$\mathbb{A}^1$-invariance in algebraic geometry

Part I: $\mathbb{A}^1$-invariance (Winter 2025–2026)

Our main goal in the first part of this seminar is to understand $\mathbb{A}^1$-homotopy of schemes and to compare the situation with that of topology. To this end, we will prove the analogues of two well-known results in topology:

  1. If $X$ is a smooth affine scheme, then there is a bijection between rank $n$ vector bundles over $X$ and $\mathbb{A}^1$-homotopy classes of maps $X\to \mathrm{Gr}_n$.
  2. Vector bundles over $R[x_1,\dots,x_n]$ are in bijection with vector bundles over $R$, for $R$ essentially finite type regular $k$-algebra.

One may view the first result as analogue of the universal property of Grassmannian in topology. Whereas, the second result is the analogue of triviality of vector bundles over contractible spaces (like $\mathbb{R}^n$). Both these facts tells us that there is an inherent relation of homotopy for schemes. Further evidence for this also appears from various other cohomology theories for schemes, like the $\mathbb{A}^1$-invariance of $K$-theory, etale cohomology and Chow rings. We will explore these topics, together with other notions like degree map around the end.

Talks

  1. Overview, Jesse Kass. 8th Jan.
  2. Patching & Zariski descent, Yuxiang Zhao. 16th Jan. Notes. Some examples (Jesse).
  3. Serre's splitting theorem, Yuxiang Zhao. 23rd Jan. Notes
  4. Extending bundles & Quillen-Suslin theorem, Animesh Renanse. 30th Jan. Notes
  5. Grothendieck topology and the Nisnevich site, Thomas Arnstein. 6th Feb. Notes
  6. Functor of points and algebraic vector bundles, Thomas Arnstein. 13th Feb. Notes
  7. Smooth and étale morphisms, Animesh Renanse. 20th Feb. Notes
  8. Bass-Quillen conjecture, Thomas Arnstein. 27th Feb. Notes
  9. Homotopy classification of algebraic vector bundles, Animesh Renanse. 6th March. Notes
  10. Degree map in $\mathbb{A}^1$-homotopy theory, Jesse Kass. 13th March.

Part II: Motivic homotopy theory (Fall 2026–2027)

In the second part we get into motivic homotopy theory proper, following Gallauer's notes [3]: the construction of the $\infty$-category of motivic spectra, bigraded spheres and the $\mathbb{A}^1$-degree, and cohomology theories such as algebraic $K$-theory and motivic cohomology.

Talks

  1. Topological and categorical preliminaries, Thomas Arnstein. 8th Oct. Notes
    Abstract

    We give the necessary definitions from classical homotopy theory in order to set up our construction of the motivic homotopy category - in particular we review Kan complexes, presheaf ($\infty$)-categories and spectra. The goal is to motivate what $\mathbb{A}^1$-homotopy invariance and a cohomology theory of schemes should be.

References

  1. Asok, A. (2019), Algebraic geometry from an A^1-homotopic viewpoint, Course notes, pdf.
  2. Asok, A., Hoyois, M., & Wendt, M. (2017). Affine representability results in A 1-homotopy theory, I: Vector bundles. pdf.
  3. Gallauer, M. (2026), Stable $\mathbb{A}^1$-motivic homotopy theory, Lecture notes, Motives in Montpellier winter school, pdf.