## 30.5 Cohomology and base change, I

Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$. Suppose further that $g : S' \to S$ is any morphism of schemes. Denote $X' = X_{S'} = S' \times _ S X$ the base change of $X$ and denote $f' : X' \to S'$ the base change of $f$. Also write $g' : X' \to X$ the projection, and set $\mathcal{F}' = (g')^*\mathcal{F}$. Here is a diagram representing the situation:

30.5.0.1
\begin{equation} \label{coherent-equation-base-change-diagram} \vcenter { \xymatrix{ \mathcal{F}' = (g')^*\mathcal{F} & X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f & \mathcal{F} \\ Rf'_*\mathcal{F}' & S' \ar[r]^ g & S & Rf_*\mathcal{F} } } \end{equation}

Here is the simplest case of the base change property we have in mind.

Lemma 30.5.1. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $f$ is affine. In this case $f_*\mathcal{F} \cong Rf_*\mathcal{F}$ is a quasi-coherent sheaf, and for every base change diagram (30.5.0.1) we have

$g^*f_*\mathcal{F} = f'_*(g')^*\mathcal{F}.$

Proof. The vanishing of higher direct images is Lemma 30.2.3. The statement is local on $S$ and $S'$. Hence we may assume $X = \mathop{\mathrm{Spec}}(A)$, $S = \mathop{\mathrm{Spec}}(R)$, $S' = \mathop{\mathrm{Spec}}(R')$ and $\mathcal{F} = \widetilde{M}$ for some $A$-module $M$. We use Schemes, Lemma 26.7.3 to describe pullbacks and pushforwards of $\mathcal{F}$. Namely, $X' = \mathop{\mathrm{Spec}}(R' \otimes _ R A)$ and $\mathcal{F}'$ is the quasi-coherent sheaf associated to $(R' \otimes _ R A) \otimes _ A M$. Thus we see that the lemma boils down to the equality

$(R' \otimes _ R A) \otimes _ A M = R' \otimes _ R M$

as $R'$-modules. $\square$

In many situations it is sufficient to know about the following special case of cohomology and base change. It follows immediately from the stronger results in Section 30.7, but since it is so important it deserves its own proof.

Lemma 30.5.2 (Flat base change). 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 with pullback $\mathcal{F}' = (g')^*\mathcal{F}$. Assume that $g$ is flat and that $f$ is quasi-compact and quasi-separated. For any $i \geq 0$

1. the base change map of Cohomology, Lemma 20.17.1 is an isomorphism

$g^*R^ if_*\mathcal{F} \longrightarrow R^ if'_*\mathcal{F}',$
2. if $S = \mathop{\mathrm{Spec}}(A)$ and $S' = \mathop{\mathrm{Spec}}(B)$, then $H^ i(X, \mathcal{F}) \otimes _ A B = H^ i(X', \mathcal{F}')$.

Proof. Using Cohomology, Lemma 20.17.1 in (1) is allowed since $g'$ is flat by Morphisms, Lemma 29.25.8. Having said this, part (1) follows from part (2). Namely, part (1) is local on $S'$ and hence we may assume $S$ and $S'$ are affine. In other words, we have $S = \mathop{\mathrm{Spec}}(A)$ and $S' = \mathop{\mathrm{Spec}}(B)$ as in (2). Then since $R^ if_*\mathcal{F}$ is quasi-coherent (Lemma 30.4.5), it is the quasi-coherent $\mathcal{O}_ S$-module associated to the $A$-module $H^0(S, R^ if_*\mathcal{F}) = H^ i(X, \mathcal{F})$ (equality by Lemma 30.4.6). Similarly, $R^ if'_*\mathcal{F}'$ is the quasi-coherent $\mathcal{O}_{S'}$-module associated to the $B$-module $H^ i(X', \mathcal{F}')$. Since pullback by $g$ corresponds to $- \otimes _ A B$ on modules (Schemes, Lemma 26.7.3) we see that it suffices to prove (2).

Let $A \to B$ be a flat ring homomorphism. Let $X$ be a quasi-compact and quasi-separated scheme over $A$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Set $X_ B = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)$ and denote $\mathcal{F}_ B$ the pullback of $\mathcal{F}$. We are trying to show that the map

$H^ i(X, \mathcal{F}) \otimes _ A B \longrightarrow H^ i(X_ B, \mathcal{F}_ B)$

(given by the reference in the statement of the lemma) is an isomorphism.

In case $X$ is separated, choose an affine open covering $\mathcal{U} : X = U_1 \cup \ldots \cup U_ t$ and recall that

$\check{H}^ p(\mathcal{U}, \mathcal{F}) = H^ p(X, \mathcal{F}),$

see Lemma 30.2.6. If $\mathcal{U}_ B : X_ B = (U_1)_ B \cup \ldots \cup (U_ t)_ B$ we obtain by base change, then it is still the case that each $(U_ i)_ B$ is affine and that $X_ B$ is separated. Thus we obtain

$\check{H}^ p(\mathcal{U}_ B, \mathcal{F}_ B) = H^ p(X_ B, \mathcal{F}_ B).$

We have the following relation between the Čech complexes

$\check{\mathcal{C}}^\bullet (\mathcal{U}_ B, \mathcal{F}_ B) = \check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A B$

as follows from Lemma 30.5.1. Since $A \to B$ is flat, the same thing remains true on taking cohomology.

