Ryomei Iwasa - 3/3 Motivic Stable Homotopy Theory

In joint work with Toni Annala and Marc Hoyois, we have developed motivic stable homotopy in broader generality than the theory initiated by Voevodsky, so that non-𝐴1-invariant theories can also be captured. I’ll describe this, bearing in mind its connection to algebraic K-theory and p-adic cohomology such as syntomic cohomology. The course is divided roughly into three parts.
Foundations: The goal of this part is to grasp the notion of 𝑃1-spectrum, which forms the basic framework of motivic stable homotopy theory.
Techniques: The goal of this part is to understand our main technique, P-homotopy invariance, which allows us to do a homotopy theory in algebraic geometry while keeping the affine line 𝐴1 non-contractible.
Applications: In this part, we apply our motivic homotopy theory to algebraic K-theory of arbitrary qcqs schemes, and prove an algebraic analogue of Snaith theorem, which says that K-theory is obtained from the Picard stack by inverting the Bott element.

Ryomei Iwasa (Université Paris-Saclay)