### WiSe 21/22 Oberseminar: Derived algebraic cobordism

**Time and place:** Tuesday 14:15-15:45 in M311 and online

Algebraic cobordism is a generalized cohomology theory for algebraic varieties originally introduced by V. Voevodsky, which is in many ways analogous to complex cobordism in topology. In particular, the special role of complex cobordism in chromatic homotopy theory was a key inspiration for Voevodsky's celebrated proof of the Bloch–Kato conjecture.

In this seminar, we aim to study a new “elementary” construction of algebraic cobordism, due to T. Annala, which uses derived algebraic geometry and is well-behaved over fields of positive characteristic (unlike the previous construction of M. Levine and F. Morel, which strongly relied on resolution of singularities). In addition to general properties such as the bivariant functoriality and the relationship to algebraic K-theory and Chow groups, we will prove the *algebraic Spivak theorem* stating that the derived cobordism groups of a perfect field are generated, up to inverting the characteristic, by cobordism classes of smooth varietes.

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

26.10 | Denis Nardin | Derived rings and schemes |

02.11 | Charanya Ravi | Quasi-projective derived schemes |

09.11 | Luca Pol | The cotangent complex, smoothness, and quasi-smoothness |

16.11 | Jeroen Hekking | Derived blow-ups |

23.11 | Marc Hoyois | Bivariant theories |

30.11 | Marc Hoyois | The section, formal group law, and strict normal crossings axioms |

07.12 | Massimo Pippi | Precobordism and precobordism with line bundles |

14.12 | Denis Nardin | The formal group law of precobordism |

21.12 | Niklas Kipp | The universal property of precobordism and cobordism |

11.01 | Pavel Sechin | Chern classes and the splitting principle |

18.01 | Charanya Ravi | The Conner–Floyd and Riemann–Roch theorems |

25.01 | — | |

01.02 | Toni Annala | The algebraic Spivak theorem |