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THAI NGUYEN UNIVERSITY OF SCIENCESTRAN DUC DUNG ON THE SEQUENTIAL POLYNOMIAL TYPE AND REDUCIBILITY INDEX OF MODULE ON COMMUTATIVE RINGS SUMMARY OF MATHEMATICS DOCTOR THESIS Thai Nguyen -

THAI NGUYEN UNIVERSITY OF SCIENCES

TRAN DUC DUNG

ON THE SEQUENTIAL POLYNOMIAL TYPE AND REDUCIBILITY INDEX OF MODULE ON

COMMUTATIVE RINGS

SUMMARY OF MATHEMATICS DOCTOR THESIS

Thai Nguyen - 2019

THAI NGUYEN UNIVERSITY OF SCIENCES

TRAN DUC DUNG

ON THE SEQUENTIAL POLYNOMIAL TYPE AND REDUCIBILITY INDEX OF MODULE ON

Let (R, m) be a Noetherian local ring and M a finitely generatedR-module with dim M = d We have depth M ≤ dim M M is a Cohen-Macaulay if depth M = dim M Cohen-Macaulay module plays a centralrole in Commutative Algebra and appears in many different areas of study

of Mathematics such as Algebraic Geomery, Combined Theory, InvariantTheory

Note that M is Cohen-Macaulay if and only if `(M/xM ) = e(x; M )for every parameter system x of M One of the important extensions of theCohen-Macaulay module class is the Buchsbaum module class introdeced

by J St¨uckrad and W Vogel, that is the class of module M satisfy the pothesis by D.A Buchsbaum: `(M/xM )−e(x; M ) is constant independent

hy-of parameter system x Then, N.T Cng, P Schenzel and N.V Trung hasintroduced a class os M module satisfactory supx(`(M/xM ) − e(x; M )) <

∞, is called generalized Cohen-Macaulay module In 1992, N.T Cuongintroduced an invariant p(M ) of M , called the polynomial type of M , inorder to measure the non-Cohen-Macaulayness of M , thereby classifyingand study structure of a finitely module over a local ring If we stipulatethe degree of zero polynomial to be −1, then M is Cohen-Macaulay if andonly if p(M ) = −1, an M is generalized Cohen-Macaulay if and only ifp(M ) ≤ 0

An important generalized of the notion of Cohen-Macaulay module

is that of sequentially Cohen-Macaulay module, introduced almost at thesame time by R.P Stanley in the graded setting and by P Schenzel inthe local setting: M is said to be sequentially Cohen-Macaulay if the quo-tient module Di/Di+1 is Cohen-Macaulay, where D0 = M and Di+1 is thelargest submodule of M of dimension less than dim Di for all i ≥ 0 Then,N.T Cuong and L.T Nhan introduced the sequential generalized Cohen-Macaulay is defined similarly to the sequential Cohen-Macaulayness exceptthat each quotient module Di/Di+1 is required to be generalized Cohen-Macaulay instead of being Cohen-Macaulay

The first purpose of thesis is introduce the notion of sequential nomial type of M , which is denote by sp(M ), in order to measure how far

poly-M is different from the sequential Cohen-poly-Macaulayness We showed thatsp(M ) is dimension of the non sequentially Cohen-Macaulay locus of M if

R is a quotient of Cohen-Macaulay local ring We study change of the quential polynomial type under localization, m-adic and an ascent-descentproperty of sequential polynomial type between M and M/xM for certainparameter x of M We describe sp(M ) in term of the deficiency modules

se-of M when R is a quotient se-of a Gorenstein local ring Note that N.T.Cuong, D.T Cuong v H.L Truong studied a new invariant of M throughmultiplicity, and ring R is a quotient of a Cohen-Macaulay local ring thenthis invariant is the sequential polynomial type of M Recently, S Goto

v L.T Nhan (2018) showed a parameter characteristics of the sequentialpolynomial type

