TOPICS
Search

Morphism of Finite Type

A morphism of finite type is a morphism f:X->S of schemes such that, for every affine scheme V=Spec(R) open in S, the inverse image f^(-1)(V) has a finite cover by affine schemes U_i=Spec(A_i) for which every A_i is generated by finitely many elements as an R-algebra.

Morphisms of finite type are preserved by composition and base change. Every proper morphism is of finite type.

See also

Affine Scheme, Base Change, Inverse Image, Morphism, Proper Morphism, Scheme

Explore with Wolfram|Alpha

WolframAlpha

More things to try:

References

Grothendieck, A. and Dieudonné, J. "Éléments de géométrie algébrique. I. Le langage des schémas." Publ. Math. IHES 4, 5-228, 1960. https://doi.org/10.1007/BF02684778.The Stacks Project Authors. "Morphisms of Finite Type." §29.15 in The Stacks Project, Tag 01T0, 2026. https://stacks.math.columbia.edu/tag/01T0.

Cite this as:

Weisstein, Eric W. "Morphism of Finite Type." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MorphismofFiniteType.html

Subject classifications