The Stacks project

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

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

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

(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

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

We have the following relation between the Čech complexes

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. $\square$

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

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

We claim the map of Čech complexes

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. $\square$