Quasi-separated morphism

We say a topological space is quasi-separated if the intersection of two open quasi-compact subsets of is quasi-compact. We say that a continuous map of topological spaces from to is quasi-separated if the inverse image along of every open quasi-separated subset of is quasi-separated. Then a scheme (resp., a morphism of schemes) is quasi-separated in the scheme-theoretic sense if and only if it is quasi-separated in the topological sense, see . ==Examples==

• If is a locally Noetherian scheme then any morphism from to any scheme is quasi-separated, and in particular is a quasi-separated scheme. • Any separated scheme or morphism is quasi-separated. • The line with two origins over a field is quasi-separated over the field but not separated. • If is an "infinite dimensional vector space with two origins" over a field then the morphism from to spec is not quasi-separated. More precisely consists of two copies of Spec glued together by identifying the nonzero points in each copy. • The quotient of an algebraic space by an infinite discrete group acting freely is often not quasi-separated. For example, if is a field of characteristic then the quotient of the affine line by the group of integers is an algebraic space that is not quasi-separated. This algebraic space is also an example of a group object in the category of algebraic spaces that is not a scheme; quasi-separated algebraic spaces that are group objects are always group schemes. There are similar examples given by taking the quotient of the group scheme by an infinite subgroup, or the quotient of the complex numbers by a lattice. ==References==