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finite morphism (Definition)

Affine schemes

Let $ X$ and $ Y$ be affine schemes, so that $ X={\mathrm{Spec}}A$ and $ Y={\mathrm{Spec}} B$. Let $ f\colon X\to Y$ be a morphism, so that it induces a homomorphism of rings $ g\colon B\to A$.

The homomorphism $ g$ makes $ A$ into a $ B$-algebra. If $ A$ is finitely-generated as a $ B$-algebra, then $ f$ is said to be a morphism of finite type.

If $ A$ is in fact finitely generated as a $ B$-module, then $ f$ is said to be a finite morphism.

For example, if $ k$ is a field, the scheme $ \mathbb{A}^n(k)$ has a natural morphism to $ {\mathrm{Spec}}k$ induced by the ring homomorphism $ k \to k[X_1,\ldots,X_n]$. This is a morphism of finite type, but if $ n>0$ then it is not a finite morphism.

On the other hand, if we take the affine scheme $ {\mathrm{Spec}} k[X,Y]/\left<Y^2-X^3-X\right>$, it has a natural morphism to $ \mathbb{A}^1$ given by the ring homomorphism $ k[X]\to k[X,Y]/\left<Y^2-X^3-X\right>$. Then this morphism is a finite morphism. As a morphism of schemes, we see that every fiber is finite.

General schemes

Now, let $ X$ and $ Y$ be arbitrary schemes, and let $ f \colon X\to Y$ be a morphism. We say that $ f$ is of finite type if there exist an open cover of $ Y$ by affine schemes $ \{U_i\}$ and a finite open cover of each $ U_i$ by affine schemes $ \{V_{ij}\}$ such that $ f\vert _{V_{ij}}$ is a morphism of finite type for every $ i$ and $ j$. We say that $ f$ is finite if there exists an open cover of $ Y$ by affine schemes $ \{U_i\}$ such that each inverse image, $ V_i=f^{-1}(U_i)$ is itself affine, and such that $ f\vert _{V_i}$ is a finite morphism of affine schemes.

Example.

Let $ X=\mathbb{P}^1(k)$ and $ Y={\mathrm{Spec}}k$. We cover $ X$ by two copies of $ \mathbb{A}^1$ and consider the natural morphisms from each of these copies to $ {\mathrm{Spec}}k$. Both of these affine morphisms are of finite type, but are not finite. The covering morphisms patch together to give a morphism from $ \mathbb{P}^1$ to $ {\mathrm{Spec}}k$. The overall morphism is of finite type, but again is not finite.

References.

D. Eisenbud and J. Harris, The Geometry of Schemes, Springer.



"finite morphism" is owned by rmilson. [ full author list (3) | owner history (2) ]
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Also defines:  affine morphism, finite type
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Cross-references: geometry, covering, cover, inverse image, open cover, fiber, morphism of schemes, ring homomorphism, induced, scheme, field, finite, finitely generated, rings, homomorphism, induces, morphism, affine schemes
There are 11 references to this entry.

This is version 6 of finite morphism, born on 2002-07-24, modified 2006-06-08.
Object id is 3199, canonical name is FiniteMorphism.
Accessed 6755 times total.

Classification:
AMS MSC: 14-00 (Algebraic geometry :: General reference works )
 14A10 (Algebraic geometry :: Foundations :: Varieties and morphisms)
 14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms)
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