locally free sheaves of finite rank

DEFINITION 3.1   Let $(X,\mathcal O_X)$ be a ringed space. An $\mathcal O_X$ -module $\mathcal{F}$ on $X$ is said to be

1. free if it is isomorphic to a direct sum of $\mathcal O_X$ .
2. locally free if there exists an open covering $\{U_\lambda\}$ of $X$ such that $\mathcal{F}\vert _{U_\lambda}$ is free for all $\lambda$ .

DEFINITION 3.2   Let $(X,\mathcal O_X)$ be a ringed space. Let $\mathcal{F}$ be a locally free sheaf of rank $r$ on $X$ . By definition, there exists an open covering $\{U_\lambda\}_{\lambda\in \Lambda}$ of $X$ such that $\mathcal{F}\vert _{U_\lambda}$ is free for all $\lambda\in Lambda$ . In other words, there is an isomorphism

$$\phi_\lambda: \mathcal{F}\cong \mathcal{O}_X^r.$$

Such $\phi_\lambda$ is called a local trivialization of $\mathcal{F}$ .

Given a set of trivializations $\{\phi_\lambda\}_{\lambda\in \Lambda}$ of $\mathcal{F}$ , We notice that for any $\lambda,\mu \in \Lambda$ there exists a $\operatorname{GL}_r$ -valued function

$$g:U_{\lambda\mu} \to \operatorname{GL}_r$$

such that for any section $s \in \mathcal{F}(U_{\lambda\mu})$ , we have

$$\phi_\lambda(s)=g_{\lambda\mu}\phi_\mu(s)$$

We call $\{g_{\lambda \mu}\}$ the transition functions.

LEMMA 3.3   The transition functions as in Definition above satisfy the following cocycle conditions.

1. $g_{\lambda\lambda}=\operatorname{id}$ .

2. $$g_{\lambda \mu} g_{\mu \nu} =g_{\lambda \nu}.$$