The Stacks project

Lemma 33.35.8. Let $k'/k$ be an extension of fields. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. Let $\mathcal{F}'$ be the pullback of $\mathcal{F}$ to $\mathbf{P}^ n_{k'}$. Then $\mathcal{F}$ is $m$-regular if and only if $\mathcal{F}'$ is $m$-regular.

Proof. This is true because

\[ H^ i(\mathbf{P}^ n_{k'}, \mathcal{F}') = H^ i(\mathbf{P}^ n_ k, \mathcal{F}) \otimes _ k k' \]

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

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