Krull, Gelfand-Kirillov, Filter, Faithful and Schur dimensionsKrull, Gelfand-Kirillov, Filter, Faithful and Schur dimensions
Faculty of Sciences. Mathematics and Computer Science

Department of Mathematics - Computer Sciences

conferenceObject

2000 2000

Mathematics

Infinite length modules

Conference on Infinite Length Modules, SEP 07-11, 1998, BIELEFELD, GERMANY

(2000), p. 149-166

3-7643-6413-0

000165048800007

E

English (eng)

University of Antwerp

The aim of the paper is to explain relations between 5 mentioned dimensions. In Section 1 a connection is established between Krull, Gelfand-Kirillov and Filter dimension of simple affine algebras, and an analog of Bernstein's inequality is proved for nonzero finitely generated modules over simple affine algebras. These results are applied to compute the Krull dimension of the ring of differential operators on a smooth affine variety. In Section 2 the results of Section 1 are generalized for affine algebras (not necessarily simple). For a new dimension, the faithful dimension, is to be introduced. In Section 3 we consider the Schur dimension and its relations with holonomic modules over the ring of differential operators on a smooth affine variety. In particular, we prove that every holonomic module M which has multiplicity e(M) = 1 over the n'th Weyl algebra A(n) has Schur dimension sd(M) = 1. In Section 4 we give a review of (recent) results on the Krull dimension of skew Laurent extensions and generalized Weyl algebras with left Noetherian coefficients. This paper can be considered as an addition to and continuation of the paper of Tom Lenagan in the present volume.

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