The Stacks project

33.35.1 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 33.35.2. 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 33.35.3. 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 33.35.4. 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 33.35.3 works for each of them. To prove this, just apply the lemma to $\bigoplus \mathcal{F}_ i$.

Remark 33.35.5. In the situation of Lemmas 33.35.2 and 33.35.3 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 20.52.2) 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).


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