quasiseparerat

Quasiseparerat (quasi-separated in English) is a standard notion in algebraic geometry describing how a scheme X sits over another scheme Y via a morphism f: X → Y. By definition, f is quasiseparerat if the diagonal morphism Δ: X → X ×_Y X is quasi-compact. Equivalently, X can be covered by open affine subsets {U_i} such that the intersections U_i ∩ U_j are quasi-compact for all i, j. When Y is a point, this specializes to X being a quasi-separated scheme.

Quasiseparerat morphisms form a stable class: they are preserved by base change and composition, and every separated morphism is quasiseparerat. If f is of finite type (and hence quasi-compact), it is automatically quasiseparerat. The property is especially important in the theory of sheaves and cohomology, because it ensures good behavior of the fiber product X ×_Y X and of the diagonal morphism, which underpins descent and many constructions in the theory of stacks.

Not every morphism is quasiseparerat; there exist pathologies in which the diagonal fails to be quasi-compact. A classical non-quasiseparerat example is the line with two origins, a standard construction used to illustrate the kind of pathology that breaks the property.

See also: diagonal morphisms, quasi-compact morphisms, separated morphisms, base change, algebraic stacks.