nLab morphism of 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 8, 2021 at 19:46:30. See the history of this page for a list of all contributions to it.