Spectrum of a commutative ring ($\mathrm{Spec}A$)

The spectrum $\mathrm{Spec}A$ of a commutative ring $A$ is the set of its prime ideals, with closed sets $V(I)=\{\mathfrak{p}:I\subseteq \mathfrak{p}\}$ and a structure sheaf whose sections locally look like fractions.

An affine scheme is a scheme isomorphic to $\mathrm{Spec}A$ for some commutative ring $A$. Homomorphisms $A\to B$ correspond contravariantly to morphisms $\mathrm{Spec}B\to \mathrm{Spec}A$.

For $f\in A$, the principal open subset $D(f)=\{\mathfrak{p}:f\notin \mathfrak{p}\}$ of $\mathrm{Spec}A$ is the affine scheme $\mathrm{Spec}{A}_f$.

The affine plane over a field $k$ is the affine scheme $\mathrm{Spec}k[x,y]$.

The punctured affine plane is obtained by deleting the closed point $(x,y)$ from $\mathrm{Spec}k[x,y]$. Its regular functions still form $k[x,y]$, whilewhich has basis represented by $x^{-a}{y}^{-b}$ for $a,b\ge 1$.