Lemma 59.91.10. To prove that cohomology commutes with base change for every proper morphism of schemes it suffices to prove it holds for the morphism \mathbf{P}^1_ S \to S for every scheme S.
Proof. Let f : X \to Y be a proper morphism of schemes. Let Y = \bigcup Y_ i be an affine open covering and set X_ i = f^{-1}(Y_ i). If we can prove cohomology commutes with base change for X_ i \to Y_ i, then cohomology commutes with base change for f. Namely, the formation of the higher direct images commutes with Zariski (and even étale) localization on the base, see Lemma 59.51.6. Thus we may assume Y is affine.
Let Y be an affine scheme and let X \to Y be a proper morphism. By Chow's lemma there exists a commutative diagram
\xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^ n_ Y \ar[dl] \\ & Y & }
where X' \to \mathbf{P}^ n_ Y is an immersion, and \pi : X' \to X is proper and surjective, see Limits, Lemma 32.12.1. Since X \to Y is proper, we find that X' \to Y is proper (Morphisms, Lemma 29.41.4). Hence X' \to \mathbf{P}^ n_ Y is a closed immersion (Morphisms, Lemma 29.41.7). It follows that X' \to X \times _ Y \mathbf{P}^ n_ Y = \mathbf{P}^ n_ X is a closed immersion (as an immersion with closed image).
By Lemma 59.91.7 it suffices to prove cohomology commutes with base change for \pi and X' \to Y. These morphisms both factor as a closed immersion followed by a projection \mathbf{P}^ n_ S \to S (for some S). By Lemma 59.91.9 the result holds for closed immersions (as closed immersions are finite). By Lemma 59.91.8 it suffices to prove the result for projections \mathbf{P}^ n_ S \to S.
For every n \geq 1 there is a finite surjective morphism
\mathbf{P}^1_ S \times _ S \ldots \times _ S \mathbf{P}^1_ S \longrightarrow \mathbf{P}^ n_ S
given on coordinates by
((x_1 : y_1), (x_2 : y_2), \ldots , (x_ n : y_ n)) \longmapsto (F_0 : \ldots : F_ n)
where F_0, \ldots , F_ n in x_1, \ldots , y_ n are the polynomials with integer coefficients such that
\prod (x_ i t + y_ i) = F_0 t^ n + F_1 t^{n - 1} + \ldots + F_ n
Applying Lemmas 59.91.7, 59.91.9, and 59.91.8 one more time we conclude that the lemma is true. \square
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