Scheme

Remarks

Example: every affine scheme $\operatorname{Spec} A$ is a scheme, with $U = X$ for every point. Example: the projective line $\mathbb{P}^1_k$ over a field $k$ is a scheme: it is covered by two open sets, each isomorphic to $\operatorname{Spec} k[t]$. It is not affine, since its ring of global functions is $k$, while $\operatorname{Spec} k$ is a single point. Non-example: the one-point space with sheaf $\Z$ is a ringed space but not a scheme. Its stalk $\Z$ is not a local ring, so it is not even locally ringed. An open subset $U$ of a scheme, with the restricted sheaf $\mathcal{O}_X|_U$, is again a scheme, called an open subscheme.