next up previous
Next: -valued points and fibers Up: Topics in Non commutative Previous: Topics in Non commutative

quasi coherent sheaves

DEFINITION 1.1   An $ \mathcal{O}_X$ -module $ \mathcal{F}$ on a scheme $ X$ is quasi coherent if there exists an affine open covering

$\displaystyle \{U_\lambda=\operatorname{Spec}(A_\lambda)\}_{\lambda \in \Lambda}
$

of $ X$ such that for each $ \lambda\in Lambda$ , $ \mathcal{F}\vert _{X_\lambda}$ is isomorphic to a $ \mathcal{O}_{\operatorname{Spec}(A_\lambda)} \otimes_{ A_\lambda} M$ for some $ A$ -module $ M$ .

It is easy to see that

LEMMA 1.2   Let $ f:X\to Y$ be a morphism of schemes. For any quasi coherent sheaf $ \mathcal{G}$ on $ Y$ , $ f^*(\mathcal{G})$ is quasi coherent.



2007-12-11