The Stacks project

33.34 Coherent sheaves on projective space

In this section we prove some results on the cohomology of coherent sheaves on $\mathbf{P}^ n$ over a field which can be found in [Mum]. These will be useful later when discussing Quot and Hilbert schemes.

33.34.1 Preliminaries

Let $k$ be a field, $n \geq 1$, $d \geq 1$, and let $s \in \Gamma (\mathbf{P}_ k^ n, \mathcal{O}(d))$ be a nonzero section. In this section we will write $\mathcal{O}(d)$ for the $d$th twist of the structure sheaf on projective space (Constructions, Definitions 27.10.1 and 27.13.2). Since $\mathbf{P}^ n_ k$ is a variety this section is regular, hence $s$ is a regular section of $\mathcal{O}(d)$ and defines an effective Cartier divisor $H = Z(s) \subset \mathbf{P}^ n_ k$, see Divisors, Section 31.13. Such a divisor $H$ is called a hypersurface and if $d = 1$ it is called a hyperplane.

Lemma 33.34.2. Let $k$ be a field. Let $n \geq 1$. Let $i : H \to \mathbf{P}^ n_ k$ be a hyperplane. Then there exists an isomorphism

\[ \varphi : \mathbf{P}^{n - 1}_ k \longrightarrow H \]

such that $i^*\mathcal{O}(1)$ pulls back to $\mathcal{O}(1)$.

Proof. We have $\mathbf{P}^ n_ k = \text{Proj}(k[T_0, \ldots , T_ n])$. The section $s$ corresponds to a homogeneous form in $T_0, \ldots , T_ n$ of degree $1$, see Cohomology of Schemes, Section 30.8. Say $s = \sum a_ i T_ i$. Constructions, Lemma 27.13.7 gives that $H = \text{Proj}(k[T_0, \ldots , T_ n]/I)$ for the graded ideal $I$ defined by setting $I_ d$ equal to the kernel of the map $\Gamma (\mathbf{P}^ n_ k, \mathcal{O}(d)) \to \Gamma (H, i^*\mathcal{O}(d))$. By our construction of $Z(s)$ in Divisors, Definition 31.14.8 we see that on $D_{+}(T_ j)$ the ideal of $H$ is generated by $\sum a_ i T_ i/T_ j$ in the polynomial ring $k[T_0/T_ j, \ldots , T_ n/T_ j]$. Thus it is clear that $I$ is the ideal generated by $\sum a_ i T_ i$. Note that

\[ k[T_0, \ldots , T_ n]/I = k[T_0, \ldots , T_ n]/(\sum a_ i T_ i) \cong k[S_0, \ldots , S_{n - 1}] \]

as graded rings. For example, if $a_ n \not= 0$, then mapping $S_ i$ equal to the class of $T_ i$ works. We obtain the desired isomorphism by functoriality of $\text{Proj}$. Equality of twists of structure sheaves follows for example from Constructions, Lemma 27.11.5. $\square$

Lemma 33.34.3. Let $k$ be an infinite field. Let $n \geq 1$. Let $\mathcal{F}$ be a coherent module on $\mathbf{P}^ n_ k$. Then there exist a nonzero section $s \in \Gamma (\mathbf{P}^ n_ k, \mathcal{O}(1))$ and a short exact sequence

\[ 0 \to \mathcal{F}(-1) \to \mathcal{F} \to i_*\mathcal{G} \to 0 \]

where $i : H \to \mathbf{P}^ n_ k$ is the hyperplane $H$ associated to $s$ and $\mathcal{G} = i^*\mathcal{F}$.