The second purpose of thesis is research some problems about ducibility index of finitely generated module on local ring A submodule

re-N of M is called an irreducible submodule if re-N can not be written as anintersection of two properly larger submodules of M The number of ir-

reducible components of an irredundant irreducible decomposition of N ,which is independence of the choice of the decompostion by E.Noether,

is called the index of reducibility of N and denoted by irM(N ) If q is aparameter ideal of M , then irM(qM ) is said to be the index of reducibility

of q on M

A uniform bound for index of reducibility of parameter for Macaulay class, Buchsbaum class, generalized Cohen-Macaulay class hasbeen resaarched by many mathematicians Recently, P.H Quy (2013)showed a uniform bound of irM(qM ) for all parameter ideals q of M inthe case where p(M ) ≤ 1 In case where p(M ) ≥ 3, a uniform bound

Cohen-of irM(qM ) for all parameter ideals q of M may not exist even whensp(M ) = −1 In fact, Goto and Suzuki (1984) constructed a sequentiallyCohen-Macaulay Noetherian local ring (R, m) such that p(R) = 3 and thesupremum of irR(q) is infinite, where q runs over all parameter ideals of R

On the other hand, the notion of good parameter ideal (such ideals exist)makes an important role in the study of modules which are not necessarilyunmixed Therefore, it is natural to ask if there exists a uniform bound

of irM(qM ) for all good parameter ideals q of M Some positive answersare given by H L Truong (2013) for the case where sp(M ) = −1, and byP.H Quy (2012) for the case where sp(M ) ≤ 0 In this thesis, we study

a uniform bound of irM(qM ) for all good parameter ideals q of M wheresp(M ) ≤ 1 On the other hand, we study the reducibility index of Artianmodule and clarify the relationship between irM(N ) and ir0R(D(M/N )),where ir0R(D(M/N )) is the sum-reducibility index of the Matlis dual ofM/N , that is the number of sum-irreducible submodules appearing in anirredundant sum-irreducible representation of D(M/N ) This is a verybasic problem that was first studied in this thesis

Regarding the approach, to study the sequential polynomial type we

exploit the properties of the dimension filtration of the module (dimensionfiltration concept introduced by P Schenzel and adjusted by NT Cuongand LT Nhan for convenience for use), the strict filter regular sequenceintroduced by N.T Cuong, M Morales and L.T Nhan and peculiar prop-erties of the Artin module, especially the local cohomology module withrespect to m In order to study a uniform bound of the index of goodparameters when sp(M ) is small, we use the theory of good parameter sys-tem introduced by N.T Cuong, T Cuong, characteristic of homogeneity

of the sequential polynomial type and the results of J.D Sally about theminimal number of generators of the module

The thesis is divided into 3 chapters Chapter 1 reiterates some sic knowledge of commutative algebra in order to base on presenting themain content of the thesis in the following chapters, including: local co-homology with respect to maximal ideal, secondary representation of theArtinian module, polynomial type, Cohen-Macaulay module, generalizedCohen-Macaulay module, sequentially Cohen-Macaulay module, sequen-tially generalized Cohen-Macaulay module

ba-Chapter 2 presents the sequential polynomial type of the module.Section 2.1 shows the relationship between dimensional filtration of M anddimensional filtration of M/xM , where x is a strict regular filter element(Prosition 2.1.8) Section 2.2 introduces the concept of the sequential poly-nomial type of M, denoted by sp(M ) to measure the non-sequential-Cohen-Macaulayness of M Proposition 2.2.4 provides the relationship betweensp(M ) and the dimension of non-sequentially Cohen-Macaulay locus of M Next, we give information about the sequential polynomial type under lo-calization and m-aic completion (Theorem 2.2.7, Theorem 2.2.9) Section2.3 provides the relationship between sp(M/xM ) and sp(M ) , where x is acertain parameter element (Theorem 2.3.4) The main result of the chap-

ter (Section 2.4) provides homogeneous characteristics of the sequentialpolynomial type (Theorem 2.4.2).

