# Category: Morphisms of schemes

Smooth morphism
In algebraic geometry, a morphism between schemes is said to be smooth if * (i) it is locally of finite presentation * (ii) it is flat, and * (iii) for every geometric point the fiber is regular. (
Quasi-compact morphism
In algebraic geometry, a morphism between schemes is said to be quasi-compact if Y can be covered by open affine subschemes such that the pre-images are quasi-compact (as topological space). If f is q
Universal homeomorphism
In algebraic geometry, a universal homeomorphism is a morphism of schemes such that, for each morphism , the base change is a homeomorphism of topological spaces. A morphism of schemes is a universal
Rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coe
Fpqc morphism
In algebraic geometry, there are two slightly different definitions of an fpqc morphism, both variations of faithfully flat morphisms. Sometimes an fpqc morphism means one that is faithfully flat and
Étale morphism
In algebraic geometry, an étale morphism (French: [etal]) is a morphism of schemes that is formally étale and locally of finite presentation. This is an algebraic analogue of the notion of a local iso
Flat morphism
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of ri
Isogeny
In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelian
Proper morphism
In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field k a complete variety. For example
Closed immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes that identifies Z as a closed subset of X such that locally, regular functions on Z can be extended to X. The latter condi