The Stacks project

Lemma 57.87.10. 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 57.51.6. 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 29.39.4). Hence $X' \to \mathbf{P}^ n_ Y$ is a closed immersion (Morphisms, Lemma 29.39.7). 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 57.87.7 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 57.87.9 the result holds for closed immersions (as closed immersions are finite). By Lemma 57.87.8 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 57.87.7, 57.87.9, and 57.87.8 one more time we conclude that the lemma is true. $\square$


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