An open embedding is a morphism that factors as

where the first map is an isomorphism and the second is the inclusion of the open set into (as a ringed space). In this situation, we often write .

A morphism of schemes is quasicompact if for every affine open subset , the preimage is quaiscompact.

A scheme is quasicompact as a topological space if its structure morphism is quasicompact.

A morphisms of schemes is quasiseperated if for every open subset , the preimage is a quasiseparated scheme, i.e the intersection of any two quasicompact open sets is quasicompact.

A scheme is quasiseparated if its structure morphism is quasiseparareted.

We call morphisms that are quasicompact and quasiseparated qcqs.

Most morphisms we encounter in nature are qcqs, since TODO

A morphism of schemes is called affine if for every affine open subset , the open subscheme is an affine scheme.

A morphism of schemes is integral if it is affine and for every affine open set with , the ring is an inegral extension of via

A morphism is locally of finite type if for every affine open subset and every affine open subset , the induces morphism (induced by makes a finitely generated -algebra. A morphism of locally finite type is of finite type if it also quasicompact.

A morphism of schemes is finite if it is affine and for every affine open , the open subscheme is the afine scheme of a finite -algebra .

Finite means integral and finite type