Lemma 33.33.4. Let $k'/k$ be an extension of fields. Let $X$ be a proper scheme over $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$. Let $\mathcal{F}'$ be the pullback of $\mathcal{F}$ to $X_{k'}$. Then $\chi (X, \mathcal{F}) = \chi (X', \mathcal{F}')$.

Proof. This is true because

$H^ i(X_{k'}, \mathcal{F}') = H^ i(X, \mathcal{F}) \otimes _ k k'$

by flat base change, see Cohomology of Schemes, Lemma 30.5.2. $\square$

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).