can use a Čech covering. Let $\mathcal{B}$ be the set of affine opens of $X$. By Hypercoverings, Lemma 25.11.4 we can find a hypercovering $K = (I, \{ U_ i\} )$ of $X$ such that each $I_ n$ is finite and each $U_ i$ is an affine open of $X$. By Hypercoverings, Lemma 25.5.3 there is a spectral sequence with $E_2$-page
\[ E_2^{p, q} = \check{H}^ p(K, \underline{H}^ q(\mathcal{F})) \]
converging to $H^{p + q}(X, \mathcal{F})$. Since each $U_ i$ is affine and $\mathcal{F}$ is quasi-coherent the value of $\underline{H}^ q(\mathcal{F})$ is zero on $U_ i$ for $q > 0$. Thus the spectral sequence degenerates and we conclude that the cohomology modules $H^ q(X, \mathcal{F})$ are computed by
\[ \prod \nolimits _{i \in I_0} \mathcal{F}(U_ i) \to \prod \nolimits _{i \in I_1} \mathcal{F}(U_ i) \to \prod \nolimits _{i \in I_2} \mathcal{F}(U_ i) \to \ldots \]
Next, note that the base change of our hypercovering to $S'$ is a hypercovering of $X' = S' \times _ S X$. The schemes $S' \times _ S U_ i$ are affine too and we have
\[ \left((f')^*\mathcal{G} \otimes _{\mathcal{O}_{S'}} (g')^*\mathcal{F}\right) (S' \times _ S U_ i) = N \otimes _ A \mathcal{F}(U_ i) \]
In this way we conclude that the cohomology modules $H^ q(X', (f')^*\mathcal{G} \otimes _{\mathcal{O}_{S'}} (g')^*\mathcal{F})$ are computed by
\[ N \otimes _ A \left( \prod \nolimits _{i \in I_0} \mathcal{F}(U_ i) \to \prod \nolimits _{i \in I_1} \mathcal{F}(U_ i) \to \prod \nolimits _{i \in I_2} \mathcal{F}(U_ i) \to \ldots \right) \]
Since $N$ is flat over $A$, we conclude that
\[ H^ q(X', (f')^*\mathcal{G} \otimes _{\mathcal{O}_{S'}} (g')^*\mathcal{F}) = N \otimes _ A H^ q(X, \mathcal{F}) \]
Since this is the translation into algebra of the statement we had to show the proof is complete.
$\square$
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