Definition 33.35.7. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. We say $\mathcal{F}$ is *$m$-regular* if

\[ H^ i(\mathbf{P}^ n_ k, \mathcal{F}(m - i)) = 0 \]

for $i = 1, \ldots , n$.

Definition 33.35.7. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. We say $\mathcal{F}$ is *$m$-regular* if

\[ H^ i(\mathbf{P}^ n_ k, \mathcal{F}(m - i)) = 0 \]

for $i = 1, \ldots , n$.

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