Primary Decomposition: Compatibility, Independence and Linear Growth

2002/09/20 by Yongwei Yao
Mathematics · #math.AC #msc:13E05 #msc:13C99 #msc:13H99

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published as Proceedings of the AMS, 130(2002) no. 6, 1629-1637 · AMS-LaTeX

arxiv created 2002/09/20 · arxiv updated 2009/11/30

Abstract

For finitely generated modules N \subsetneq M over a Noetherian ring R, we study the following properties about primary decomposition: (1) The Compatibility property, which says that if \ass (M/N)=\P₁, P₂, ..., P_s\ and Q_i is a P_i-primary component of N \subsetneq M for each i=1,2,...,s, then N =Q₁ ∩ Q₂ ∩ ... ∩ Q_s; (2) For a given subset X=\P₁, P₂, ..., P_r \ ⊆ \ass(M/N), X is an open subset of \ass(M/N) if and only if the intersections Q₁ ∩ Q₂∩ ... ∩ Q_r= Q₁' ∩ Q₂' ∩ ... ∩ Q_r' for all possible P_i-primary components Q_i and Q_i' of N\subsetneq M; (3) A new proof of the `Linear Growth' property, which says that for any fixed ideals I₁, I₂, ..., I_t of R, there exists a k ∈ \mathbb N such that for any n₁, n₂, ..., n_t ∈ \mathbb N there exists a primary decomposition of I₁^n₁I₂^n₂... I_t^n_tM ⊂ M such that every P-primary component Q of that primary decomposition contains P^{k(n₁+n₂+...+n_t)}M.

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