scheme
Idea
According to Nullstellensatz, there is a one-to-one correspondence…
… generalize varieties so that on the left we may take arbitrary commutative rings.
Definition
Let $R$ be a commutative ring, and $X$ a locally ringed space.
- $X$ is an affine scheme if it isomorphic to $\text{Spec}(R)$ (for some $R$).
- $X$ is a scheme if it admits a covering ${U_i}$ such that each $U_i$ is an affine scheme.
Examples
1 (\mathbb{A}^n_k, affine) The maximal ideals of $k[x_1,\dots,x_n]$ are of the form $x_i-\alpha$ for some $\alpha\in\mathbb{A}^n_k$. Hence $\mathbb{A}^n_k$ is the spectrum of the commutative ring $k[x_1,\dots,x_n]$, and is therefore an affine scheme.
In fact one may take this as the definition of affine $n$-space, and for any commutative ring $R$ define affine $n$-space as $\text{Spec}(R[x_1,\dots,x_n])$.