Morita Invariance of the Filter Dimension and of the Inequality of Bernstein

Abstract

It is proved that the filter dimension 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 \({\cal D} (X)\) of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand–Kirillov dimension \( {\rm GK} (M)\geq n = \frac{{\rm GK} (A)}{2}\) for all nonzero finitely generated A-modules M. In fact, a stronger result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. A direct consequence of this fact is that an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras H c for integral c: \({\rm GK} (M)\geq n =\frac{{\rm GK} (H_c)}{2}\) for all nonzero finitely generated H c -modules M. For these class of algebras, it gives an affirmative answer to a question of Ken Brown about symplectic reflection algebras.