Primary decomposition II: Primary components and linear growth

Abstract

We study the following properties about primary decomposition over a Noetherian ring R: (1) For finitely generated modules N ⊆ M and a given subset X = { P 1 , P 2 , … , P r } ⊆ Ass ( M / N ) , we define an X-primary component of N ⊊ M to be an intersection Q 1 ∩ Q 2 ∩ ⋯ ∩ Q r for some P i -primary components Q i of N ⊆ M and we study the maximal X-primary components of N ⊆ M ; (2) We give a proof of the ‘linear growth’ property of Ext and Tor , which says that for finitely generated modules N and M, any fixed ideals I 1 , I 2 , … , I t of R and any fixed integer i ∈ N , there exists a k ∈ N such that for any n ̲ = ( n 1 , n 2 , … , n t ) ∈ N t there exists a primary decomposition of 0 in E n ̲ = Ext R i ( N , M / I 1 n 1 I 2 n 2 ⋯ I t n t M ) (or 0 in T n ̲ = Tor i R ( N , M / I 1 n 1 I 2 n 2 ⋯ I t n t M ) ) such that every P-primary component Q of that primary decomposition contains P k | n ̲ | E n ̲ (or P k | n ̲ | T n ̲ ), where | n ̲ | = n 1 + n 2 + ⋯ + n t .