Motives and rigidity

This talk is about the theory of motives and the expected rigidity phenomena for the p-primary étale part. When p is nilpotent on the base, Voevodsky’s theory fails to capture the picture, because its p-primary étale part is zero. Annala, Hoyois, and I have proposed a natural extension of Voevodsky’s theory, “non-A^1-invariant motives”, which addresses this issue. I’ll explain its foundations, then I’ll discuss my ongoing project with Wolf, which aims to establish a comparison with prismatic theory.