In case $X$ is quasi-separated, choose an affine open covering $\mathcal{U} : X = U_1 \cup \ldots \cup U_ t$. We will use the Čech-to-cohomology spectral sequence Cohomology, Lemma 20.11.5. The reader who wishes to avoid this spectral sequence can use Mayer-Vietoris and induction on $t$ as in the proof of Lemma 30.4.5. The spectral sequence has $E_2$-page $E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F}))$ and converges to $H^{p + q}(X, \mathcal{F})$. Similarly, we have a spectral sequence with $E_2$-page $E_2^{p, q} = \check{H}^ p(\mathcal{U}_ B, \underline{H}^ q(\mathcal{F}_ B))$ which converges to $H^{p + q}(X_ B, \mathcal{F}_ B)$. Since the intersections $U_{i_0 \ldots i_ p}$ are quasi-compact and separated, the result of the second paragraph of the proof gives $\check{H}^ p(\mathcal{U}_ B, \underline{H}^ q(\mathcal{F}_ B)) = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \otimes _ A B$. Using that $A \to B$ is flat we conclude that $H^ i(X, \mathcal{F}) \otimes _ A B \to H^ i(X_ B, \mathcal{F}_ B)$ is an isomorphism for all $i$ and we win. $\square$

Lemma 30.5.3 (Finite locally free base change). Consider a cartesian diagram of schemes

$\xymatrix{ Y \ar[d]_{g} \ar[r]_ h & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(B) \ar[r] & \mathop{\mathrm{Spec}}(A) }$

Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module with pullback $\mathcal{G} = h^*\mathcal{F}$. If $B$ is a finite locally free $A$-module, then $H^ i(X, \mathcal{F}) \otimes _ A B = H^ i(Y, \mathcal{G})$.

Warning: Do not use this lemma unless you understand the difference between this and Lemma 30.5.2.

Proof. In case $X$ is separated, choose an affine open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ and recall that

$\check{H}^ p(\mathcal{U}, \mathcal{F}) = H^ p(X, \mathcal{F}),$

see Lemma 30.2.6. Let $\mathcal{V} : Y = \bigcup _{i \in I} g^{-1}(U_ i)$ be the corresponding affine open covering of $Y$. The opens $V_ i = g^{-1}(U_ i) = U_ i \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(B)$ are affine and $Y$ is separated. Thus we obtain

$\check{H}^ p(\mathcal{V}, \mathcal{G}) = H^ p(Y, \mathcal{G}).$

We claim the map of Čech complexes

$\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F}) \otimes _ A B \longrightarrow \check{\mathcal{C}}^\bullet (\mathcal{V}, \mathcal{G})$

is an isomorphism. Namely, as $B$ is finitely presented as an $A$-module we see that tensoring with $B$ over $A$ commutes with products, see Algebra, Proposition 10.89.3. Thus it suffices to show that the maps $\Gamma (U_{i_0 \ldots i_ p}, \mathcal{F}) \otimes _ A B \to \Gamma (V_{i_0 \ldots i_ p}, \mathcal{G})$ are isomorphisms which follows from Lemma 30.5.1. Since $A \to B$ is flat, the same thing remains true on taking cohomology.

In the general case we argue in exactly the same way using affine open covering $\mathcal{U} : X = \bigcup _{i \in I} U_ i$ and the corresponding covering $\mathcal{V} : Y = \bigcup _{i \in I} V_ i$ with $V_ i = g^{-1}(U_ i)$ as above. We will use the Čech-to-cohomology spectral sequence Cohomology, Lemma 20.11.5. The spectral sequence has $E_2$-page $E_2^{p, q} = \check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F}))$ and converges to $H^{p + q}(X, \mathcal{F})$. Similarly, we have a spectral sequence with $E_2$-page $E_2^{p, q} = \check{H}^ p(\mathcal{V}, \underline{H}^ q(\mathcal{G}))$ which converges to $H^{p + q}(Y, \mathcal{G})$. Since the intersections $U_{i_0 \ldots i_ p}$ are separated, the result of the previous paragraph gives isomorphisms $\Gamma (U_{i_0 \ldots i_ p}, \underline{H}^ q(\mathcal{F})) \otimes _ A B \to \Gamma (V_{i_0 \ldots i_ p}, \underline{H}^ q(\mathcal{G}))$. Using that $- \otimes _ A B$ commutes with products and is exact, we conclude that $\check{H}^ p(\mathcal{U}, \underline{H}^ q(\mathcal{F})) \otimes _ A B \to \check{H}^ p(\mathcal{V}, \underline{H}^ q(\mathcal{G}))$ is an isomorphism. Using that $A \to B$ is flat we conclude that $H^ i(X, \mathcal{F}) \otimes _ A B \to H^ i(Y, \mathcal{G})$ is an isomorphism for all $i$ and we win. $\square$

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