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

$${𝔬}_x:=\{f/g\in k(U)\mid f,g\in k[U],g(x)\ne 0\},$$

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 $X\hookrightarrow \mathrm{Spec}(k[X])$ gives rise to a lattice–preserving bijection between the open sets of $X$ and of $\mathrm{Spec}(k[X])$, and the sections of the structure sheaf on $X$ are isomorphic to the sections of the sheaf $\mathrm{Spec}(k[X])$.