Affine scheme

An affine scheme is a locally ringed space isomorphic to

for some commutative ring $A$. Here $\operatorname{Spec}A$ is the prime spectrum with its Zariski topology, and $\mathcal O_{\operatorname{Spec}A}$ is its structure sheaf. The ring is recovered from global sections:

For example, if $k$ is a field, $\operatorname{Spec}k$ is a one-point affine scheme whose local ring is $k$. The affine line $\operatorname{Spec}k[x]$ is another affine scheme, but it contains more than the familiar $k$-valued points: it also has points corresponding to other prime ideals, including a generic point. General schemes are assembled by gluing affine schemes along open subsets.