nLab morphism of finite type


This article is about homomorphisms between schemes which are locally of finite type. For the notion of locally finite type in dependent type theory, see locally finite type.



A homomorphism between schemes is said to be (locally) of finite type if it behaves like a finite covering space.

For schemes

A morphism f:XYf : X \to Y of schemes is locally of finite type if

  • for every open cover {U iY}\{U_i \to Y\} by affine schemes, U iSpecB iU_i \simeq Spec B_i;

  • and every cover {U ij iX}\{U_{i j_i} \to X\} by affine schemes U ij i=SpecA ij iU_{i j_i} = Spec A_{i j_i}, fitting into a commuting diagram (this always exists, see coverage)

    U ij i U i X f Y \array{ U_{i j_i} &\to& U_i \\ \downarrow && \downarrow \\ X &\stackrel{f}{\to}& Y }

    for all i,ji,j,

we have that the morphism of algebras B iA ijB_i \to A_{i j} formally dual to U ijU iU_{i j} \to U_i exhibits A ijA_{i j} as a finitely generated algebra over B iB_i.

If for fixed ii the j ij_i range only over a finite set, then the morphism is said to be of finite type.


Introductory disucssoon over the complex numbers (with an eye towards GAGA) is in

  • Amnon Neeman, section 3.10 Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345, 2007 (publisher)

Last revised on July 25, 2023 at 16:09:33. See the history of this page for a list of all contributions to it.