Primary decomposition II: Primary components and linear growth

We study the following properties about primary decomposition over a Noetherian ring R  : (1) For finitely generated modules N⊆MN⊆M and a given subset X={P1,P2,…,Pr}⊆Ass(M/N)X={P1,P2,…,Pr}⊆Ass(M/N), we define an X  -primary component of N⊊MN⊊M to be an intersection Q1∩Q2∩⋯∩QrQ1∩Q2∩⋯∩Qr for some PiPi-primary components QiQi of N⊆MN⊆M and we study the maximal X  -primary components of N⊆MN⊆M; (2) We give a proof of the ‘linear growth’ property of ExtExt and TorTor, which says that for finitely generated modules N and M  , any fixed ideals I1,I2,…,ItI1,I2,…,It of R   and any fixed integer i∈Ni∈N, there exists a k∈Nk∈N such that for any n̲=(n1,n2,…,nt)∈Nt there exists a primary decomposition of 0 in En̲=ExtRi(N,M/I1n1I2n2⋯ItntM) (or 0 in Tn̲=ToriR(N,M/I1n1I2n2⋯ItntM)) such that every P-primary component Q   of that primary decomposition contains Pk|n̲|En̲ (or Pk|n̲|Tn̲), where |n̲|=n1+n2+⋯+nt.