structure sheaf

Let X be an irreduciblePlanetmathPlanetmathPlanetmath algebraic variety over a field k, together with the Zariski topologyMathworldPlanetmath. Fix a point x∈X and let UβŠ‚X be any affine open subset of X containing x. Define

𝔬x:={f/g∈k⁒(U)∣f,g∈k⁒[U],g⁒(x)β‰ 0},

where k⁒[U] is the coordinate ring of U and k⁒(U) is the fraction field of k⁒[U]. The ring 𝔬x is independent of the choice of affine open neighborhood U of x.

The structure sheaf on the varietyMathworldPlanetmathPlanetmath X is the sheaf of rings whose sectionsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath on any open subset UβŠ‚X are given by


and where the restrictionPlanetmathPlanetmath map for VβŠ‚U is the inclusion mapMathworldPlanetmath π’ͺX⁒(U)β†ͺπ’ͺX⁒(V).

There is an equivalence of categories under which an affine varietyMathworldPlanetmath X with its structure sheaf corresponds to the prime spectrum of the coordinate ring k⁒[X]. In fact, the topological embedding Xβ†ͺSpec⁑(k⁒[X]) gives rise to a latticeMathworldPlanetmath–preserving bijection11Those who are fans of topos theory will recognize this map as an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of topos. between the open sets of X and of Spec⁑(k⁒[X]), and the sections of the structure sheaf on X are isomorphic to the sections of the sheaf Spec⁑(k⁒[X]).

Title structure sheaf
Canonical name StructureSheaf
Date of creation 2013-03-22 12:38:20
Last modified on 2013-03-22 12:38:20
Owner djao (24)
Last modified by djao (24)
Numerical id 4
Author djao (24)
Entry type Definition
Classification msc 14A10