Về kiểu đa thức dãy và chỉ số khả quy của môđun trên vành giao hoán tt tiếng anh

THAI NGUYEN UNIVERSITY 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 THAI NGUYEN UNIVERSITY OF SCIENCES TRAN DUC DUNG ON THE SEQUENTIAL POLYNOMIAL TYPE AND REDUCIBILITY INDEX OF MODULE ON COMMUTATIVE RINGS Major: Algebra and number theory Code: 46 01 04 SUMMARY OF MATHEMATICS DOCTOR THESIS Suppervisors: Prof Dr Sc Nguyen Tu Cuong Prof Dr Le Thi Thanh Nhan Thai Nguyen - 2019 Preliminaries Let (R, m) be a Noetherian local ring and M a finitely generated R-module with dim M = d We have depth M ≤ dim M M is a CohenMacaulay if depth M = dim M Cohen-Macaulay module plays a central role in Commutative Algebra and appears in many different areas of study of Mathematics such as Algebraic Geomery, Combined Theory, Invariant Theory 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 the Cohen-Macaulay module class is the Buchsbaum module class introdeced by J Stă uckrad and W Vogel, that is the class of module M satisfy the hypothesis by D.A Buchsbaum: (M/xM )−e(x; M ) is constant independent of parameter system x Then, N.T Cng, P Schenzel and N.V Trung has introduced a class os M module satisfactory supx ( (M/xM ) − e(x; M )) < ∞, is called generalized Cohen-Macaulay module In 1992, N.T Cuong introduced an invariant p(M ) of M , called the polynomial type of M , in order to measure the non-Cohen-Macaulayness of M , thereby classifying and study structure of a finitely module over a local ring If we stipulate the degree of zero polynomial to be −1, then M is Cohen-Macaulay if and only if p(M ) = −1, an M is generalized Cohen-Macaulay if and only if p(M ) ≤ 2 An important generalized of the notion of Cohen-Macaulay module is that of sequentially Cohen-Macaulay module, introduced almost at the same time by R.P Stanley in the graded setting and by P Schenzel in the local setting: M is said to be sequentially Cohen-Macaulay if the quotient module Di /Di+1 is Cohen-Macaulay, where D0 = M and Di+1 is the largest submodule of M of dimension less than dim Di for all i ≥ Then, N.T Cuong and L.T Nhan introduced the sequential generalized CohenMacaulay is defined similarly to the sequential Cohen-Macaulayness except that each quotient module Di /Di+1 is required to be generalized CohenMacaulay instead of being Cohen-Macaulay The first purpose of thesis is introduce the notion of sequential polynomial type of M , which is denote by sp(M ), in order to measure how far M is different from the sequential Cohen-Macaulayness We showed that sp(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 sequential polynomial type under localization, m-adic and an ascent-descent property of sequential polynomial type between M and M/xM for certain parameter x of M We describe sp(M ) in term of the deficiency modules of M when R is a quotient of a Gorenstein local ring Note that N.T Cuong, D.T Cuong v H.L Truong studied a new invariant of M through multiplicity, and ring R is a quotient of a Cohen-Macaulay local ring then this invariant is the sequential polynomial type of M Recently, S Goto v L.T Nhan (2018) showed a parameter characteristics of the sequential polynomial type The second purpose of thesis is research some problems about reducibility index of finitely generated module on local ring A submodule N of M is called an irreducible submodule if N can not be written as an intersection 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 a parameter 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 CohenMacaulay class, Buchsbaum class, generalized Cohen-Macaulay class has been resaarched by many mathematicians Recently, P.H Quy (2013) showed a uniform bound of irM (qM ) for all parameter ideals q of M in the case where p(M ) ≤ In case where p(M ) ≥ 3, a uniform bound of irM (qM ) for all parameter ideals q of M may not exist even when sp(M ) = −1 In fact, Goto and Suzuki (1984) constructed a sequentially Cohen-Macaulay Noetherian local ring (R, m) such that p(R) = and the supremum 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 necessarily unmixed 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 answers are given by H L Truong (2013) for the case where sp(M ) = −1, and by P.H