Purity in motivic homotopy theory

• Applicants: Professor Dr. Oliver Röndigs; Professor Dr. Markus Spitzweck
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539085450

The aim of the project is to prove purity results within the Morel-Voevodsky motivic stable homotopy category, analogous to Gabber's purity results for etale cohomology. Such purity results are identifications of the operation i^! E with a suspension of the pullback i^* E, for suitable morphisms i of schemes and suitable motivic spectra E. Morphisms of interest are regular closed embeddings of regular schemes. Motivic spectra of relevance are the motivic Eilenberg-MacLane spectrum representing motivic cohomology with integer coefficients, the Thom spectrum representing algebraic cobordism, and the sphere spectrum. This supplies not only answers to a conjecture of Deglise, but also interesting computations of motivic cohomology, algebraic cobordism and stable homotopy groups via localization sequences.

• DFG Programme: Research Grants