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 arbitrary field of characteristic $p>0$ and $\mathcal D (P_n)$ the ring of differential operators on a polynomial algebra $P_n$ in $n$ variables. A long anticipated analogue of the inequality of Bernstein is proved for the ring $\mathcal D (P_n)$. In fact, three different proofs are given of this inequality (two of which are essentially characteristic free): the first one is based on the concept of the filter dimension, the second, on the concept of a set of holonomic subalgebras with multiplicity, and the third works only for finitely presented modules and follows from a description of these modules (obtained in the paper). On the way, analogues of the concepts of (Gelfand-Kirillov) dimension, multiplicity, holonomic modules are found in prime characteristic (giving answers to old questions of how to find such analogs). The idea is very simple to find characteristic free generalizations (and proofs) which in characteristic zero give known results, and in prime characteristic, generalizations. An analogue of Quillen’s Lemma is proved for simple finitely presented $\mathcal D (P_n)$-modules. Moreover, for each such module $L$, $\textrm {End}_{\mathcal D (P_n)}(L)$ is a finite separable field extension of $K$ and $\dim _K(\textrm {End}_{\mathcal D (P_n)}(L))$ is equal to the multiplicity $e(L)$ of $L$. In contrast to the characteristic zero case where the Gelfand-Kirillov dimension of a nonzero finitely generated $\mathcal D (P_n)$-module $M$ can be any natural number from the interval $[n,2n]$, in the prime characteristic, the (new) dimension $\textrm {Dim}(M)$ can be any real number from the interval $[n,2n]$. It is proved that every holonomic module has finite length, but in contrast to the characteristic zero case it is not true that neither a nonzero finitely generated module of dimension $n$ is holonomic nor that a holonomic module is finitely presented. Some of the surprising results are: $(i)$ each simple finitely presented $\mathcal D (P_n)$-module $M$ is holonomic having the multiplicity which is a natural number (in characteristic zero rather the opposite is true, i.e. $\textrm {GK} (M)=2n-1$, as a rule), $(ii)$ the dimension $\textrm {Dim}(M)$ of a nonzero finitely presented $\mathcal D (P_n)$-module $M$ can be any natural number from the interval $[n,2n]$, $(iii)$ the multiplicity $e(M)$ exists for each finitely presented $\mathcal D (P_n)$-module $M$ and $e(M)\in \mathbb {Q}$, the multiplicity $e(M)$ is a natural number if $\textrm {Dim}(M)=n$, and can be an arbitrarily small rational number if $\textrm {Dim}(M)>n$.