Chapter 3 presents some problems of the reducibility index of ule Section 3.2 proof the existence of a uniform bound of good parameterparameters q of M with sp(M ) ≤ 1 (Theorem 3.2.6) Section 3.3 investi-gates the index of reducibility in the Artinian module category and gives

mod-a compmod-arison between the index of the submodule of M with the index

of Matlis duality of the corresponding quotient module of M (Theorem3.3.10)

CHAPTER 1

Preparation knowledge

In this chapter, we recall some basic knowledge of commutative algebra inorder to base on presenting the main content of the thesis in the follow-ing chapters, including: local cohomology with respect to maximal ideal,secondary representation of the Artinian module, polynomial type, Cohen-Macaulay module and its extensions

The notion of the polynomial type p(M ) was introduced by N.T.Cuong (1992), in order to measure how far the module M is from belonging

to the class of Cohen-Macaulay modules For each system of parameters

x = (x1, , xd) of M and each tuple of d positive integers n = (n1, , nd),

we consider the difference

Definition 1.2.1 The least degree of all polynomials boundingabove the function IM,x(n), which does not depend on the choice of x, iscalled the polynomial type of M and denoted by p(M )

The concept of dimensional filtration is introduced by P Schenzel(1998) Then N.T Cuong and L.T Nhan (2003) has slightly adjusted thisdefinition by removing repeating components to make it more convenientfor use.

Definition 1.3.1 A filtration Hm0(M ) = Dt ⊂ ⊂ D1 ⊂ D0 =

M of submodules of M is said to be the dimesion filtration of M , if foreach 1 ≤ i ≤ t, Di is the largest submodule of M of dimension less thandimRDi −1

The notion of sequentially Cohen-Macaulay module was introduced

by R Stanley in the graded setting and by P Schenzel in the local setting.This notion was extended to the concept of sequentially generalized Cohen-Macaulay module in a natural way

Definition 1.3.2 Let Hm0(M ) = Dt ⊂ ⊂ D1 ⊂ D0 = M bethe dimension filtration of M The module M is said to be sequentiallyCohen-Macaulay if each quotient module Di−1/Di is Cohen-Macaulay If

Di−1/Di is generalized Cohen-Macaulay for all i = 1, , t, then M is said

to be sequentially generalized Cohen-Macaulay

The concept of good parameter system introduced by N.T Cuongand T Cuong to study the class of Cohen-Macaulay modules and itsextension

Definition 1.3.3 A filtration M = H0 ⊃ H1 ⊃ ⊃ Hn ofsubmodules of M is said to satisfy the dimension condition if dimRHi <dimRHi−1 for all i ≤ n A parameter ideal q = (x1, , xd) of M is said to

be a good parameter ideal with respect to such a filtration if (xhi+1, , xd)M ∩

Hi = 0 for all i ≤ n, where hi = dimRHi If q is good with respect to thedimension filtration, then it is simply called a good parameter ideal of M

2.1 Dimension filtration and strict filter regular

Prosition 2.1.8 Suppose that R is a quotient of Cohen-Macaulay localring Let Hm0(M ) = Dt ⊂ ⊂ D1 ⊂ D0 = M is dimension filtration

of M and x ∈ m be a strict f -element of Di−1/Di for all i ≤ t Set

Di0 = (Di+xM )/xM for i ≤ t Let Hm0(M/xM ) = Lt0 ⊂ ⊂ L0 = M/xM

is dimension of M/xM Then we have

(i) t0 ≤ t ≤ t0 + 1 Concretely, t = t0 if dt−1 ≥ 2 and t = t0+ 1 if dt−1 = 1

(ii) D0i ⊆ Li v `(Li/D0i) < ∞ for all i ≤ t0

2.2 Sequentialy polynomial type: Localization and

completion

Throughout this section, let Hm0(M ) = Dt ⊂ ⊂ D1 ⊂ D0 = M be

the dimension filtration of M and di := dim Di for all i ≤ t

Definition 2.2.1 The sequential polynomial type of M , denoted by sp(M )

is defined throghout by the polynomial type of a quotient module in

di-mension filtration of M :

sp(M ) = max{p(Di−1/Di) | i = 1, , t}

It is clear that sp(M ) = −1 if and only if M is sequentially

Cohen-Macaulay Moreover, sp(M ) ≤ 0 if and only if M is sequentially generalized

Cohen-Macaulay In general, sp(M ) measures how far M is different from

the sequential Cohen-Macaulayness Let

nSCM(M ) := {p ∈ Spec(R) | Mp is not a sequentially Cohen-Macaulay Rp-module}denote the non sequentially Cohen-Macaulay locus of M

