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Vector Bundles on Projective Space

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Vector Bundles on Projective Space Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X. Lemma 1.1. For a vector bundle π : E ! X and any nondisjoint open subsets Uα;Uβ ⊂ X with isomorphisms 'α;'β satisfying the diagram (1.1) above with the −1 r r same value for r, the map 'β ◦ 'α : Uαβ × A ! Uαβ × A is given by an r × r invertible matrix in OX (Uαβ), where Uαβ := Uα \ Uβ. −1 Proof. Note that 'β ◦ 'α defines an automorphism of the vector bundle pr1 : Uαβ × r A ! Uαβ. It therefore suffices to show that an automorphism of a trivial vector bundle, i.e., the vector bundle for which (1.1) commutes for U = X, is given by an r × r invertible matrix in O(X). r Let f denote our automorphism of the trivial bundle pr1 : X × A ! X, and let r r e1; : : : ; er be a basis for A , with corresponding coordinates ξ1; : : : ; ξr in O(A ). The r r ∗ r second projection pr2 : X × A ! A give elements xi = pr2(ξi) 2 O(X × A ). A r point α = fxg × A for fixed x 2 X is then uniquely determined by the values xi(α). ∗ r So f is uniquely determined by specifying f (xi) 2 X × A . ∼ r Defining 'i as the composite X ! X × feig ,! X × A , and setting aij = ∗ ∗ 'i (f (xj)) 2 O(X), we have ∗ X f (xj) = aijxi; (1.2) r r since fx : fxg × A ! fxg × A must be linear, and (aij) is an r × r invertible matrix in O(X) by repeating the process above for the inverse morphism f −1. Conversely, any such matrix defines an automorphism of the trivial bundle by (1.2). Since a vector bundle is locally trivial, we can construct one by gluing together S trivial bundles over an open cover of V . Let V = Uα be an open cover such that −1 ∼ π : E ! V is trivial on each Uα. For each Uα, first fix an isomorphism 'α : π (Uα) ! r −1 Uα × A . Over Uαβ := Uα \ Uβ, then, we have that 'α ◦ 'β gives an automorphism r of the trivial vector bundle Uαβ × A , and is therefore given by an invertible r × r matrix Cαβ in OX (Uαβ) by Lemma 1.1. These matrices satisfy the conditions Cαβ = id and Cαγ = Cαβ ◦ Cβγ on Uαβγ := Uα \ Uβ \ Uγ: (1.3) We see conversely that specifying such matrices with entries in OX (Uαβ) define a r vector bundle provided they satisfy (1.3) by gluing together the varieties Uα × A along these isomorphisms. We now show how different choices of 'α give different matrices. Any other 0 0 r isomorphism 'α is of the form 'α = fα'α where fα is an automorphism of Uα ×A ! 2 Uα. By Lemma 1.1, fα can be expressed as a matrix Bα with entries in OX (Uα) with an inverse matrix of the same form; thus, we have new transition matrices 0 −1 Cαβ = BαCαβBβ . Conversely, any such change of the matrices Cαβ leads to different isomorphisms 'α. Any such change, however, leads to isomorphic vector bundles via −1 0 the morphism of varieties defined by the 'α ◦ 'α. 2 Vector bundles on An We would like to prove the following theorem: Theorem 2.1. Every vector bundle over An is trivial. There are two ways to go about this proof: if k = C we can use the analytic topology on An and E and prove this using topological methods. On the other hand, we can also use purely algebraic methods, and so the lemma will hold for algebraic vector bundles over any field k. We will present the latter. We will not prove the most general case. 2.1 Vector bundles and locally free sheaves Vector bundles over X, in fact, correspond to locally free sheaves of OX -modules of finite rank. Definition. A section of a vector bundle π : E ! X is a morphism σ : X ! E such that π ◦ σ = id on X. Denote Γ(U; E) to be the set of sections of the restriction EjU . Proposition 2.2 ([Ser55, no 4]). There is a one-to-one correspondence between vec- tor bundles over a variety X and locally free OX -modules of finite rank, and between trivial bundles and free OX -modules of finite rank. Proof. We assume X is irreducible, for if it is not, it suffices to show the proposition on each irreducible component. Note this means that both the rank of a vector bundle and the rank of a locally free OX -module are constant. If σ(x) 2 Γ(U; E) is a section on U, and f(x) 2 OX (U), then f(x) · σ(x) is another section in Γ(U; E), using the vector space structure on Ex. Similarly, if 0 0 σ (x) 2 Γ(U; E), then σ(x) + σ (x) is another section. Thus, Γ(U; E) is an OX (U)- module for all U, and S (E): U 7! Γ(U; E) is a presheaf of OX (U)-modules, and is a sheaf since morphisms of varieties are defined locally. Since π : E ! X is locally isomorphic to the trivial bundle U ×Ar, we see that the sheaf S is locally isomorphic r to OX . 3 Conversely, let F be a locally free OX -module of finite rank. Then, there exists ∼ r an open cover fUαg of X such that there are isomorphisms 'α : F jUα !OX jUα . Then, ' ◦ '−1 : O jr !O∼ jr α β X Uαβ X Uαβ is an isomorphism of OX jUαβ -modules, given by a matrix Cαβ = (cij) with cij 2 OX (Uαβ). These obviously satisfy (1.3), and so we get a vector bundle E(F ). Since these two operations S (−) and E(−) clearly define a correspondence be- tween trivial bundles and free OX -modules of finite rank, and since vector bundles (resp. sheaves) can be uniquely glued together using the matrices Cαβ, we see that we have a correspondence between vector bundles and locally free OX -modules. This reduces questions about vector bundles to questions about locally free sheaves of finite rank. 2.2 Serre's correspondence We can reduce this question about locally free sheaves of finite rank further for affine spaces to questions about modules using Serre's correspondence [FAC, Ch. II, x4]. Let X = Spec A be the affine variety of a noetherian k-algebra A, and OX its structure sheaf. Definition (Sheaf associated to a module). Let M be an A-module. A defines a constant sheaf of rings on X denoted by A , and M defines a constant sheaf of A - modules denoted by M . Put A (M) = OX ⊗A M , the tensor product sheaf. We call A (M) the sheaf associated to M. If ': M ! M 0 is an A-homomorphism, then we get a morphism A (') := id ⊗': A (M) ! A (M 0), and so A (−) is a functor. Definition (Module associated to a sheaf). Let F be an OX -module. Define the module associated to F as Γ(F ) := Γ(X; F ); Γ(F ) is then an A-module. If : F ! G is a morphism of OX -modules, we get a morphism Γ( ) by taking global sections. If M is finitely generated, A (M) is a coherent OX -module; conversely, if F o is a coherent OX -module, Γ(F ) is finitely generated [FAC, n s 48, 49]. The two operations A (−) and Γ(−) are then \inverse" to each other: Theorem 2.3 ([FAC, no 49, Thm. 1]). (a) If M is a finitely generated A-module, Γ(A (M)) is canonically isomorphic to M. (b) If F is a coherent OX -module, A (Γ(F )) is canonically isomorphic to F . 4 Using this correspondence, we find that vector bundles have a characterization based on their global sections. Recall that an A-module M is projective if it is the r direct summand of a free A-module, or equivalently if Mp is isomorphic to Ap for some r for all p 2 X [FAC, no 50, Prop.
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