Proof. The map $\mathcal{F}(-1) \to \mathcal{F}$ comes from Constructions, Equation (27.10.1.2) with $n = 1$, $m = -1$ and the section $s$ of $\mathcal{O}(1)$. Let's work out what this map looks like if we restrict it to $D_{+}(T_0)$. Write $D_{+}(T_0) = \mathop{\mathrm{Spec}}(k[x_1, \ldots , x_ n])$ with $x_ i = T_ i/T_0$. Identify $\mathcal{O}(1)|_{D_{+}(T_0)}$ with $\mathcal{O}$ using the section $T_0$. Hence if $s = \sum a_ iT_ i$ then $s|_{D_{+}(T_0)} = a_0 + \sum a_ ix_ i$ with the identification chosen above. Furthermore, suppose $\mathcal{F}|_{D_{+}(T_0)}$ corresponds to the finite $k[x_1, \ldots , x_ n]$-module $M$. Via the identification $\mathcal{F}(-1) = \mathcal{F} \otimes \mathcal{O}(-1)$ and our chosen trivialization of $\mathcal{O}(1)$ we see that $\mathcal{F}(-1)$ corresponds to $M$ as well. Thus restricting $\mathcal{F}(-1) \to \mathcal{F}$ to $D_{+}(T_0)$ gives the map

\[ M \xrightarrow {a_0 + \sum a_ ix_ i} M \]

To see that the arrow is injective, it suffices to pick $a_0 + \sum a_ ix_ i$ outside any of the associated primes of $M$, see Algebra, Lemma 10.63.9. By Algebra, Lemma 10.63.5 the set $\text{Ass}(M)$ of associated primes of $M$ is finite. Note that for $\mathfrak p \in \text{Ass}(M)$ the intersection $\mathfrak p \cap \{ a_0 + \sum a_ i x_ i\} $ is a proper $k$-subvector space. We conclude that there is a finite family of proper sub vector spaces $V_1, \ldots , V_ m \subset \Gamma (\mathbf{P}^ n_ k, \mathcal{O}(1))$ such that if we take $s$ outside of $\bigcup V_ i$, then multiplication by $s$ is injective over $D_{+}(T_0)$. Similarly for the restriction to $D_{+}(T_ j)$ for $j = 1, \ldots , n$. Since $k$ is infinite, a finite union of proper sub vector spaces is never equal to the whole space, hence we may choose $s$ such that the map is injective. The cokernel of $\mathcal{F}(-1) \to \mathcal{F}$ is annihilated by $\mathop{\mathrm{Im}}(s : \mathcal{O}(-1) \to \mathcal{O})$ which is the ideal sheaf of $H$ by Divisors, Definition 31.14.8. Hence we obtain $\mathcal{G}$ on $H$ using Cohomology of Schemes, Lemma 30.9.8. $\square$

Remark 33.34.4. Let $k$ be an infinite field. Let $n \geq 1$. Given a finite number of coherent modules $\mathcal{F}_ i$ on $\mathbf{P}^ n_ k$ we can choose a single $s \in \Gamma (\mathbf{P}^ n_ k, \mathcal{O}(1))$ such that the statement of Lemma 33.34.3 works for each of them. To prove this, just apply the lemma to $\bigoplus \mathcal{F}_ i$.

Remark 33.34.5. In the situation of Lemmas 33.34.2 and 33.34.3 we have $H \cong \mathbf{P}^{n - 1}_ k$ with Serre twists $\mathcal{O}_ H(d) = i^*\mathcal{O}_{\mathbf{P}^ n_ k}(d)$. For every $d \in \mathbf{Z}$ we have a short exact sequence

\[ 0 \to \mathcal{F}(d - 1) \to \mathcal{F}(d) \to i_*(\mathcal{G}(d)) \to 0 \]

Namely, tensoring by $\mathcal{O}_{\mathbf{P}^ n_ k}(d)$ is an exact functor and by the projection formula (Cohomology, Lemma 20.51.2) we have $i_*(\mathcal{G}(d)) = i_*\mathcal{G} \otimes \mathcal{O}_{\mathbf{P}^ n_ k}(d)$. We obtain corresponding long exact sequences

\[ H^ i(\mathbf{P}^ n_ k, \mathcal{F}(d - 1)) \to H^ i(\mathbf{P}^ n_ k, \mathcal{F}(d)) \to H^ i(H, \mathcal{G}(d)) \to H^{i + 1}(\mathbf{P}^ n_ k, \mathcal{F}(d - 1)) \]

This follows from the above and the fact that we have $H^ i(\mathbf{P}^ n_ k, i_*\mathcal{G}(d)) = H^ i(H, \mathcal{G}(d))$ by Cohomology of Schemes, Lemma 30.2.4 (closed immersions are affine).

