structure sheaf

Let $X$ be an irreducible algebraic variety over a field $k$ , together with the Zariski topology. Fix a point $x \in X$ and let $U \subset X$ be any affine open subset of $X$ containing $x$ . Define $$ \o_x := \{f/g \in k(U) \mid f,g \in k[U],\ g(x) \neq 0\}, $$ where $k[U]$ is the coordinate ring of $U$ and $k(U)$ is the fraction field of $k[U]$ . The ring $\o_x$ is independent of the choice of affine open neighborhood $U$ of $x$ .

The structure sheaf on the variety $X$ is the sheaf of rings whose sections on any open subset $U \subset X$ are given by $$ \O_X(U) := \bigcap_{x \in U} \o_x, $$ and where the restriction map for $V \subset U$ is the inclusion map $\O_X(U) \hookrightarrow \O_X(V)$ .

There is an equivalence of categories under which an affine variety $X$ with its structure sheaf corresponds to the prime spectrum of the coordinate ring $k[X]$ . In fact, the topological embedding $X \hookrightarrow \operatorname{Spec}(k[X])$ gives rise to a lattice-preserving bijection ¹ between the open sets of $X$ and of $\operatorname{Spec}(k[X])$ , and the sections of the structure sheaf on $X$ are isomorphic to the sections of the sheaf $\operatorname{Spec}(k[X])$ .

Footnotes

... bijection¹

Those who are fans of topos theory will recognize this map as an isomorphism of topos.