Show that any non-constant morphism \(\AA^1\to \AA^1\) or \(\AA^1\smz\to \AA^1\smz\) is finite.
Solution
Let \(k\) be the algebraically closed ground field. A morphism of affine varieties is finite when the coordinate ring of the source is a finite module over the coordinate ring of the target via the comorphism.
1. Every non-constant morphism\[ f:\AA^1\longrightarrow\AA^1 \]is finite.
Choose coordinates \(t\) on the target and \(x\) on the source. The comorphism is \[ f^*:k[t]\longrightarrow k[x], \qquad t\longmapsto p(x) \] for some polynomial \[ p(x)=a_dx^d+a_{d-1}x^{d-1}+\cdots+a_0. \] Non-constancy means \(d\geq1\) and \(a_d\ne0\).
In the \(k[t]\)-algebra \(k[x]\), the element \(x\) satisfies the monic equation \[ T^d +\frac{a_{d-1}}{a_d}T^{d-1} +\cdots +\frac{a_1}{a_d}T +\frac{a_0-t}{a_d} =0, \] because substituting \(T=x\) and \(t=p(x)\) gives zero. Thus \(x\) is integral over the image \(k[p(x)]\).
Since \[ k[x]=k[p(x)][x], \] the ring \(k[x]\) is generated as a \(k[p(x)]\)-module by \[ 1,x,\ldots,x^{d-1}. \] Hence \(f\) is finite.
2. The units of the Laurent polynomial ring are exactly\[ k[x,x^{-1}]^\times = \{c x^n:c\in k^\times,\ n\in\ZZ\}. \]
For a nonzero Laurent polynomial \[ u=\sum_{i=r}^s a_ix^i, \qquad a_r,a_s\ne0, \] define its width by \[ w(u)=s-r. \] If \(u,v\ne0\), the lowest and highest nonzero terms of \(uv\) are obtained by multiplying the corresponding lowest and highest terms, so \[ w(uv)=w(u)+w(v). \]
If \(u\) is a unit, choose \(v\) with \(uv=1\). Then \[ 0=w(1)=w(u)+w(v), \] and both widths are nonnegative. Hence \(w(u)=0\), so \(u\) has exactly one nonzero term: \[ u=cx^n. \] Conversely every such Laurent monomial is a unit.
3. Every non-constant morphism\[ g:\AA^1\smz\longrightarrow\AA^1\smz \]is finite.
The punctured affine line has coordinate ring \[ k[x,x^{-1}]. \] A morphism to \(\AA^1\smz\) is determined by the image of the target coordinate \(t\), which must be a unit. By step 2, \[ g^*(t)=c x^n \] for some \(c\in k^\times\) and \(n\in\ZZ\).
The morphism is constant exactly when \(n=0\), so non-constancy gives \[ r:=|n|\geq1. \] The image of the target coordinate ring is \[ k[cx^n,c^{-1}x^{-n}] = k[x^r,x^{-r}]. \] Put \[ R=k[x^r,x^{-r}]. \] Every Laurent monomial \(x^m\) can be written uniquely as \[ x^m=(x^r)^q x^i \] with \(q\in\ZZ\) and \(0\leq i<r\). Therefore \[ k[x,x^{-1}] = \bigoplus_{i=0}^{r-1}R x^i. \] Thus \(k[x,x^{-1}]\) is a finite, indeed free, \(R\)-module of rank \(r\). Hence \(g\) is finite.
4. Q.E.D.
Step 1 proves the affine-line case, and step 3 proves the punctured-line case after the unit classification in step 2.