affine morphisms

DEFINITION 8.1   Let $X$ be a scheme. Let $\mathcal A$ be a quasi coherent sheaf with an $\mathcal{O}_X$ -bilinear multiplication so that it is a sheaf of (unital associative) algebras. Then we may construct a scheme $\operatorname{Spec}(\mathcal A)$ over $X$ . The morphism $\operatorname{Spec}(\mathcal A)\to X$ is called an explicit affine morphism. A morphism which is isomorphic to an explicit affine morphism so defined is called an affine morphism.