Open subscheme/Affine and extended ideal/Finitely generated/Fact/Proof

Proof

We only give a sketch. (1). There always exists a natural scheme morphism

and is affine if and only if this morphism is an isomorphism. It is always an open embedding (because it is an isomorphism on the , ), and the image is . This is everything if and only if the extended ideal is the unit ideal.

(2). We write and consider the natural morphism

corresponding to the ring inclusion . This morphism is again an open embedding and its image is everything.