Group schemes with strict $\mathcal{O}$-action

Let $\mathcal{O}$ denote the ring of integers in a $p$-adic local field. Recall that $\mathcal{O}$-modules are formal groups with an $\mathcal{O}$-action such that the induced action on the Lie algebra is via scalars. In the paper this notion is generalised to finite flat group schemes. It is shown that the usual properties carry over. For example, Cartier duality holds with the multiplicative group replaced by the Lubin–Tate group. We also show that liftings over $\mathcal{O}$-divided powers are controlled by Dieudonné modules or, better, by complexes. For these facts new proofs have to be invented, because the classical recipe of embedding into abelian varieties is not available.