Morita invariance of the filter dimension and of the inequality of Bernstein

Abstract

It is proved that the filter dimenion is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra $A$ is Morita equivalent to the ring $\CD (X)$ of differential operators on a smooth irreducible affine algebraic variety $X$ of dimension $n\geq 1$ over a field of characteristic zero then the Gelfand-Kirillov dimension $ \GK (M)\geq n = \frac{\GK (A)}{2}$ for all nonzero finitely generated $A$-modules $M$. In fact, a more strong result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. As a direct consequence of this fact an affirmative answer is given to the question/conjecture posed by Ken Brown several years ago of whether an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras $H_c$ for integral $c$: $\GK (M)\geq n =\frac{\GK (H_c)}{2}$ for all nonzero finitely generated $H_c$-modules $M$.