Quy (2012) for the case where sp(M ) ≤ In this thesis, we study a uniform bound of irM (qM ) for all good parameter ideals q of M where sp(M ) ≤ On the other hand, we study the reducibility index of Artian module and clarify the relationship between irM (N ) and irR (D(M/N )), where irR (D(M/N )) is the sum-reducibility index of the Matlis dual of M/N , that is the number of sum-irreducible submodules appearing in an irredundant sum-irreducible representation of D(M/N ) This is a very basic 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 (dimension filtration concept introduced by P Schenzel and adjusted by NT Cuong and LT Nhan for convenience for use), the strict filter regular sequence introduced by N.T Cuong, M Morales and L.T Nhan and peculiar properties of the Artin module, especially the local cohomology module with respect to m In order to study a uniform bound of the index of good parameters when sp(M ) is small, we use the theory of good parameter system introduced by N.T Cuong, T Cuong, characteristic of homogeneity of the sequential polynomial type and the results of J.D Sally about the minimal number of generators of the module The thesis is divided into chapters Chapter reiterates some basic knowledge of commutative algebra in order to base on presenting the main content of the thesis in the following chapters, including: local cohomology with respect to maximal ideal, secondary representation of the Artinian module, polynomial type, Cohen-Macaulay module, generalized Cohen-Macaulay module, sequentially Cohen-Macaulay module, sequentially generalized Cohen-Macaulay module Chapter presents the sequential polynomial type of the module Section 2.1 shows the relationship between dimensional filtration of M and dimensional 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 polynomial type of M, denoted by sp(M ) to measure the non-sequential-CohenMacaulayness of M Proposition 2.2.4 provides the relationship between sp(M ) and the dimension of non-sequentially Cohen-Macaulay locus of M Next, we give information about the sequential polynomial type under localization and m-aic completion (Theorem 2.2.7, Theorem 2.2.9) Section 2.3 provides the relationship between sp(M/xM ) and sp(M ) , where x is a certain parameter element (Theorem 2.3.4) The main result of the chap- ter (Section 2.4) provides homogeneous characteristics of the sequential polynomial type (Theorem 2.4.2) Chapter presents some problems of the reducibility index of module Section 3.2 proof the existence of a uniform bound of good parameter parameters q of M with sp(M ) ≤ (Theorem 3.2.6) Section 3.3 investigates the index of reducibility in the Artinian module category and gives a comparison between the index of the submodule of M with the index of Matlis duality of the corresponding quotient module of M (Theorem 3.3.10) 6 CHAPTER Preparation knowledge In this chapter, we recall some basic knowledge of commutative algebra in order to base on presenting the main content of the thesis in the following chapters, including: local cohomology with respect to maximal ideal, secondary representation of the Artinian module, polynomial type, CohenMacaulay 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 IM,x (n) = n1 nd R (M/(x1 , , xd )M ) − n1 nd e(x, M ) as a function in n1 , , nd , where e(x, M ) denotes the multiplicity of M with respect to x In general, IM,x (n) is not a polynomial for n1 , , nd large enough, but it takes non-negative values and it is bounded above by polynomials Definition 1.2.1 The least degree of all polynomials bounding above the function IM,x (n), which does not depend on the choice of x, is called the polynomial type of M and denoted by p(M ) 7 The concept of dimensional filtration is introduced by P Schenzel (1998) Then N.T Cuong and L.T Nhan (2003) has slightly adjusted this definition by removing repeating components to make it more convenient for 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 for each ≤ i ≤ t, Di is the largest submodule of M of dimension less than dimR Di −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 CohenMacaulay module in a natural way Definition 1.3.2 Let Hm0 (M ) = Dt ⊂ ⊂ D1 ⊂ D0 = M be the dimension filtration