Vector Bundles. Let \(Y\) be a scheme. \(A\) (geometric) vector bundle of rank \(n\)
over \(Y\) is a scheme \(X\) and a morphism \(f: X \rightarrow Y\), together with
additional data consisting of an open covering \(\left\\{U_{i}\right\\}\) of
\(Y\), and isomorphisms \(\psi_{i}: f^{-1}\left(U_{i}\right) \rightarrow
\mathbf{A}_{U_{i}}^{n}\) such that for any \(i, j,\) and for any open affine
subset \(V=\operatorname{Spec} A \subseteq U_{i} \cap U_{j}\) the automorphism
\(\psi=\psi_{j} \circ \psi_{i}^{-1}\) of \(\mathbf{A}_{V}^{n}=\operatorname{Spec}
A\left[x_{1}, \ldots, x_{n}\right]\) is given by a
linear automorphism \(\theta\) of \(A\left[x_{1}, \ldots, x_{n}\right],\) i.e.,
\(\theta(a)=a\) for any \(a \in A,\) and \(\theta\left(x_{i}\right)=\) \(\sum a_{i j}
x_{j}\) for suitable \(a_{i j} \in A\)
An isomorphism \(g:\left(X,
f,\left\\{U_{i}\right\\},\left\\{\psi_{i}\right\\}\right)
\rightarrow\left(X^{\prime},
f^{\prime},\left\\{U_{i}^{\prime}\right\\},\left\\{\psi_{i}^{\prime}\right\\}\right)\)
of one vector bundle
of rank \(n\) to another one is an isomorphism \(g: X \rightarrow X^{\prime}\) of
the underlying schemes, such that \(f=f^{\prime} \circ g,\) and such that \(X,
f,\) together with the covering of \(Y\) consisting of all the \(U_{i}\) and
\(U_{i}^{\prime},\) and the isomorphisms \(\psi_{i}\) and \(\psi_{i}^{\prime} \circ
g,\) is also a vector bundle structure on \(X\)
(a) Let \(\mathscr{E}\) be a locally free sheaf of rank \(n\) on a scheme \(Y\). Let
\(S(\mathscr{E})\) be the symmetric algebra on \(\mathscr{E},\) and let
\(X=\operatorname{Spec} S(\mathscr{E}),\) with projection morphism \(f: X
\rightarrow Y\) For each open affine subset \(U \subseteq Y\) for which
\(\left.\mathscr{E}\right|_{U}\) is free, choose a basis of \(\mathscr{E}\) and
let \(\psi: f^{-1}(U) \rightarrow \mathbf{A}_{U}^{n}\) be the isomorphism
resulting from the identification of \(S(\mathscr{E}(U))\) with
\(\mathscr{O}(U)\left[x_{1}, \ldots, x_{n}\right] .\) Then \((X,
f,\\{U\\},\\{\psi\\})\) is a vector bundle of rank \(n\) over \(Y\), which (up to
isomorphism) does not depend on the bases of \(\mathscr{E}_{U}\) chosen. We call
it the geometric vector bundle associated to \(\delta,\) and denote it by
\(\mathbf{V}(\mathscr{E})\).
(b) For any morphism \(f: X \rightarrow Y\), a section of \(f\) over an open set
\(U \subseteq Y\) is a morphism \(s: U \rightarrow X\) such that \(f \circ s=\) id
\(_{U} .\) It is clear how to restrict sections to smaller open sets, or how to
glue them together, so we see that the presheaf \(U \mapsto\\{\text { set of
sections of } f \text { over } U\\}\) is a sheaf of sets on \(Y\), which we
denote by \(\mathscr{S}(X / Y) .\) Show that if \(f: X \rightarrow Y\) is a vector
bundle of \(\operatorname{rank} n,\) then the sheaf of sections \(\mathscr{S}(X /
Y)\) has a natural structure of \(\mathscr{O}_{Y}\) -module, which makes it a
locally free \(\mathscr{O}_{Y}\) -module of rank \(n\). [Hint: It is enough to
define the module structure locally, so we can assume \(Y=\operatorname{Spec}
A\) is affine, and \(X=\mathbf{A}_{Y}^{n} .\) Then a section \(s: Y \rightarrow X\)
comes from an \(A\) -algebra homomorphism \(\theta: A\left[x_{1}, \ldots,
x_{n}\right] \rightarrow\) \(A,\) which in turn determines an ordered \(n\) -tuple
\(\left\langle\theta\left(x_{1}\right), \ldots,
\theta\left(x_{n}\right)\right\rangle\) of elements of \(A .\) Use this
correspondence between sections \(s\) and ordered \(n\) -tuples of elements of \(A
\text { to define the module structure. }]\)
(c) Again let \(\delta\) be a locally free sheaf of rank \(n\) on \(Y\), let
\(X=\mathbf{V}(\delta)\), and let \(\mathscr{S}=\) \(\mathscr{S}(X / Y)\) be the
sheaf of sections of \(X\) over \(Y\). Show that \(\mathscr{S} \cong
\mathscr{E}^{\curlyvee},\) as follows. Given a section \(s \in \Gamma\left(V,
\delta^{\curlyvee}\right)\) over any open set \(V\), we think of \(s\) as an
element of \(\operatorname{Hom}\left(\left.\mathscr{E}\right|_{V},
\mathcal{O}_{V}\right) .\) So \(s\) determines an \(\mathscr{O}_{V^{-} \text
{algebra homomorphism }} S\left(\left.\mathscr{E}\right|_{V}\right)
\rightarrow \mathcal{O}_{V}\)
This determines a morphism of spectra \(V=\operatorname{Spec} O_{V} \rightarrow
\operatorname{Spec} S\left(\left.\mathscr{E}\right|_{V}\right)=\)
\(f^{-1}(V),\) which is a section of \(X / Y .\) Show that this construction gives
an isomorphism of \(\mathscr{E}^{\curlyvee}\) to \(\mathscr{S}\)
(d) Summing up, show that we have established a one-to-one correspondence
between isomorphism classes of locally free sheaves of rank \(n\) on \(Y\), and
isomorphism classes of vector bundles of rank \(n\) over \(Y\). Because of this,
we sometimes use the words "locally free sheaf" and "vector bundle"
interchangeably, if no confusion seems likely to result.
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