Monogenic bialgebras over finite fields and rings of Witt vectors (Q5946832): Difference between revisions

Let \(k=\mathbb{F}_{p^l}\) be a finite field. The author studies Hopf algebras over \(k\), which are monogenic, i.e. generated as algebras by a single element, and local-local, i.e. both \(H\) and its linear dual \(H^*\) are local rings. The first result gives a complete classification of these Hopf algebras: There is a one-to-one correspondence between monogenic local-local Hopf algebras over \(k\) and the set NEWLINE\[NEWLINE\mathbb{Z}^+\cup\{(n,r,z)\mid n\geq 1,\;r\leq n-1,\;0\leq z\leq p^d-2,\;d=\gcd(l,r+1)\}.NEWLINE\]NEWLINE The proof uses the one-to-one correspondence between finite local-local cocommutative bialgebras and Dieudonné modules. The author then studies lifts of these algebras to characteristic \(0\). He shows that \(H\) lifts to the ring \(W(k)\) of Witt vectors with coefficients in \(k\) if and only if \(H\) corresponds under the above classification to a triple \((n,r,z)\) with \(r+1\) dividing \(n\). Furthermore, if a lift is possible, the author computes the precise number of lifts. The proof uses the correspondence between lifts and finite Honda systems attached to Dieudonné modules and finite submodules of \(W(k)\) [cf. \textit{J.-M. Fontaine}, C. R. Acad. Sci., Paris, Sér. A 280, 1423-1425 (1975; Zbl 0331.14023)].