## A computation of prismatic Dieudonné module

### Volume 205 / 2022

#### Abstract

We establish a natural isomorphism between two fundamental invariants: the second prismatic cohomology of the projective line $\mathbb {P}^1$ and the prismatic Dieudonné module of the $p$-divisible group $\mu _{p^\infty }$, as defined in the work of Anschütz and Le Bras. We call this the “Chern–Dieudonné isomorphism”. Our construction of this isomorphism is essentially “motivic”, in the sense that it is obtained purely via geometric principles. To achieve this, we use the geometric reconstruction of Dieudonné modules proven by the author and the theory of prismatic Chern classes due to Bhatt–Lurie. As a consequence, we can compute the Dieudonné module of $\mu _{p^\infty }$ over a general quasi-regular semiperfectoid algebra $S$ (and therefore the associated prismatic Dieudonné crystal), which was left open in the work of Anschütz and Le Bras.