Marc Hoyois (Regensburg): Non-A^1-invariant motivic spectra

This talk was part of the workshop "Motivic and non-commutative aspects of enumerative geometry" held in July 2023 at Radboud University, Nijmegen, The Netherlands.

Abstract: Recently, there have been several attempts at developing extensions of motivic homotopy theory that include non-A^1-invariant phenomena, such as the algebraic K-theory of singular schemes or de Rham cohomology in positive characteristic. These are usually based on extensions of the category of schemes itself, such as schemes with modulus or log structures. In joint work with Toni Annala and Ryomei Iwasa, we consider a very naive extension of stable motivic homotopy theory, in which we simply remove the A^1-invariance axiom. We show that a surprising number of basic results in A^1-homotopy theory can be proved in this context, using the invertibility of P^1 in a clever way. We construct in particular a non-A^1-invariant refinement of algebraic cobordism, which is related to algebraic K-theory by a Conner-Floyd isomorphism.