33.34.6 Regularity

Here is the definition.

Definition 33.34.7. 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 $i = 1, \ldots , n$.

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 33.34.8. Let $k \subset k'$ 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 33.34.9. In the situation of Lemma 33.34.3, 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 33.34.5. The lemma follows. $\square$

Lemma 33.34.10. 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 33.34.8 we may replace $k$ by an infinite overfield and assume $k$ is infinite. Thus we may apply Lemma 33.34.3. By Lemma 33.34.9 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 33.34.5 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 33.34.11. 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

\[ H^0(\mathbf{P}^ n_ k, \mathcal{F}(m)) \otimes _ k H^0(\mathbf{P}^ n_ k, \mathcal{O}(1)) \longrightarrow H^0(\mathbf{P}^ n_ k, \mathcal{F}(m + 1)) \]

is surjective.

Proof. Let $k \subset k'$ be an extension of fields. Let $\mathcal{F}'$ be as in Lemma 33.34.8. 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 33.34.3 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 33.34.5. 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 33.34.2 for example). After substracting 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 33.34.12. 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 33.34.11 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 $\mathcal{F}(m)$. $\square$

33.34.13 Hilbert polynomials

The following lemma will be made obsolete by the more general Lemma 33.44.1.

Lemma 33.34.14. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. The function

\[ d \longmapsto \chi (\mathbf{P}^ n_ k, \mathcal{F}(d)) \]

is a polynomial.

Proof. We prove this by induction on $n$. If $n = 0$, then $\mathbf{P}^ n_ k = \mathop{\mathrm{Spec}}(k)$ and $\mathcal{F}(d) = \mathcal{F}$. Hence in this case the function is constant, i.e., a polynomial of degree $0$. Assume $n > 0$. By Lemma 33.32.4 we may assume $k$ is infinite. Apply Lemma 33.34.3. Applying Lemma 33.32.2 to the twisted sequences $0 \to \mathcal{F}(d - 1) \to \mathcal{F}(d) \to i_*\mathcal{G}(d) \to 0$ we obtain

\[ \chi (\mathbf{P}^ n_ k, \mathcal{F}(d)) - \chi (\mathbf{P}^ n_ k, \mathcal{F}(d - 1)) = \chi (H, \mathcal{G}(d)) \]

See Remark 33.34.5. Since $H \cong \mathbf{P}^{n - 1}_ k$ by induction the right hand side is a polynomial. The lemma is finished by noting that any function $f : \mathbf{Z} \to \mathbf{Z}$ with the property that the map $d \mapsto f(d) - f(d - 1)$ is a polynomial, is itself a polynomial. We omit the proof of this fact (hint: compare with Algebra, Lemma 10.58.5). $\square$

Definition 33.34.15. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$. The function $d \mapsto \chi (\mathbf{P}^ n_ k, \mathcal{F}(d))$ is called the Hilbert polynomial of $\mathcal{F}$.

The Hilbert polynomial has coefficients in $\mathbf{Q}$ and not in general in $\mathbf{Z}$. For example the Hilbert polynomial of $\mathcal{O}_{\mathbf{P}^ n_ k}$ is

\[ d \longmapsto {d + n \choose n} = \frac{d^ n}{n!} + \ldots \]

This follows from the following lemma and the fact that

\[ H^0(\mathbf{P}^ n_ k, \mathcal{O}_{\mathbf{P}^ n_ k}(d)) = k[T_0, \ldots , T_ n]_ d \]

(degree $d$ part) whose dimension over $k$ is ${d + n \choose n}$.

Lemma 33.34.16. Let $k$ be a field. Let $n \geq 0$. Let $\mathcal{F}$ be a coherent sheaf on $\mathbf{P}^ n_ k$ with Hilbert polynomial $P \in \mathbf{Q}[t]$. Then

\[ P(d) = \dim _ k H^0(\mathbf{P}^ n_ k, \mathcal{F}(d)) \]

for all $d \gg 0$.

