Appears in collection : 2023 IHES Summer School – Recent Advances in Algebraic K-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.