of M The module M is said to be sequentially Cohen-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 Cuong and T Cuong to study the class of Cohen-Macaulay modules and its extension Definition 1.3.3 A filtration M = H0 ⊃ H1 ⊃ ⊃ Hn of submodules of M is said to satisfy the dimension condition if dim R Hi < dim R Hi−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 = for all i ≤ n, where hi = dim R Hi If q is good with respect to the dimension filtration, then it is simply called a good parameter ideal of M CHAPTER The sequential polynomial type of module Through this chapter, let (R, m) be a Noether local ring and M a finitely generated R-modulem with dim M = d, A be an R-module Artinian We denoted R and M the m-adic completion of R and M respectively 2.1 Dimension filtration and strict filter regular sequence By N.T Cuong, M Morales and L.T Nhan, an element x ∈ m is called strict filter regular element (strict f-element for short) of M if x ∈ / p vi mi p∈( AttR Hmj (M )) \ {m} The main result in this section is to show the j≤d relationship between dimension filtration of M and dimension filtration of M/xM , where x is strict f-element of Di−1 /Di for every i = 1, , t Prosition 2.1.8 Suppose that R is a quotient of Cohen-Macaulay local ring 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 Di = (Di +xM )/xM for i ≤ t Let Hm0 (M/xM ) = Lt ⊂ ⊂ L0 = M/xM is dimension of M/xM Then we have (i) t ≤ t ≤ t + Concretely, t = t if dt−1 ≥ and t = t + if dt−1 = (ii) Di ⊆ Li v (Li /Di ) < ∞ for all i ≤ t 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 dimension 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 CohenMacaulay Moreover, sp(M ) ≤ 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 ∈ SuppR M Suppose that R is catenary (i) If dim(R/p) > sp(M ) then Mp is a sequentially Cohen-Macaulay Rp - 10 module (ii) If dim(R/p) ≤ sp(M ) then sp(Mp ) ≤ sp(M ) − dim(R/p) Note that p(M ) = p(M ), however we not have such a relationship between sp(M ) and sp(M ) Example 2.2.8 Let (R, m) be the Noetherian local domain of dimension constructed by D Ferrand and M Raynaud such that R has an embedded associated prime P of dimension Then sp(R) = but sp(R) = −1 Following M Nagata, R is called unmixed if dim R/p = dim R for every p ∈ Ass R The following results show the relationship between sp(M ) and sp(M ), at the same time we give a criterion for sp(M ) and sp(M ) to be the same Theorem 2.2.9 sp(M ) ≤ sp(M ) The equality holds true if R/p is unmixed for all associated primes p of M Without the unmixedness of associated primes, sp(M ) and sp(M ) may be different Example 2.2.10 For any integer r ≥ 0, there exists a Noether local domain (R∗ , m∗ ) which is university catenary such that sp(R∗ ) = −1 and sp(R∗ ) = r + 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 ) > Let x ∈ m be a strict f -element of Di−1 /Di for all i ≤ t Then sp(M/xM ) ≤ sp(M ) − The equality holds if R is a quotient of local Cohen-Macaulay 11 The equality sp(M/xM ) = sp(M ) − in Theorem 2.3.4 may not be valid if we drop the assumption that R is a quotient of local CohenMacaulay Example 2.3.6 For each integer r ≥ 0, there exists a Noetherian doamin (R , m ) and a strict f -element a ∈ m of R∗ such that sp(R∗ ) = r + and sp(R∗ /aR∗ ) = −1 2.4 A homological characterization of sequential polynomial 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 = Set Λ(M ) = {d0 , , dt } Note that Λ(M ) \ {−1} = {dim(R/p) | p ∈ AssR M } Suppose that R is a quotient of a Gorenstein local ring Set q1 := max dim(K j (M )) and q2 := max p(K j (M )) The following theorem, j ∈Λ(M / ) j∈Λ(M ) shows that sp(M ) can be computed in term of the deficiency modules K j (M ) Theorem 2.4.2 If R is a quotient of a Gorenstein local ring, then sp(M ) = max{q1 , q2 } 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 K j (M ) ≤ r for all j ∈ / Λ(M ) and dim K j (M ) = j with p(K j (M )) ≤ r for all j ∈ Λ(M ) 12 CHAPTER Index of reducibility of module Through this chapter, let (R, m) be a Noether local ring and M a finitely generated R-modulem with dim M = d, N be a submodule of M , A be an R-module Artinian We denoted R and M 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 A submodule N of M is called