StructureSheaf 2002-05-14 22:00:29 2002-05-14 22:00:29 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them \usepackage{euscript} % define commands here \renewcommand{\o}{\mathfrak{o}} \renewcommand{\O}{\mathcal{O}} 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 {\em 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\footnote{Those 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 $\operatorname{Spec}(k[X])$.