# Lecture plan and exercises

This the material covered in the course. It follows mostly the structure of the notes by Steffen Oppermann.

Exercises will be posted here every week. You will be expected to have worked on them **before** the next exercise session, which will be held in class during the last hour of the Monday lecture.

Week | Monday | Friday | Exercises |
---|---|---|---|

34 | Introduction | Categories: definition, functors | |

35 | No Lecture | Categories: natural transformations, equivalences of categories | |

36 | Categories: equivalence of categories | Categories: adjoint functors, limits | Exercise session 1 |

37 | Categories: limits and adjoint functors | Additive and abelian categories: definitions, kernels and cokernels | Exercise session 2 |

38 | Additive and abelian categories: exact sequences | Additive and abelian categories: pullbacks and pushouts | Exercise session 3 |

39 | Additive and abelian categories: diagram lemmas | Hom and tensor products: definitions, projectives and injectives | |

40 | Hom and tensor products: hom-tensor adjunction | Morita theory: progenerators | Exercise session 4 |

41 | Morita theory: Eilenberg-Watts theorem, Morita theorems | Complexes and homology: definitions, long exact sequence | Exercise session 5 |

42 | Complexes and homology: homotopy | Complexes and homology: cones and quasi-isomorphisms | Exercise session 6 |

43 | Complexes and homology: projective and injective resolutions | Derived functors: definitions | Exercise session 7 |

44 | Derived functors: first properties | Derived functors: syzygies and dimension shift, Yoneda extensions | Exercise session 8 |

45 | Derived functors: balancing Tor and Ext | Triangulated categories: definitions | Exercise session 9 |

46 | Triangulated categories: homotopy categories | Triangulated categories: derived categories, localisation | |

47 | Derived categories: Hom(A,B[n]) | Derived categories: derived functors | Exercise session 10 |

48 | Extra Lecture: Big exercise session | No Lecture | Exercise session 11 |