Proof. This follows from the vanishing of cohomology of high enough twists of $\mathcal{F}$. See Cohomology of Schemes, Lemma 30.14.1. $\square$

33.34.17 Boundedness of quotients

In this subsection we bound the regularity of quotients of a given coherent sheaf on $\mathbf{P}^ n$ in terms of the Hilbert polynomial.

Lemma 33.34.18. Let $k$ be a field. Let $n \geq 0$. Let $r \geq 1$. Let $P \in \mathbf{Q}[t]$. There exists an integer $m$ depending on $n$, $r$, and $P$ with the following property: if

\[ 0 \to \mathcal{K} \to \mathcal{O}^{\oplus r} \to \mathcal{F} \to 0 \]

is a short exact sequence of coherent sheaves on $\mathbf{P}^ n_ k$ and $\mathcal{F}$ has Hilbert polynomial $P$, then $\mathcal{K}$ is $m$-regular.

Proof. We prove this by induction on $n$. If $n = 0$, then $\mathbf{P}^ n_ k = \mathop{\mathrm{Spec}}(k)$ and any coherent module is $0$-regular and any surjective map is surjective on global sections. Assume $n > 0$. Consider an exact sequence as in the lemma. Let $P' \in \mathbf{Q}[t]$ be the polynomial $P'(t) = P(t) - P(t - 1)$. Let $m'$ be the integer which works for $n - 1$, $r$, and $P'$. By Lemmas 33.34.8 and 33.32.4 we may replace $k$ by a field extension, hence we may assume $k$ is infinite. Apply Lemma 33.34.3 to the coherent sheaf $\mathcal{F}$. The Hilbert polynomial of $\mathcal{F}' = i^*\mathcal{F}$ is $P'$ (see proof of Lemma 33.34.14). Since $i^*$ is right exact we see that $\mathcal{F}'$ is a quotient of $\mathcal{O}_ H^{\oplus r} = i^*\mathcal{O}^{\oplus r}$. Thus the induction hypothesis applies to $\mathcal{F}'$ on $H \cong \mathbf{P}^{n - 1}_ k$ (Lemma 33.34.2). Note that the map $\mathcal{K}(-1) \to \mathcal{K}$ is injective as $\mathcal{K} \subset \mathcal{O}^{\oplus r}$ and has cokernel $i_*\mathcal{H}$ where $\mathcal{H} = i^*\mathcal{K}$. By the snake lemma (Homology, Lemma 12.5.17) we obtain a commutative diagram with exact columns and rows

\[ \xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & \mathcal{K}(-1) \ar[r] \ar[d] & \mathcal{O}^{\oplus r}(-1) \ar[r] \ar[d] & \mathcal{F}(-1) \ar[d] \ar[r] & 0 \\ 0 \ar[r] & \mathcal{K} \ar[r] \ar[d] & \mathcal{O}^{\oplus r} \ar[r] \ar[d] & \mathcal{F} \ar[d] \ar[r] & 0\\ 0 \ar[r] & i_*\mathcal{H} \ar[r] \ar[d] & i_*\mathcal{O}_ H^{\oplus r} \ar[r] \ar[d] & i_*\mathcal{F}' \ar[r] \ar[d] & 0 \\ & 0 & 0 & 0 } \]

Thus the induction hypothesis applies to the exact sequence $0 \to \mathcal{H} \to \mathcal{O}_ H^{\oplus r} \to \mathcal{F}' \to 0$ on $H \cong \mathbf{P}^{n - 1}_ k$ (Lemma 33.34.2) and $\mathcal{H}$ is $m'$-regular. Recall that this implies that $\mathcal{H}$ is $d$-regular for all $d \geq m'$ (Lemma 33.34.10).

