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. $\square$

Comments (0)

There are also:

  • 4 comment(s) on Section 57.87: The proper base change theorem

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0A4F. Beware of the difference between the letter 'O' and the digit '0'.