Height one Hopf algebras in low ramification. (Q1774349)

Let \(k\) be a perfect field of characteristic \(p>0\). The author studies finite local Abelian \(k\)-Hopf algebras \(H\) with local duals such that the Frobenius map \(x\mapsto x^p\) annihilates the augmentation ideal of \(H\). He first classifies these height one Hopf algebras via their Dieudonné modules. Using the structure theory for finitely generated modules over a skew polynomial ring, he shows that there is a one-to-one correspondence between height one \(k\)-Hopf algebras of rank \(p^n\) and partitions of \(n\). Then, using the theory of finite Honda systems of \textit{B. Conrad} [Compos. Math. 119, No. 3, 239-320 (1999; Zbl 0984.14015)], he shows that if \(R\) is a valuation ring with residue field \(k\), \(W(k)\) is the Witt ring of \(k\), and the ramification index \(e\) of \(R\) over \(W(k)\) satisfies \(1<e\leq p-1\), then every height one \(k\)-Hopf algebra lifts to \(R\), and the author determines all such lifts. If \(e=1\), then no height one \(k\)-Hopf algebra lifts.