Classical Affine Varieties: Coordinate Rings, Morphisms, and Rational Maps — Examples

The polynomial ring k[t] is a domain: for two nonzero polynomials the product of their nonzero leading coefficients is the nonzero leading coefficient of their product. Thus its zero ideal is radical. F1 gives $I({\mathbf{A}}^1)=\sqrt{(0)}=(0)$, so F2 gives $k[X]=k[t]\mathrm{/}(0)=k[t]$. F4 gives $k(X)=\mathrm{Frac}(k[t])=k(t)$.

Evaluation at $a\in k$ has kernel $(t-a)$: for $p(t)={\sum }_{j=0}^r{c}_j{t}^j$, the identity $p(t)-p(a)=(t-a){\sum }_{j=1}^r{c}_j{\sum }_{i=0}^{j-1}{t}^{j-1-i}{a}^i$ proves divisibility when $p(a)=0$, and the converse follows by evaluation. F3 therefore identifies the local ring with fractions $p\mathrm{/}q$ where $q(a)\mathrm{\ne }0$. Its maximal ideal consists of fractions with $p(a)=0$ and its residue map is $p\mathrm{/}q\mapsto p(a)\mathrm{/}q(a)$. Constants make this map onto k.

For f=t, $D(t)=k\setminus \{0\}$ contains 1. F5 and F6 give its coordinate ring as $k[t{]}_t=\{p(t)\mathrm{/}{t}^r:r\ge 0\}=k[t,t^{-1}]$. The graph realization is $(t,z)$ with $tz=1$, the inverse of projection being $t\mapsto (t,t^{-1})$. F8 identifies its fraction field with k(t). For f=0 the open is empty and its function algebra zero; for f=1 the ring is k[t].

The polynomial map $\varphi (t)=t^2$ is the morphism associated by F7 to $k[u]\to k[t]$, $u\mapsto t^2$. Explicitly $({\sum }_j{c}_j{u}^j{)}^{\ast }={\sum }_j{c}_j{t}^{2j}$; in particular $u^{\ast }=t^2$, $(u-a{)}^{\ast }=t^2-a$, $0^{\ast }=0$ and $1^{\ast }=1$. No division by 2 occurs, so this verification includes characteristic 2.