Examples of Finite Morphisms

A morphim $\pi: X \to Y$ is finite if for every affine open $\text{Spec} B$ of $Y$, $\pi^{-1}(\text{Spec}B)$ is the spectrum of a $B$-algebra that is a finitely generated $B$-module.

Notice that finite morphisms are in particular affine.

Claim: Finite-ness is affine local on target. Thus it suffices to check that $\pi^{-1}(U_{\alpha})$ satisfies the above for one affine cover $\{U_{\alpha} \}_{\alpha}$ of $Y$.

In particular, a morphism $\pi: \text{Spec} A \to \text{Spec} B$ is a finite iff the induced homomorphism $B \to A$ gives $A$ the structure of a finitely generated $B$-module.

Claim: Finite morphisms are always closed, and they always have finite fibers. 

Example 1. Finite Field Extenion.

 If $L/K$ is a field extension, then $\text{Spec} L \to \text{Spec} K$ is finite iff $L/K$ is finite.

Example 2. Branched Cover.

Let $\pi: \text{Spec} k[t] \to \text{Spec} k[u]$ be the morphism induced by $u \mapsto p[t]$, where $p(t)$ is a polynomial of degree $n$.

Thus the $k[u]$ algebra structure on $k[t]$ is given by $u \cdot f(t) := p(t) f(t)$.
Therefore $k[t]$ is generated as a $k[u]= k[p(t)]$-module by $1, t, t^2, \ldots, t^{n-1}$ (i.e. $[k[t]: k[p(t)]] = n.$) so $\pi$ is finite.

Indeed, let $M$ be the $k[p(t)]$-submodule of $k[t]$ generated by $1, t, t^2, \ldots, t^{n-1}.$. Then clearly $M$ contains all polynomial in $t$ of degree less than or equal to $n$. Suppose $k[t] \backslash M \neq \emptyset$. Then there is a minimal degree among all the elements in $k[t]$ outside of $M$. Suppose $g$ is such a polynomial with minimal degree.  Dividing $g$ by $p(t)$, we get $g = fp + r(t)$ for some $r$ either equal to $0$ or of degree less than $p$. As $r \in M$, we must have $fp \not \in M, in particular$f$must not be in$M$, contradicting minimality of$g$.

Example 3. Closed Embedding.

The morphism $\text{Spec}(A/I) \to \text{Spec} A$ is finite since $A/I$ is a finitely generated $A$ module (generated by $1 \in A/I$.)

In particular, the embedding $\text{Spec} k \to \text{Spec} k[t]$ is finite. (Here we take $I= (t)$).

Example 4. Normalization. 

Let $\pi: \text{Spec} k[t] \to \text{Spec} k[x,y]/(y^2 - x^2 - x^3)$ induced by
$x \mapsto t^2 - 1$ and $y \mapsto t^3 - t^2$.

This is finite as $k[t]$ is a finite $k[t^2-1, t^3- t^2]$-module generated by $1$ and $t$.

This is an isomorphism between $D(t^2 -1)$ and $D(x)$ (i.e isomorphism away from the node of the target).