Dimension, multiplicity, holonomic modules, and an analogue of the inequality of Bernstein for rings of differential operators in prime characteristic

Abstract

Let $K$ be an {\em arbitrary} field of characteristic $p>0$ and $\CD (P_n)$ be the ring of differential operators on a polynomial algebra $P_n$ in $n$ variables. A long anticipated {\em analogue of the inequality of Bernstein} is proved for the ring $\CD (P_n)$. On the way, analogues of the concepts of (Gelfand-Kirillov) {\em dimension, multiplicity, holonomic modules} are found in prime characteristic (giving answers to old questions of finding such analogs).An analogue of the {\em Quillen's Lemma} is proved for simple {\em finitely presented} $\CD (P_n)$-modules. In contrast to the characteristic zero case where the Geland-Kirillov dimension of a nonzero finitely generated $\CD (P_n)$-module $M$ can be {\em any natural} number from the interval $[n,2n]$, in the prime characteristic, the (new) dimension $\Dim (M)$ can be \underline{\em any real} number from the interval $[n,2n]$.