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Mixed multiplicities for arbitrary ideals and generalized Buchsbaum‐Rim multiplicities
Author(s) -
Callejas-Bedregal R.,
Pérez V. H. Jorge
Publication year - 2007
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/jdm054
Subject(s) - multiplicity (mathematics) , noetherian , mathematics , local ring , pure mathematics , sequence (biology) , combinatorics , algebra over a field , ring (chemistry) , mathematical analysis , chemistry , organic chemistry , biology , genetics
We introduce first the notion of mixed multiplicities for arbitrary ideals in a local d ‐dimensional Noetherian ring (A,m) which, in some sense, generalizes the concept of mixed multiplicities for m‐primary ideals. We also generalize Teissier's product formula for a set of arbitrary ideals and extend the notion of the Buchsbaum‐Rim multiplicity (BR‐multiplicity) of a submodule of a free module to the case where the submodule no longer has finite colength. For a submodule M of A p , we introduce a sequence of multiplicities e k BR (M), k = 0, …, d + p − 1 which in the case of an ideal (p = 1) coincides with the multiplicity sequence c0(I,A),…, c d (I,A) defined for an arbitrary ideal I of A by Achilles and Manaresi. In the case where M has finite colength in Ap and is totally decomposable, we prove that our BR‐multiplicity sequence essentially falls into the standard BR‐multiplicity of M.