# Serre duality

The most general version of Serre duality states that on certain schemes $X$ of dimension^{} $n$, including all projective varieties over any algebraically closed field $k$, there is a natural perfect^{} pairing (http://planetmath.org/BilinearMap)

$${\mathrm{Ext}}^{i}(\mathcal{F},\omega )\times {H}^{n-i}(X,\mathcal{F})\to k,$$ |

where $\mathcal{F}$ is any coherent sheaf on $X$ and $\omega $ is a sheaf, called the *dualizing sheaf*. Here “perfect” means that the natural map above induces an isomorphism^{}

$${\mathrm{Ext}}^{i}(\mathcal{F},\omega )\cong \mathrm{Hom}({H}^{n-i}(X,\mathcal{F}),k).$$ |

In special cases, this reduces to more approachable forms. If $X$ is nonsingular^{} (or more generally, Cohen-Macaulay), then $\omega $ is simply ${\bigwedge}^{n}\mathrm{\Omega}$, where $\mathrm{\Omega}$ is the sheaf of differentials on $X$.

If $\mathcal{F}$ is locally free, then

$${\mathrm{Ext}}^{i}(\mathcal{F},\omega )\cong {\mathrm{Ext}}^{i}({\mathcal{O}}_{X},{\mathcal{F}}^{*}\otimes \omega )\cong {H}^{i}(X,{\mathcal{F}}^{*}\otimes \omega ),$$ |

so that we obtain the somewhat more familiar looking fact that there is a perfect pairing ${H}^{i}(X,{\mathcal{F}}^{*}\otimes \omega )\times {H}^{n-i}(X,\mathcal{F})\to k$.

While Serre duality is not in a strict sense a generalization^{} of Poincaré duality, they are philosophically similar, and both fit into a larger pattern on duality results.

Title | Serre duality |
---|---|

Canonical name | SerreDuality |

Date of creation | 2013-03-22 13:51:24 |

Last modified on | 2013-03-22 13:51:24 |

Owner | mps (409) |

Last modified by | mps (409) |

Numerical id | 12 |

Author | mps (409) |

Entry type | Definition |

Classification | msc 14F25 |

Related topic | DualityInMathematics |

Defines | dualizing sheaf |