Let $i \geq 2$ and $d \geq m'$. It follows from the long exact cohomology sequence associated to the left column of the diagram above and the vanishing of $H^{i - 1}(H, \mathcal{H}(d))$ that the map

\[ H^ i(\mathbf{P}^ n_ k, \mathcal{K}(d - 1)) \longrightarrow H^ i(\mathbf{P}^ n_ k, \mathcal{K}(d)) \]

is injective. As these groups are zero for $d \gg 0$ (Cohomology of Schemes, Lemma 30.14.1) we conclude $H^ i(\mathbf{P}^ n_ k, \mathcal{K}(d))$ are zero for all $d \geq m'$ and $i \geq 2$.

We still have to control $H^1$. First we observe that all the maps

\[ H^1(\mathbf{P}^ n_ k, \mathcal{K}(m' - 1)) \to H^1(\mathbf{P}^ n_ k, \mathcal{K}(m')) \to H^1(\mathbf{P}^ n_ k, \mathcal{K}(m' + 1)) \to \ldots \]

are surjective by the vanishing of $H^1(H, \mathcal{H}(d))$ for $d \geq m'$. Suppose $d > m'$ is such that

\[ H^1(\mathbf{P}^ n_ k, \mathcal{K}(d - 1)) \longrightarrow H^1(\mathbf{P}^ n_ k, \mathcal{K}(d)) \]

is injective. Then $H^0(\mathbf{P}^ n_ k, \mathcal{K}(d)) \to H^0(H, \mathcal{H}(d))$ is surjective. Consider the commutative diagram

\[ \xymatrix{ H^0(\mathbf{P}^ n_ k, \mathcal{K}(d)) \otimes _ k H^0(\mathbf{P}^ n_ k, \mathcal{O}(1)) \ar[r] \ar[d] & H^0(\mathbf{P}^ n_ k, \mathcal{K}(d + 1)) \ar[d] \\ H^0(H, \mathcal{H}(d)) \otimes _ k H^0(H, \mathcal{O}_ H(1)) \ar[r] & H^0(H, \mathcal{H}(d + 1)) } \]

By Lemma 33.34.11 we see that the bottom horizontal arrow is surjective. Hence the right vertical arrow is surjective. We conclude that

\[ H^1(\mathbf{P}^ n_ k, \mathcal{K}(d)) \longrightarrow H^1(\mathbf{P}^ n_ k, \mathcal{K}(d + 1)) \]

is injective. By induction we see that

\[ H^1(\mathbf{P}^ n_ k, \mathcal{K}(d - 1)) \to H^1(\mathbf{P}^ n_ k, \mathcal{K}(d)) \to H^1(\mathbf{P}^ n_ k, \mathcal{K}(d + 1)) \to \ldots \]

are all injective and we conclude that $H^1(\mathbf{P}^ n_ k, \mathcal{K}(d - 1)) = 0$ because of the eventual vanishing of these groups. Thus the dimensions of the groups $H^1(\mathbf{P}^ n_ k, \mathcal{K}(d))$ for $d \geq m'$ are strictly decreasing until they become zero. It follows that the regularity of $\mathcal{K}$ is bounded by $m' + \dim _ k H^1(\mathbf{P}^ n_ k, \mathcal{K}(m'))$. On the other hand, by the vanishing of the higher cohomology groups we have

\[ \dim _ k H^1(\mathbf{P}^ n_ k, \mathcal{K}(m')) = - \chi (\mathbf{P}^ n_ k, \mathcal{K}(m')) + \dim _ k H^0(\mathbf{P}^ n_ k, \mathcal{K}(m')) \]

Note that the $H^0$ has dimension bounded by the dimension of $H^0(\mathbf{P}^ n_ k, \mathcal{O}^{\oplus r}(m'))$ which is at most $r{n + m' \choose n}$ if $m' > 0$ and zero if not. Finally, the term $\chi (\mathbf{P}^ n_ k, \mathcal{K}(m'))$ is equal to $r{n + m' \choose n} - P(m')$. This gives a bound of the desired type finishing the proof of the lemma. $\square$


Comments (2)

Comment #4063 by Yicheng Zhou on

Just a small typo: in the paragraph between Definition 08AD and Lemma 08AE, a is missing in the left side of


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