Institut für Mathematik Universität Zürich | ||

Algebra and Topology

course by Lorenzo Mantovani and Christian Dahlhausen in spring term 2020 with exercise class by Emil Jacobsen.

Tuesday, 15.15-17.00, and Thursday, 15.15-17.00, online via BigBlueButton.

Links

- The official page on the math institute's homepage where you find the exercise and homework sheets.
- The course page on Lorenzo Mantovani's website. There you can find a course description and a board diary for the first half of the course.
- The course on BigBlueButton.
- The exercise session on BigBlueButton.

Lecture notes

Here you find lecture notes for the second half of the course: version: 28th May 2020Board diary (for the second half of the course)

Date | Content |

07.04. | Motivation for cohomology, abelian categories, injective objects, sheaves have enough injectives |

21.04. | Existence and properties of right-derived functors |

23.04. | Derived functors via F-acyclic resolutions, injective abelian groups |

28.04. | Projective objects, left-derived functors, enough projective sheaves on finitely generated spaces, sheaf cohomology |

30.04. | Flasque sheaves are acyclic, cohomology on open/closed subspaces, Mayer-Vietoris sequences |

05.05. | Godement resolution, Čech cohomology |

07.05. | Čech cohomology equals sheaf cohomology on paracompact spaces, first cohomology on good covers |

12.05. | an example, first cohomology in terms of torsors |

14.05. | Higher direct images, homotopy invariance of sheaf cohomology |

19.05. | Sheaves in terms of étalé spaces, étalé spaces of locally constant sheaves |

21.05. | complements on (locally) constant sheaves, fundamental groupoid and its monodromy representation |

26.05. | equivalence of the monodromy functor, classification of locally constant sheaves on spheres and on the torus |

28.05. | alternative proof of homotopy invariance of sheaf cohomology for locally constant sheaves, outlook |