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 $i = 1, \ldots , n$.
33.35.6 Regularity
Here is the definition.
Note that $\mathcal{F} = \mathcal{O}(d)$ is $m$-regular if and only if $d \geq -m$. This follows from the computation of cohomology groups in Cohomology of Schemes, Equation (30.8.1.1). Namely, we see that $H^ n(\mathbf{P}^ n_ k, \mathcal{O}(d)) = 0$ if and only if $d \geq -n$.
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. $\square$
Lemma 33.35.9. In the situation of Lemma 33.35.3, if $\mathcal{F}$ is $m$-regular, then $\mathcal{G}$ is $m$-regular on $H \cong \mathbf{P}^{n - 1}_ k$.
Proof. Recall that $H^ i(\mathbf{P}^ n_ k, i_*\mathcal{G}) = H^ i(H, \mathcal{G})$ by Cohomology of Schemes, Lemma 30.2.4. Hence we see that for $i \geq 1$ we get
\[ H^ i(\mathbf{P}^ n_ k, \mathcal{F}(m - i)) \to H^ i(H, \mathcal{G}(m - i)) \to H^{i + 1}(\mathbf{P}^ n_ k, \mathcal{F}(m - 1 - i)) \]
by Remark 33.35.5. The lemma follows. $\square$
Lemma 33.35.10. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. If $\mathcal{F}$ is $m$-regular, then $\mathcal{F}$ is $(m + 1)$-regular.
Proof. We prove this by induction on $n$. If $n = 0$ every sheaf is $m$-regular for all $m$ and there is nothing to prove. By Lemma 33.35.8 we may replace $k$ by an infinite overfield and assume $k$ is infinite. Thus we may apply Lemma 33.35.3. By Lemma 33.35.9 we know that $\mathcal{G}$ is $m$-regular. By induction on $n$ we see that $\mathcal{G}$ is $(m + 1)$-regular. Considering the long exact cohomology sequence associated to the sequence
\[ 0 \to \mathcal{F}(m - i) \to \mathcal{F}(m + 1 - i) \to i_*\mathcal{G}(m + 1 - i) \to 0 \]
as in Remark 33.35.5 the reader easily deduces for $i \geq 1$ the vanishing of $H^ i(\mathbf{P}^ n_ k, \mathcal{F}(m + 1 - i))$ from the (known) vanishing of $H^ i(\mathbf{P}^ n_ k, \mathcal{F}(m - i))$ and $H^ i(\mathbf{P}^ n_ k, \mathcal{G}(m + 1 - i))$. $\square$
Lemma 33.35.11. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. If $\mathcal{F}$ is $m$-regular, then the multiplication map is surjective.
\[ H^0(\mathbf{P}^ n_ k, \mathcal{F}(m)) \otimes _ k H^0(\mathbf{P}^ n_ k, \mathcal{O}(1)) \longrightarrow H^0(\mathbf{P}^ n_ k, \mathcal{F}(m + 1)) \]
Proof. Let $k'/k$ be an extension of fields. Let $\mathcal{F}'$ be as in Lemma 33.35.8. By Cohomology of Schemes, Lemma 30.5.2 the base change of the linear map of the lemma to $k'$ is the same linear map for the sheaf $\mathcal{F}'$. Since $k \to k'$ is faithfully flat it suffices to prove the lemma over $k'$, i.e., we may assume $k$ is infinite.
Assume $k$ is infinite. We prove the lemma by induction on $n$. The case $n = 0$ is trivial as $\mathcal{O}(1) \cong \mathcal{O}$ is generated by $T_0$. For $n > 0$ apply Lemma 33.35.3 and tensor the sequence by $\mathcal{O}(m + 1)$ to get
\[ 0 \to \mathcal{F}(m) \xrightarrow {s} \mathcal{F}(m + 1) \to i_*\mathcal{G}(m + 1) \to 0 \]
see Remark 33.35.5. Let $t \in H^0(\mathbf{P}^ n_ k, \mathcal{F}(m + 1))$. By induction the image $\overline{t} \in H^0(H, \mathcal{G}(m + 1))$ is the image of $\sum g_ i \otimes \overline{s}_ i$ with $\overline{s}_ i \in \Gamma (H, \mathcal{O}(1))$ and $g_ i \in H^0(H, \mathcal{G}(m))$. Since $\mathcal{F}$ is $m$-regular we have $H^1(\mathbf{P}^ n_ k, \mathcal{F}(m - 1)) = 0$, hence long exact cohomology sequence associated to the short exact sequence
\[ 0 \to \mathcal{F}(m - 1) \xrightarrow {s} \mathcal{F}(m) \to i_*\mathcal{G}(m) \to 0 \]
shows we can lift $g_ i$ to $f_ i \in H^0(\mathbf{P}^ n_ k, \mathcal{F}(m))$. We can also lift $\overline{s}_ i$ to $s_ i \in H^0(\mathbf{P}^ n_ k, \mathcal{O}(1))$ (see proof of Lemma 33.35.2 for example). After subtracting the image of $\sum f_ i \otimes s_ i$ from $t$ we see that we may assume $\overline{t} = 0$. But this exactly means that $t$ is the image of $f \otimes s$ for some $f \in H^0(\mathbf{P}^ n_ k, \mathcal{F}(m))$ as desired. $\square$
Lemma 33.35.12. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. If $\mathcal{F}$ is $m$-regular, then $\mathcal{F}(m)$ is globally generated.
Proof. For all $d \gg 0$ the sheaf $\mathcal{F}(d)$ is globally generated. This follows for example from the first part of Cohomology of Schemes, Lemma 30.14.1. Pick $d \geq m$ such that $\mathcal{F}(d)$ is globally generated. Choose a basis $f_1, \ldots , f_ r \in H^0(\mathbf{P}^ n_ k, \mathcal{F})$. By Lemma 33.35.11 every element $f \in H^0(\mathbf{P}^ n_ k, \mathcal{F}(d))$ can be written as $f = \sum P_ if_ i$ for some $P_ i \in k[T_0, \ldots , T_ n]$ homogeneous of degree $d - m$. Since the sections $f$ generate $\mathcal{F}(d)$ it follows that the sections $f_ i$ generate $\mathcal{F}(m)$. $\square$
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