PlanetMath: structure sheaf

(more info)

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structure sheaf

(Definition)

Let be an irreducible algebraic variety over a field , together with the Zariski topology. Fix a point $x \in X$ and let $U \subset X$ be any affine open subset of containing . Define

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

where

is the coordinate ring of

and

is the fraction field of

. The ring $\o _x$ is independent of the choice of affine open neighborhood

The structure sheaf on the variety is the sheaf of rings whose sections on any open subset $U \subset X$ are given by

$\displaystyle \O _X(U) := \bigcap_{x \in U} \o _x,$

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