Exercise 5.18(d), chapter 2. Hartshorne

Question

In this exercise, suppose we start with a locally free sheaf $ \mathscr E $ of rank n over a scheme $\mathbf Y $, then corresponding to that we can associate $\mathbf V(\mathscr E^ \lor) $ and when we come back we get its sheaf of sections which is isomorphic to $ \mathscr {E^ \lor}^\lor$ $ \cong \mathscr E $ ( by exercise 5.18(c) and exercise 5.1(a)). So one way is okay. Now suppose we start with a vector bundle $ \mathbf X $ over $ \mathbf Y $ of rank n. Then its sheaf of sections will be a locally free sheaf of rank n, say $ \mathscr E $ and the corresponding vector bundle is $\mathbf V(\mathscr E^ \lor) $. Now I am not able to show that $ \mathbf X \cong \mathbf V(\mathscr E^ \lor) $. Can anyone please help me out. I can get isomorphisms locally, but am not able to see why they glue.

Answer 1

Below is an amateur's attempt to prove that $X$ and $\mathbf{V}(\mathscr{E}^\vee)$ are isomorphic. Please take it with a grain of salt, as I'm not an algebraic geometer.

Notation: Given a morphism of affine schemes $\phi : U \to W$, I will denote the induced homomorphism between their 'coordinate rings' by $\phi^\natural : \mathscr{O}_W(W) \to \mathscr{O}_U(U)$. If $U$ is an open subset of $W$ and $f \in \mathscr{O}_W(W)$, I will denote the element $\left.f\right|_{U} \in \mathscr{O}_W(U)$ again simply by $f$, for the sake of better legibility of the formulae. Instead of $\mathbf{X}$ and $\mathbf{Y}$, I will write $X$ and $Y$, respectively.