We have the following relation between sp(M ) and the dimension of the

non-sequentially Cohen-Macaulay locus of M

Prosition 2.2.4 If R is catenary then sp(M ) ≥ dim(nSCM(M )) The

equality holds true if R is a quotient of local Cohen-Macaulay

Next we study the sequential polynomial type under localization

Theorm 2.2.7 Let p ∈ SuppRM Suppose that R is catenary

(i) If dim(R/p) > sp(M ) then Mp is a sequentially Cohen-Macaulay Rp

(ii) If dim(R/p) ≤ sp(M ) then sp(Mp) ≤ sp(M ) − dim(R/p)

Note that p(M ) = p( cM ), however we do not have such a relationshipbetween sp(M ) and sp( cM )

Example 2.2.8 Let (R, m) be the Noetherian local domain of dimension

2 constructed by D Ferrand and M Raynaud such that bR has an embeddedassociated prime P of dimension 1 Then sp(R) = 1 but sp( bR) = −1

Following M Nagata, R is called unmixed if dim bR/bp = dim bR forevery bp ∈ Ass bR The following results show the relationship betweensp(M ) and sp( cM ), at the same time we give a criterion for sp(M ) andsp( cM ) to be the same

Theorem 2.2.9 sp( cM ) ≤ sp(M ) The equality holds true if R/p isunmixed for all associated primes p of M

Without the unmixedness of associated primes, sp( cM ) and sp(M )may be different

Example 2.2.10 For any integer r ≥ 0, there exists a Noether localdomain (R∗, m∗) which is university catenary such that sp(cR∗) = −1 andsp(R∗) = r + 2

2.3 A relation between sp(M ) and sp(M/xM ) where x

is a parameter element

In this section, we show a relation between sp(M ) and sp(M/xM ),where x is a certain parameter of M Let Hm0(M ) = Dt ⊂ ⊂ D0 = M

be the dimension filtration of M and di := dim Di for all i ≤ t

Theorem 2.3.5 Gi s sp(M ) > 0 Let x ∈ m be a strict f -element of

Di−1/Di for all i ≤ t Then sp(M/xM ) ≤ sp(M ) − 1 The equality holds

if R is a quotient of local Cohen-Macaulay

The equality sp(M/xM ) = sp(M ) − 1 in Theorem 2.3.4 may not

be valid if we drop the assumption that R is a quotient of local Macaulay

Cohen-Example 2.3.6 For each integer r ≥ 0, there exists a Noetherian doamin(R0, m0) and a strict f -element a ∈ m of R∗ such that sp(R∗) = r + 1 andsp(R∗/aR∗) = −1

2.4 A homological characterization of sequential

poly-nomial type

Let Hm0(M ) = Dt ⊂ ⊂ D0 = M be a dimension filtration of M and

di := dim Di for all i ≤ t We stipulate dim Dt = −1 whenever Dt = 0.Set Λ(M ) = {d0, , dt} Note that Λ(M ) \ {−1} = {dim(R/p) | p ∈AssRM } Suppose that R is a quotient of a Gorenstein local ring Set q1 :=max

j / ∈Λ(M )dim(Kj(M )) and q2 := max

Corollary 2.4.3 Let r ≥ −1 be an integer Suppose that R is a quotient

of a Gorenstein local ring Then sp(M ) ≤ r if and only if dim Kj(M ) ≤ rfor all j /∈ Λ(M ) and dim Kj(M ) = j with p(Kj(M )) ≤ r for all j ∈ Λ(M )

CHAPTER 3

Index of reducibility of module

Through this chapter, let (R, m) be a Noether local ring and M afinitely generated R-modulem with dim M = d, N be a submodule of M ,

A be an R-module Artinian We denoted bR and cM the m-adic completion

of R and M respectively

3.1 Index of reducibility of module Noetherian

Firstly, we recall the concept of index of redcibility of module Asubmodule N of M is called an irreducible submodule if N can not be writ-ten as an intersection of two properly larger submodules of M Following

E Noether, the number of irreducible components of an irredundant reducible decomposition of N , which is independence of the choice of thedecompostion

ir-Definition 3.1.2 The number of irreducible components of an dant irreducible decomposition of N , which is independence of the choice

irredun-of the decompostion by E.Noether, is called the index irredun-of reducibility irredun-of Nand denoted by irM(N ) If q is a parameter ideal of M , then irM(qM ) issaid to be the index of reducibility of q on M

We recall some results of J D Sally about the minimal number