The hyperbola xy = 1 is isomorphic to the punctured affine line

Assume the Axiom of Choice and let $k$ be an algebraically closed field. Let $H:=V(xy-1)\subseteq {\mathbf{A}}_k^2\mathrm{.}$ The substitutions $x\mapsto t$, $y\mapsto t^{-1}$ and $t\mapsto x$, $t^{-1}\mapsto y$ define inverse $k$-algebra homomorphisms $k[H]=k[x,y]\mathrm{/}(xy-1)\cong k[t,t^{-1}]\mathrm{.}$ The Laurent polynomial ring is a domain, so A classical affine variety has a domain as its coordinate ring, and conversely shows that $H$ is a classical affine variety.

Give $D_{{\mathbf{A}}^1}(t)$ its principal-open regular-function structure. Projection to the first coordinate gives a morphism $p:H\to D_{{\mathbf{A}}^1}(t),(x,y)\mapsto x,$ because the coordinate function $x$ is regular and can never be $0$ on $H$.

Conversely, $q:D_{{\mathbf{A}}^1}(t)\to H,u\mapsto (u,u^{-1})$ is regular on the principal open $D_{{\mathbf{A}}^1}(t)$, since $u^{-1}$ is regular there by Regular functions on a principal open are the principal localization of the coordinate ring, and its coordinates satisfy $u\cdot u^{-1}=1$, so it lands in $H$. Direct substitution gives $p(q(u))=u\text{and}q(p(x,y))=(x,x^{-1})=(x,y),$ where the last equality uses $xy=1$. Thus $p$ and $q$ are inverse regular maps. They exhibit $H$ as the affine model of the punctured line and realize the displayed coordinate-ring isomorphism.