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Hilbert-Samuel Functions and Multiplicity of a Finitely Generated A-module M with Respect to a Filtration or a Bifiltration of M : Generalizations of a Theorem of Bhattacharya to the Cross Product of Filtrations on a Nœtherian Semi-local Ring


Saint Blanc Amani KOUA 1), Abdoulaye ASSANE 2) and Daouda SANGARE 3)
UFR-SFA Laboratoire de Mathématiques et Informatiques
Université Nangui Abrogoua
02 BP 801 Abidjan 02, Côte d’Ivoire


Mathematics Subject Classification: (MSC 2010): 13 A02 A15 13 A30 13J30 16W50 16W70
Key words: filtration, bifiltration, (I,J)-good bifiltration, cross product of filtrations, function of polynomial type, Hilbert-Samuel functions, Hilbert- Samuel polynomials, multiplicity.

Abstract:

In this paper we give generalizations of a Bhattacharya theorem to the cross product of filtrations on a nœtherian semi-local ring. If A is a semi-local nœtherian ring, M a non-zero finitely generated A-module, f = (In)n a filtration of A and φ = (Mn)n a filtration of M, we prove under certain conditions that: the Hilbert-Samuel function of the A-module M with respect to the bifiltration (resp. the filtration ), defined by HS(M,fφ)(m,n) = lA(MInMm) m,n (resp. HS(M,fφ)(n) = lA(MInMn) n ) is of polynomial type and that its degree is equal to the Krull dimension of the A-module M. Also, we show that the multiplicities of the A-module M with respect to the bifiltration and the filtration are non negative integers.

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