Quasiseparated

Quasi-separated is a property used in algebraic geometry to describe schemes and morphisms between them. For a scheme X, X is quasi-separated if the intersection of any two affine open subschemes is quasi-compact. Equivalently, the diagonal morphism Δ: X → X × X is a quasi-compact morphism. For a morphism of schemes f: X → S, f is called quasi-separated if the diagonal morphism Δ_{X/S}: X → X ×_S X is quasi-compact. This condition ensures that certain base change constructions and fiber products behave well enough for sheaf and cohomology theories to be developed in a robust way.

Equivalent characterizations and relations to other properties include: a scheme is quasi-separated if it can be covered by open affines whose pairwise intersections are quasi-compact; in particular, the diagonal criterion provides a global way to verify quasi-separatedness. A morphism is quasi-separated precisely when its diagonal is quasi-compact, which ties the property to the geometry of fiber products.

Basic properties include: any affine scheme is quasi-separated, since the intersection of two affine opens is again affine and hence quasi-compact; Noetherian schemes are always quasi-separated because every open subset is quasi-compact in a Noetherian space. The property is stable under base change and under composition of morphisms, and the class of quasi-separated morphisms is stable under open immersions and products: if X and Y are quasi-separated over a base, their product is quasi-separated over that base.

Significance and examples: quasi-separatedness is a standard assumption in many foundational treatments of schemes, including descent theory, sheaf theory, and derived categories, because it guarantees that certain natural constructions yield well-behaved topological and algebraic structures. While most routinely used schemes (such as Noetherian or finite-type schemes) are quasi-separated, there exist schemes that are not, illustrating that the condition can be essential in pathological contexts. References to quasi-separatedness appear throughout standard texts and the Stacks Project.