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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

$\displaystyle \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

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

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

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


next up previous
Next: Ultra filter Up: Topics in Non commutative Previous: -valued points and fibers
2007-12-11