# structure sheaf

Let $X$ be an irreducible^{} algebraic variety over a field $k$, together with the Zariski topology^{}. Fix a point $x\beta \x88\x88X$ and let $U\beta \x8a\x82X$ be any affine open subset of $X$ containing $x$. Define

$${\mathrm{\pi \x9d\x94\neg}}_{x}:=\{f/g\beta \x88\x88k\beta \x81\u2019(U)\beta \x88\pounds f,g\beta \x88\x88k\beta \x81\u2019[U],g\beta \x81\u2019(x)\beta \x890\},$$ |

where $k\beta \x81\u2019[U]$ is the coordinate ring of $U$ and $k\beta \x81\u2019(U)$ is the fraction field of $k\beta \x81\u2019[U]$. The ring ${\mathrm{\pi \x9d\x94\neg}}_{x}$ 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 $U\beta \x8a\x82X$ are given by

$${\mathrm{\pi \x9d\x92\u037a}}_{X}\beta \x81\u2019(U):=\underset{x\beta \x88\x88U}{\beta \x8b\x82}{\mathrm{\pi \x9d\x94\neg}}_{x},$$ |

and where the restriction^{} map for $V\beta \x8a\x82U$ is the inclusion map^{} ${\mathrm{\pi \x9d\x92\u037a}}_{X}\beta \x81\u2019(U)\beta \x86\u037a{\mathrm{\pi \x9d\x92\u037a}}_{X}\beta \x81\u2019(V)$.

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 $k\beta \x81\u2019[X]$. In fact, the topological embedding $X\beta \x86\u037a\mathrm{Spec}\beta \x81\u2018(k\beta \x81\u2019[X])$ gives rise to a lattice^{}βpreserving bijection^{1}^{1}Those who are fans of topos theory will recognize this map as an isomorphism^{} of topos. between the open sets of $X$ and of $\mathrm{Spec}\beta \x81\u2018(k\beta \x81\u2019[X])$, and the sections of the structure sheaf on $X$ are isomorphic to the sections of the sheaf $\mathrm{Spec}\beta \x81\u2018(k\beta \x81\u2019[X])$.

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 |