Dieudonn\'e Theory via Cohomology of Classifying Stacks.

We prove that if $G$ is a finite flat group scheme or a $p$-divisible group, then the second crystalline cohomology of its classifying stack $H^2_{crys}(BG)$ recovers the Dieudonne module of $G$. We also prove mixed characteristic analogues of this result using prismatic cohomology.