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HEIGHT PLUS DIFFERENTIAL DIMENSION IN COMMUTATIVE NOETHERIAN RINGS
Author(s) -
GOODEARL K. R.,
LENAGAN T. H.,
ROBERTS P. C.
Publication year - 1984
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-30.1.15
Subject(s) - krull dimension , mathematics , codimension , noetherian , local ring , dimension (graph theory) , pure mathematics , global dimension , noetherian ring , ring (chemistry) , prime (order theory) , differential (mechanical device) , regular local ring , commutative property , discrete mathematics , combinatorics , algebra over a field , physics , chemistry , organic chemistry , thermodynamics
Differential dimension and differential codimension for prime ideals in a commutative noetherian ring R equipped with commuting derivations δ 1 ,…,δ u may be defined in terms of ranks of Jacobian matrices. Given prime ideals P ⊆ Q in R such that char ( R/Q ) = 0, it is proved that diff.codim.( Q )−diff.codim.( P ) ⩽ ht( Q/P ) and that ht( P ) + diff.dim.( P ) ⩽ ht( Q ) + diff.dim.( Q ). Using the latter inequality, the known formulas for the global dimension and the Krull dimension of the formal differential operator ring R [θ 1 ,…,θ u ] are simplified.