Structure sheaf - Knowlpedia

A structure sheaf on a space $X$ is a sheaf of rings $\mathcal O_X$ whose sections are regarded as functions on open subsets of $X$ . A scheme is not merely its topological space: its structure sheaf carries the local algebra that distinguishes it from other schemes with the same points.

For the prime spectrum $X=\operatorname{Spec}A$ , the structure sheaf is characterized on a basic Zariski open set

D(f)=\{\mathfrak p\in\operatorname{Spec}A:f\notin\mathfrak p\}

\mathcal O_X(D(f))=A_f,

the localization obtained by inverting $f$ . At a point $\mathfrak p$ , its stalk is $\mathcal O_{X,\mathfrak p}\cong A_{\mathfrak p}$ , a local ring.

Examples

On $\operatorname{Spec}k[x]$ , the section $1/x$ is regular on $D(x)$ , although it is not a global polynomial.