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
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