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

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

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

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.