2023 IHES Summer School - Recent Advances in Algebraic K-theory

Motivic Stable Homotopy Theory (3/3)

21 juil. 2023, 09:15

1h 15m

Marilyn and James Simons Conference Center (IHES)

Orateur

Prof. Ryomei Iwasa (Université Paris-Saclay)

Description

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-$A^1$-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 $P^1$-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 $A^1$ 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.

Documents de présentation

Vidéo