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).
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 dth 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.
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 20.54.2) 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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