SoSe 25 Seminar: A¹-invariance in algebraic geometry
Time and place: Wednesday 16-18 M103
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 |
|---|---|---|
| 23.04 | Affine varieties, A¹-homotopies, and the naive A¹-homotopy category | |
| 30.04 | Projective modules I: basic properties and Zariski descent | |
| 07.05 | Projective modules II: Serre’s splitting theorem | |
| 14.05 | Picard groups and normal rings | |
| 21.05 | A¹-invariance of vector bundles I: the Quillen–Suslin theorem | |
| 28.05 | Liva Diler | Grothendieck topologies and sheaves |
| 04.06 | Alissa Doggwiler | Schemes, functors of points, and algebraic vector bundles |
| 11.06 | Smooth and étale morphisms | |
| 18.06 | A¹-invariance of vector bundles II: Lindel’s theorem and the Bass–Quillen conjecture | |
| 25.06 | — | |
| 02.07 | The A¹-homotopical classification of algebraic vector bundles | |
| 09.07 | Symmetric bilinear forms and Grothendieck–Witt groups | |
| 16.07 | A¹-homotopy classes of endomorphisms of the projective line | |
| 23.07 | Marc Hoyois | Outlook |