Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic

Details Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic

2008-04-07

arxiv.org/abs/0804.1091v2

Mathematics

Rings and Algebras

24 pages

Scientific paper

Let $K$ be a field of characteristic $p>0$. It is proved that each automorphism $\s \in \Aut_K(\CDPn)$ of the ring $\CDPn$ of differential operators on a polynomial algebra $P_n= K[x_1, ..., x_n]$ is {\em uniquely} determined by the elements $\s (x_1), ... ,\s (x_n)$, and the set $\Frob (\CDPn)$ of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of $\CDPn$, like $P_n$, is equal to $\Aut_K(\CDPn) \cdot \CF$ where $\CF$ is the set of all the extensions of the Frobenius from $P_n$ to $\CDPn$ that leave invariant the subalgebra of scalar differential operators. The set $\CF$ is found explicitly, it is large (a typical extension depends on {\em countably} many independent parameters).

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