Let $f: X \to Y$ and $g : \mathbf{V}(\mathscr{E}^\vee) \to Y$ be the canonical projections, and let $\{U_i\}_{i \in I}$ be an affine open covering of $Y$ together a family of trivialisations $\psi_i : f^{-1}(U_i) \to \mathbb{A}^n_{U_i} = \operatorname{Spec}\mathscr{O}_Y(U_i)[x_1,\ldots,x_n]$ for $f:X\to Y$. Given $i \in I$, consider the 'fibre-wise linear' functions $\xi_k^{(i)} := \psi_i^\natural(x_k) \in \mathscr{O}_X(f^{-1}U_i)$, and define homomorphisms of $\mathscr{O}_{U_i}$-modules $\Xi_k^{(i)} : \left.\mathscr{E}\right|_{U_i} \to \mathscr{O}_{U_i}$ via the rule $\Xi_k^{(i)}(\sigma) := \sigma^\natural(\xi_k^{(i)})$ for all $\sigma \in \mathscr{E}(V)$ with $V$ an open subset of $U_i$, and $k = 1,\ldots,n$. Note that the maps $\Xi_1^{(i)},\ldots, \Xi_n^{(i)}$ together define an isomorphism $\Xi^{(i)} : \left.\mathscr{E}\right|_{U_i} \to \left(\mathscr{O}_{U_i}\right)^{\oplus n}$ (see the hint to part (b) of Exercise 5.18). Thus, we shall regard $\Xi_1^{(i)},\ldots, \Xi_n^{(i)}$ as a basis of $\mathscr{E}^\vee(U_i)$ over $\mathscr{O}_Y(U_i)$. We also have an isomorphism $\phi_i : g^{-1}(U_i) \to \mathbb{A}^n_{U_i}$ for which $\phi_i^\natural : \mathscr{O}_Y(U_i)[x_1,\ldots,x_n] \to \mathbf{S}(\mathscr{E}^\vee(U_i))$ is determined by the requirement $\phi_i^\natural(x_k^{(i)}) = \Xi_k^{(i)}$ for $k = 1,\ldots,n$. This enables us to define a family of isomorphisms $\tau_i : f^{-1}(U_i) \to g^{-1}(U_i)$ by putting $\tau_i := \phi_i^{-1} \circ \psi_i$ for $i \in I$. With this definition, we have $$\tau_i^\natural(\Xi_k^{(i)}) = \xi_k^{(i)} \quad (k=1,\ldots,n).$$ Now, suppose that $V$ is an affine open subset of $U_i \cap U_j$ for some $i,j \in I$, and let $a = (a_{k\ell})_{k,\ell = 1}^n$ be a matrix in $\operatorname{GL}_n(\mathscr{O}_Y(V))$ such that the isomorphism $\psi_{ij} = \psi_i\psi_j^{-1} : \mathbb{A}^n_V \to \mathbb{A}^n_V$, where as before $\mathbb{A}^n_V = \operatorname{Spec}\mathscr{O}_Y(V)[x_1,\ldots,x_n]$, is given by the rule $$ \psi_{ij}^\natural(x_k) = \sum_{\ell=1}^n a_{k\ell} x_\ell \quad (k=1,\ldots,n).$$ Since $\psi_i^\natural$ and $\psi_j^\natural$ are isomorphisms of $\mathscr{O}_Y(V)$-algebras, we have $$ \xi_k^{(i)} = \psi_i^\natural(x_k) = \psi_j^\natural(\psi_{ij}^\natural(x_k)) = \sum a_{k\ell} \psi_j^\natural(x_\ell) = \sum a_{k\ell} \xi_\ell^{(j)}.$$ Therefore, since $\sigma^\natural : \mathscr{O}_X(f^{-1}W) \to \mathscr{O}_Y(W)$ is also a homomorphism of $\mathscr{O}_Y(W)$-algebras for all open $W \subset V$ and $\sigma \in \mathscr{E}(W)$, we have $$ \Xi_k^{(i)}(\sigma) = \sigma^\natural(\xi_k^{(i)}) = \sum a_{k\ell}\sigma^\natural(\xi_\ell^{(j)}) = \sum a_{k\ell} \Xi_\ell^{(j)}(\sigma)$$ for all such sections $\sigma$, whence we deduce that $\Xi_k^{(i)} = \sum a_{k\ell} \Xi_\ell^{(j)}$ as sections of $\mathscr{E}^\vee(V)$. Now, we are in a position to show that $\tau_j^\natural(\Xi_k^{(i)}) = \tau_i^\natural(\Xi_k^{(i)})$ for all $k = 1,\ldots,n$. Indeed, we have $$ \tau_j^\natural(\Xi_k^{(i)}) = \sum a_{k\ell} \tau_j^\natural(\Xi_\ell^{(j)}) = \sum a_{k\ell} \xi_\ell^{(j)} = \xi_k^{(i)} = \tau_i^\natural(\Xi_k^{(i)}).$$ Since $\Xi_1^{(i)},\ldots,\Xi_n^{(i)}$ generate the polynomial algebra $\mathbf{S}(\mathscr{E}^\vee(V))$ over $\mathscr{O}_Y(V)$, we deduce that the isomorphisms $\mathbf{S}(\mathscr{E}^\vee(V)) \to \mathscr{O}_X(f^{-1}V)$ induced by $\tau_i^\natural$ and $\tau_j^\natural$ are the same, and hence, that $\tau_i$ and $\tau_j$ agree on $V$. Covering $U_i \cap U_j$ with affine open sets such as $V$, we conclude that $\tau_i = \tau_j$ on $U_i \cap U_j$. Therefore, the maps $\tau_i$ glue together to give a global isomorphism $\tau : X \to \mathbf{V}(\mathscr{E}^\vee)$ of schemes over $Y$. As the argument/construction above shows, $f: X\to Y$ and $g: \mathbf{V}(\mathscr{E}^\vee)\to Y$ admit a common trivialising covering $\{U_i\}_{i\in I}$ of $Y$, with trivialising maps $\psi_i : f^{-1}(U_i)\to \mathbb{A}^n_{U_i}$ and $\phi_i : g^{-1}(U_i)\to \mathbb{A}^n_{U_i}$ carried into one another by $\tau$. Hence, $\tau$ is an isomorphism of vector bundles over $Y$.