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Krull Dimension of Differential Operator Rings IV—Multiple Derivations
Author(s) -
Goodearl K. R.,
Lenagan T. H.
Publication year - 1983
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-47.2.306
Subject(s) - krull dimension , mathematics , regular local ring , noetherian ring , global dimension , noetherian , dimension (graph theory) , local ring , pure mathematics , prime ideal , differential operator , ring (chemistry) , commutative algebra , prime (order theory) , associated prime , commutative property , discrete mathematics , combinatorics , algebra over a field , chemistry , organic chemistry
This paper is concerned with the Krull dimension (in the sense of Gabriel and Rentschler) of a differential operator ring T = R [θ 1 ,…,θ u ], where R is a commutative noetherian ring with finite Krull dimension, equipped with u commuting derivations. The main theorem states that K.dim( T ) is the maximum of all the values height( M )+ u and height( P )+differential‐dimension( P ), where M ranges over those maximal ideals of R with char(R/M) > 0 and P ranges over those prime ideals of R with char( R / P )=0. As applications, the Krull dimension of the Weyl algebra A u ( S ) is computed for any commutative noetherian ring S with finite Krull dimension, and when T has finite global dimension, gl.dim( T ) is shown to coincide with K.dim( T ). In the case where R is a finitely generated algebra over a field of characteristic zero, the formula for K.dim( T ) is reduced to the maximum of the values height( M )+differential−dimension( M ), where M ranges over just the maximal ideals of R . This formula is also proved for arbitrary commutative noetherian 2‐differential Q‐algebras, and a slightly weaker formula, where M ranges over the prime ideals of R of depth at most 1, is proved for the case when R is a localization of a finitely generated algebra over a field of characteristic zero.