On the H_0^{A^1} of classifying spaces of algebraic groups

Weinzierl, Simon Maximilian (2024): On the H_0^{A^1} of classifying spaces of algebraic groups. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics

Abstract

The homotopy sheaves in $\A^1$-homotopy theory, which was developed by Morel and Voevodsky, are, similarly to the homotopy groups in topology, notoriously difficult to calculate explicitly. Here it seems that in contrast to the latter the base case of $\pi_0^{\A^1}$ has a special quality, which we can see from the fact that Morel was able to derive $\A^1$-invariance for the higher homotopy sheaves, but not for $\pi_0^{\A^1}$. In special cases, Choudhury and Elemanto, Kulkarni, and Wendt were able to prove $\A^1$-invariance, but counterexamples due to Ayoub show that the general conjecture is not correct. In the present work, we consider an abelian variant of the $\A^1$-homotopy theory, the $\A^1$-derived category, which was likewise introduced by Morel. Using the spectrum of a field as a base scheme, it is already known that the zeroth homology is strictly $\A^1$-invariant and it follows that $\H_0^{\A^1}$ has the quality of a free strictly $\A^1$-invariant functor. In the light of the recently published results of Elmanto, Kulkarni and Wendt on the determination of $\pi_0^{\A^1}(\Bet G)$, for reductive algebraic groups $G$, as sheafified étale cohomology, we calculate the associated zeroth $\A^1$-homology for classifying spaces of some algebraic groups. For this we first develop tools, in particular we extend theorems about unramified sheaves, which were introduced by Morel, and treat among others the cases of (special) orthogonal groups, unitary groups, split groups of type $G_2$, and spin groups of low dimension. The arguments used are based on the well-elaborated theory of cohomological invariants of these groups dissemenated by Garibaldi, Merkurjev and Serre.

• Item Type: Theses (Dissertation, LMU Munich)
• Subjects: 500 Natural sciences and mathematics; 500 Natural sciences and mathematics > 510 Mathematics
• Faculties: Faculty of Mathematics, Computer Science and Statistics
• Language: English
• Date of oral examination: 5. June 2024
• 1. Referee: Morel, Fabien
• Signature of the printed copy: 0001/UMC 30525