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
Comments (8)
Comment #936 by correction_bot on
Comment #957 by Johan on
Comment #2343 by Daniel on
Comment #2412 by Johan on
Comment #4682 by Nicolas Müller on
Comment #4807 by Johan on
Comment #5458 by Du on
Comment #5676 by Johan on