Sheaf cohomology

(the importance of being coherent)

(the importance of being coherent)

Sommersemester 2018, JGU Mainz

The lectures are given on

**Wednesdays, 10 - 12** in **Room 04-432** and on

**Thursdays, 10 - 12** in **Room 04-432**.

Sheaf cohomology — and, in particular coherent sheaf cohomology — is a fundamental tool in modern algebraic geometry. Not only does it allow to prove and even state basic theorems such as the Serre Duality Theorem and the Riemann—Roch Theorem, but it also provides crucial invariants for many classification problems.

Following the third chapter of Hartshorne’s book, we will first introduce sheaf cohomology from the point of view of derived functors. Then we will study the connection to Čech cohomology. As applications, after computing the cohomology of twisted line bundles on the projective space, we will prove the Serre Duality Theorem and we will study flat morphisms of schemes. If time permits, we will also review the Theorem on Formal Functions and the Semicontinuity Theorem, with a view towards their classical corollaries, namely Zariski’s Main Theorem, Stein Factorization and Grauert’s Theorem.

We shall follow Hartshorne's *Algebraic Geometry* [HAG], the third chapter.

Additional references are the books by Ueno, *Algebraic geometry 2 & 3*.

The exam will be an oral exam at the end of the course.

Date | Topics | Lecturer | Homework |
---|---|---|---|

We. 18 April | Section III.1 | Dino | |

Th. 19 April | Section III.2 | Dino | Exercises III.2.1a, 2, 7a, II.1.16 |

We. 25 April | Section III.2 | Dino | |

Th. 26 April | Discussion on the exercises | Dino | |

We. 2 May | Section III.3 | Dino | |

Th. 3 May | Section III.3 | Dino | Exercises III.3.1, 2, 8. |

We. 9 May | Discussion on the exercises | Dino | |

We. 16 May | Section III.4 | Dino | Exercises III.4.1, 3, 7. |

Th. 17 May | Discussion on the exercises | Dino | |

We. 23 May | Section III.5 | Dino | |

Fr. 25 May | Section III.5 | Dino | |

We. 30 May | Section III.6 | Davide | Exercises |

We. 6 June | Section III.7 | Davide | Solutions |

Th. 7 June | Section III.7 | Davide | |

We. 13 June | Section III.8 | Davide | Exercises |

Th. 14 June | Cohen-Macaulay rings | Davide | Solutions |

We. 20 June | Section III.9 | Davide | |

Th. 21 June | Section III.9 | Davide | Exercises |

We. 27 June | Section III.12 | Davide | |

Th. 28 June | Section III.12 | Davide | |

We. 4 July | Section III.11 | Davide | |

Th. 5 July | Discussion on the exercises | Davide | Solutions |