an irreducible submodule if N can not be written as an intersection of two properly larger submodules of M Following E Noether, the number of irreducible components of an irredundant irreducible decomposition of N , which is independence of the choice of the decompostion Definition 3.1.2 The number of irreducible 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 a parameter ideal of M , then irM (qM ) is said to be the index of reducibility of q on M We recall some results of J D Sally about the minimal number 13 of generators of module Denote µ(N ) := dim R/m (N/mN ) the minimal number of generators of N Set c(M ) = sup {µ(N ) | N is a submodule of M } J D Sally proved that c(R) < ∞ if and only if dim R ≤ Then S Goto v N Suzuki improved this result for modules In a natural way, P.H Qu defined the following analogous notation for Artinian modules Notation 3.1.6 For an Artinian R-module A, set r(A) = sup {dimR/m Soc(A/B) | B is a submodule ofA} For a finitely generated R-module N , set N ∗ = HomR (N, E(R/m)), the Matlis duality of N Then N ∗ is an Artinian R-module and c(N ) = r(N ∗ ) For an Artinian R-module A, if R = R then A∗ = HomR (A, E(R/m)) is a finitely generated R-module and r(A) = c(A∗ ) Note that r(A) < ∞ if and only if dimR A ≤ 3.2 Index of reducibility with the small sequential polynomial type P H Quy (2013) showed a uniform bound of irM (qM ) for all parame- ter ideals q of M in the case where p(M ) ≤ If p(M ) = 2, the question of whether there exists a uniform bound of irM (qM ) for all parameter ideals is still open In case where p(M ) ≥ 3, a uniform bound of irM (qM ) for all parameter ideals q of M may not exist even when sp(M ) = −1 In fact, Goto and Suzuki (1984) constructed a sequentially Cohen-Macaulay Noetherian local ring (R, m) such that p(R) = and the supremum 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 necessarily unmixed 14 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 answers are given by H L Truong (2013) for the case where sp(M ) = −1, and by P H Quy (2012) for the case where sp(M ) ≤ The answer for the case sp(M ) ≤ will be solve in section of this chapter Before proving the main theorem, we need some lemmas Lemma 3.2.1.Let sp(M ) ≤ and R = R Let H be a submodule of M such that dim R H < d and p(M/H) ≤ Let x ∈ m be a parameter of M such that xH = Then sp(M/xM ) ≤ For each Artinian R-module A with dim R (A) ≤ 1, the number r(A) is defined as in Notation 3.1.8 Lemma 3.2.2 Let R = R and µ = µ(m) the minimal number of generators of m Let H be a submodule of M such that dim R H < d and p(M/H) ≤ Let x ∈ m be a parameter of M such that xH = Then dim R/m Soc(Hmd−1 (M/xM )) ≤ dim R/m Soc(Hmd (M )) + dim R/m Soc(Hmd−1 (H)) + (µ + 1) r(Hmd−1 (M/H)) + µ r(Hmd−2 (M/H)) From now on we use the following notations Notation 3.2.3 Let M = H0 ⊃ H1 ⊃ ⊃ Hn be a filtration of submodules of M satisfying the dimension condition Let (x1 , , xd ) be a good parameter ideal of M with respect to this filtration Let M/xd M = H0 ⊃ H1 ⊃ ⊃ Hm be the filtration of submodules of M/xd M, where m and Hi are defined as follows: If dim R H1 < d − 1, then we set m = n and Hi = (Hi + xd M )/xd M ∼ = Hi ; If dim R H1 = d − 1, then we set m = n − and Hi = (Hi+1 + xd M )/xd M ∼ = Hi+1 , for i = 1, , m Lemma 3.2.4 Let R = R Let M = H0 ⊃ H1 ⊃ ⊃ Hn be a filtration 15 of submodules of M satisfying the dimension condition such that p(Hn ) ≤ and p(Hi−1 /Hi ) ≤ for all i ≤ n Let (x1 , , xd ) be a good parameter ideal with respect to this filtration Then (x1 , , xd−1 ) is a good parameter ideal of M/xd M with respect to the filtration M/xd M = H0 ⊃ H1 ⊃ ⊃ Hm Moreover, p(Hm ) ≤ 1, dim R Hi < dim R Hi−1 and p(Hi−1 /Hi ) ≤ for all i = 1, , m In the next lemma, let µ = µ(m) be the minimal number of generators of m Lemma 3.2.5 Let sp(M ) ≤ Let M = H0 ⊃ H1 ⊃ ⊃ Hn ⊃ Hn+1 = be a filtration of submodules of M satisfying the dimension condition Assume that p(Hi /Hi+1 ) ≤ for all i ≤ n and Hn satisfies the following condition: Hn = M when d ≤ 2, and dim R Hn ≥ when d > Set hi = dim R Hi for i ≤ n Then for any good parameter ideal q = (x1 , , xd ) of M with respect to this filtration, we have n n d irM (qM ) ≤ µ r Hmj (Hi /Hi+1 ) dim R/m Soc(Hmhi (Hi )) + i=0 j