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. $\square$

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