Primary decomposition: Compatibility, independence and linear growth
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- by Yongwei Yao PDF
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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 $\operatorname {Ass} (M/N)=\{ P_1, P_2, \dots , P_s\}$ and $Q_i$ is a $P_i$-primary component of $N \subsetneq M$ for each $i=1,2,\dots ,s$, then $N =Q_1 \cap Q_2 \cap \cdots \cap Q_s$; (2) For a given subset $X=\{ P_1, P_2, \dots , P_r \} \subseteq \operatorname {Ass}(M/N)$, $X$ is an open subset of $\operatorname {Ass}(M/N)$ if and only if the intersections $Q_1 \cap Q_2\cap \cdots \cap Q_r= Q_1’ \cap Q_2’ \cap \cdots \cap Q_r’$ 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 $I_1, I_2, \dots , I_t$ of $R$ there exists a $k \in \mathbb N$ such that for any $n_1, n_2, \dots , n_t \in \mathbb N$ there exists a primary decomposition of $I_1^{n_1}I_2^{n_2}\cdots I_t^{n_t}M \subset M$ such that every $P$-primary component $Q$ of that primary decomposition contains $P^{k(n_1+n_2+\cdots +n_t)}M$.References
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Additional Information
- Yongwei Yao
- Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66045
- MR Author ID: 685921
- Email: yyao@math.ukans.edu
- Received by editor(s): October 5, 2000
- Received by editor(s) in revised form: January 12, 2001
- Published electronically: November 15, 2001
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1629-1637
- MSC (2000): Primary 13E05; Secondary 13C99, 13H99
- DOI: https://doi.org/10.1090/S0002-9939-01-06284-0
- MathSciNet review: 1887009