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.

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Idea

In algebraic geometry, a morphism between schemes is said to be of finite type if it behaves like a map with finite dimensional fibers, and is said to be locally of finite type if each of its fibers can built by gluing together a (potentially very large) family of finite dimensional pieces.

For schemes

A morphism $f \colon X \to Y$ of schemes is locally of finite type if

• for every open cover $\{U_i \to Y\}$ by affine schemes, $U_i \simeq Spec B_i$;

• and every open cover $\{U_{i j_i} \to X\}$ by affine schemes $U_{i j_i} = Spec A_{i j_i}$, fitting into a commuting diagram (this always exists, cf. coverage)

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

  for all $i,j$,

we have that the homomorphism of algebras $B_i \longrightarrow A_{i j}$ formally dual to $U_{i j} \to U_i$ exhibits $A_{i j}$ as a finitely generated algebra over $B_i$.

If for fixed $i$ the $j_i$ range only over a finite set, then the morphism is said to be of finite type.

Related concepts

• morphism of finite presentation

References

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

• Amnon Neeman, section 3.10 of: Algebraic and analytic geometry, London Math. Soc. Lec. Note Series 345, Cambridge University Press (2007) [ISBN:9780521709835]