### SoSe 20 Seminar: A¹-invariance in algebraic geometry

**Time and place:** Wednesday 14-16 M102

An A¹-homotopy is an algebraic analogue of a homotopy in topology, where the unit interval [0,1] is replaced by the algebraic affine line A¹. As in topology, it turns out that many interesting invariants of algebraic varieties are A¹-invariant, i.e., they do not see the difference between A¹-homotopic maps. An important example is étale cohomology, which is an algebro-geometric analogue of singular cohomology. The goal of this seminar is to learn the necessary background and study some elementary A¹-homotopical phenomena in algebraic geometry. In particular, we will discuss algebraic vector bundles and symmetric bilinear forms. The main results we will obtain are the following:

1) The A¹-homotopical classification of vector bundles: if X is a smooth affine variety, there is a bijection between isomorphism classes of vector bundles on X and A¹-homotopy classes of maps to the Grassmannian.

2) There is a bijection between the set of pointed A¹-homotopy classes of endomorphisms of the projective line and equivalence classes of nondegenerate symmetric bilinear forms.

Date | Speaker | Topic |
---|---|---|

22.04 | Yuhao Zhang | Affine varieties, A¹-homotopies, and the naive A¹-homotopy category |

29.04 | Marc Hoyois | Projective modules I: basic properties, patching and Zariski descent |

06.05 | — | — |

13.05 | Matthias Uschold | Picard groups and normal rings |

20.05 | Matteo Durante | A¹-invariance of vector bundles I: the Quillen–Suslin theorem |

27.05 | Benjamin Dünzinger | Grothendieck topologies and sheaves |

03.06 | Hongmin Yin | Projective modules II: Serre’s splitting theorem |

10.06 | Benjamin Albersdörfer | Schemes, functors of points, and algebraic vector bundles |

17.06 | Benedikt Aubeck | Smooth and étale morphisms |

24.06 | Nicolai Palm | A¹-invariance of vector bundles II: Lindel’s theorem and the Bass–Quillen conjecture |

01.07 | Zhelun Chen | The A¹-homotopical classification of algebraic vector bundles |

08.07 | Peter Hanukaev | Symmetric bilinear forms and Grothendieck–Witt groups |

15.07 | Niklas Kipp | Symmetric bilinear forms and A¹-homotopy endomorphisms of P¹ |

22.07 | Marc Hoyois | Outlook |