# 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

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

where $k[U]$ is the coordinate ring of $U$ and $k(U)$ is the fraction field of $k[U]$. The ring $\mathfrak{o}_{x}$ 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

 $\mathcal{O}_{X}(U):=\bigcap_{x\in U}\mathfrak{o}_{x},$

and where the restriction map for $V\subset U$ is the inclusion map $\mathcal{O}_{X}(U)\hookrightarrow\mathcal{O}_{X}(V)$.

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 bijection11Those who are fans of topos theory will recognize this map as an isomorphism of topos. 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 $\operatorname{Spec}(k[X])$.

Title structure sheaf StructureSheaf 2013-03-22 12:38:20 2013-03-22 12:38:20 djao (24) djao (24) 4 djao (24) Definition msc 14A10