I have some questions about locally free sheaves when reading The Rising Sea: Foundation of Algebraic Geometry:
Given a locally free sheaf $\mathscr{F}$ of rank $n$ on a scheme $X$, we can take an open cover $\{U_i\}$ of $X$ such that for each $U_i$, $\phi_i:\mathscr{F}|_{U_i}\cong \mathcal{O}_{U_i}^{\oplus n}$. Then this open cover determines transition functions $T_{i j}\in GL_n(\mathcal{O}(U_i\cap U_j))$, which satisfy the cocycle condition. Moreover, this data determines the locally free sheaf.
My question is:
How to show that transition functions determine the locally free sheaf? Clearly $\mathscr{F}$ is equivalent to the data $\{U_i\}$ and $\phi_{i j}:\mathcal{O}_{U_i\cap U_j}^{\oplus n}\cong \mathcal{O}_{U_i\cap U_j}^{\oplus n}$ satisfying the cocycle condition. And when $U_i\cap U_j $ is affine, $\phi_{i j}:\mathcal{O}_{U_i\cap U_j}^{\oplus n}\cong \mathcal{O}_{U_i\cap U_j}^{\oplus n}$ is equivalent to $\phi_{ij}(U_i\cap U_j)\in GL_n(\mathcal{O}(U_i\cap U_j))$ via quasi-coherence. But I wonder why they are equivalent in general case.
Best Answer
Since $GL_n$ is affine, a morphism $X \to GL_n$ (for any scheme $X$) is the same as a ring map $\Gamma(GL_n, \mathcal{O}_{GL_n}) \to \Gamma(X, \mathcal{O}_X)$. (See Vakil 7.3.G.)
Since $\Gamma(GL_n, \mathcal{O}_{GL_n}) \cong \mathbb{Z}[x_{ij} : 1 \leq i, j \leq n][1/\det]$, this is equivalent to giving an invertible matrix of elements of $\Gamma(X, \mathcal{O}_X)$, which in turn is equivalent to an automorphism of the $\mathcal{O}_X^{\oplus n}$ as an $\mathcal{O}_X$-module.