Primary Decomposition Ii: Primary Components and Linear Growth
نویسنده
چکیده
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 = {P1, P2, . . . , Pr} ⊆ Ass(M/N), we define an X-primary component of N ( M to be an intersection Q1 ∩ Q2 ∩ · · · ∩ Qr for some Pi-primary components Qi 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 I1, I2, . . . , It of R and any fixed integer i ∈ N, there exists a k ∈ N such that for any n = (n1, n2, . . . , nt) ∈ Nt there exists a primary decomposition of 0 in En = ExtR(N, M/I n1 1 I n2 2 · · · I nt t M) (or 0 in Tn = Tori (N, M/I n1 1 I n2 2 · · · I nt t M)) such that every P -primary component Q of that primary decomposition contains P k|n|En (or P k|n|Tn), where |n| = n1 + n2 + · · ·+ nt.
منابع مشابه
Classical quasi-primary submodules
In this paper we introduce the notion of classical quasi-primary submodules that generalizes the concept of classical primary submodules. Then, we investigate decomposition and minimal decomposition into classical quasi-primary submodules. In particular, existence and uniqueness of classical quasi-primary decompositions in finitely generated modules over Noetherian rings are proved. More...
متن کاملQuasi-Primary Decomposition in Modules Over Proufer Domains
In this paper we investigate decompositions of submodules in modules over a Proufer domain into intersections of quasi-primary and classical quasi-primary submodules. In particular, existence and uniqueness of quasi-primary decompositions in modules over a Proufer domain of finite character are proved. Proufer domain; primary submodule; quasi-primary submodule; classical quasi-primary; decompo...
متن کاملInfinite-dimensional versions of the primary, cyclic and Jordan decompositions
The famous primary and cyclic decomposition theorems along with the tightly related rational and Jordan canonical forms are extended to linear spaces of infinite dimensions with counterexamples showing the scope of extensions.
متن کاملPrimary Decomposition: Compatibility, Independence and Linear Growth
For finitely generated modules N ( 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, . . . , Ps} and Qi is a Pi-primary component of N ( M for each i = 1, 2, . . . , s, then N = Q1 ∩Q2 ∩ · · · ∩Qs; (2) For a given subset X = {P1, P2, . . . , Pr} ⊆ Ass(M/N), X is an open subset of Ass(...
متن کاملThe Compatibility, Independence, and Linear Growth Properties
The first part is about primary decomposition. After reviewing the basic definitions, we survey the compatibility, independence, and linear growth properties that have been known. Then, we prove the linear growth property of primary decomposition for a new family of modules. In the remaining sections, we study secondary representation, which can be viewed as a dual of primary decomposition. Cor...
متن کامل