Về môđun cohen macaulay suy rộng chính tắc và một số quỹ tích không cohen macaulay trên vành noether địa phương tt tieng anh

MINISTRY OF EDUCATION AND TRAINING THAI NGUYEN UNIVERSITY LUU PHUONG THAO ON CANONICAL GENERALIZED COHEN-MACAULAY MODULES AND SOME NON-COHEN-MACAULAY LOCI OVER NOETHERIAN LOCAL RINGS SUMMARY OF MATHEMATICS DOCTOR THESIS THAI NGUYEN - 2020 MINISTRY OF EDUCATION AND TRAINING THAI NGUYEN UNIVERSITY LUU PHUONG THAO ON CANONICAL GENERALIZED COHEN-MACAULAY MODULES AND SOME NON-COHEN-MACAULAY LOCI OVER NOETHERIAN LOCAL RINGS Major: Algebra and number theory Code: 46 01 04 SUMMARY OF MATHEMATICS DOCTOR THESIS Supervisors: Prof Dr Le Thi Thanh Nhan Dr Tran Nguyen An THAI NGUYEN - 2020 Preface Let (R, m) be a commutative Noetherian local ring with the unique maximal ideal m, and M a finitely generated R-module of Krull dimension d The relation between invariants depth and dimension of M is given by the formula depth M ≤ dim M If depth M = dim M then M is called a CohenMacaulay module When R is a Cohen-Macaulay R-module, we say that R is a Cohen-Macaulay ring The class of Cohen-Macaulay modules and its generalizations have attracted many researchers all over the world The structure of these modules is characterized via most well-known theories of commutative algebra such as multiplicity, local cohomology, localization, completion, etc These modules are studied in many different branches of mathematics including homology algebra, invariant theory, combinatory and algebraic geometry The thesis related to two directions of generalization of the class of CohenMacaulay modules as follows The first generalization is based on the difference I(x; M ) between the length (M/xM ) and the multiplicity e(x; M ) with respect to a system of parameters x of M Note that M is Cohen-Macaulay if and only if I(x; M ) = for a (for all) system of parameters x Therefore, a conjecture was given by D A Buchsbaum in 1965 as follows: I(x; M ) := (M/xM ) − e(x; M ) is a constant not depending on the system of parameters x of M The negative answer for this conjecture was given by W Vogel and J Stă uckrad in 1973, and they studied the class of rings and modules satisfying the conditions of this conjecture called Buchsbaum rings and modules In 1978, N T Cuong, P Schenzel and N V Trung introduced a generalization of Buchsbaum modules, that is the class of modules M satisfying the condition sup I(x; M ) < ∞, where the supremum runs over every parameters system x of M , and they called them generalized Cohen-Macaulay modules Nowadays, the notion Buchsbaum module and generalized Cohen-Macaulay module have become well-known in Commutative Algebra Continue on generalizing in this direction, we gain a class of Cohen-Macaulay modules in dimension > s, where s ≥ −1 is an integer We say that M is a Cohen-Macaulay module in dimension > s if every system of parameters of M is a regular M -sequence in dimension > s Note that M is Cohen-Macaulay module if and only if it is Cohen-Macaulay in dimension > −1 In the case R is a quotient of a Cohen-Macaulay ring, M is generalized Cohen-Macaulay if and only if M is Cohen-Macaulay in dimension > The second generalization of the class of Cohen-Macaulay modules is based on structure of the canonical modules, in the case R is a homomorphic image of a Gorenstein local ring (R , m ) of dimension n For each integer i ≥ 0, n −i i i is finitely generated R-module and it is (M, R ) Then KM := ExtR set KM d by KM and called i-th deficiency module of M When i = d, we denote KM call it the canonical module of M If KM is Cohen-Macaulay, we say that M is canonical Cohen-Macaulay Note that if M is a Cohen-Macaulay module then so is KM Thus, the class of canonical Cohen-Macaulay modules is a generalization of the class of Cohen-Macaulay modules The notions of canonical Cohen-Macaulay rings and modules was originated from following problem: Let (R, m) be an integral domain and Q(R) the quotient ring of R A natural question is whether exists or not an intermediate R ⊆ B ⊆ Q(R) such that B is a finitely generated R-module and B is a Cohen-Macaulay ring? A such ring B (if exist) is called a birational Macaulayfication of R This is an important problem in Commutative Algebra In 2004, P Schenzel proved that an integral domain local Noetherian R have a birational Macaulayfication if and only if R is canonical Cohen-Macaulay ring In 2006, L T Nhan gave a characterization for canonical Cohen-Macaulay modules through the vanishing of residual lengths of local cohomology modules with respect to strict f-sequence systems of parameters, which was introduced by N T Cuong, M Morales, L T Nhan In 2012, M Brodmann and L T Nhan showed that, in the case d ≥ and x is a strict f-element of parameter, M is canonical Cohen-Macaulay if and only if M/xM is canonical Cohen-Macaulay Naturally, N T H Loan and L T Nhan introduced the class of canonical generalized Cohen-Macaulay modules, which are modules M such that their canonical modules are generalized Cohen-Macaulay They characterized this class of modules through the existence of a uniform bound for residual lengths of local cohomology modules with respect to strict f-sequence systems of pa- rameters Note that if M is generalized Cohen-Macaulay, then M is canonical generalized Cohen-Macaulay The thesis studies the class of canonical generalized Cohen-Macaulay modules and some non Cohen-Macaulay loci on Noetherian local ring The first purpose of the thesis is characterizing structure of class of canonical generalized Cohen-Macaulay modules when R is a homomorphic image of a Gorenstein local ring The second purpose is clarifying the relative between the non Cohen-Macaulay locus of KM and that of M The third purpose is studying set of attached primes, dimension and multiplicity of Artinian local cohomology modules via certain flat Rp → RP , where P ∈ Spec(R), p = P ∩ R and R is arbitrary, not necessarily a factor of a Gorenstein ring, upon which we give the formula to compute dimension of non Cohen-Macaulay in dimension > s About the research method, to characterize the class of canonical generalized Cohen-Macaulay modules, we exploit specific properties of Artinian local cohomology modules and use the strict f-sequence parameters system flexibly On the relation between two non Cohen-Macaulay loci nCM(KM ) and nCM(M ), we need the Structure of Buchsbaum ring Theorem proved by S Goto in 1980, the Theorem of Structure of canonical modules via flat homomorphism proved by Y Aoyama and S Goto in 1985, and the formula of dimension of deficiency module under the action of the formal power series extension To study local cohomology modules under the impact of certain flat homomorphism Rp → RP , we apply effectively the Shifted localization Principle and Shifted completion Principle for attached primes of local cohomology modules given by L T Nhan and P H Quy in 2014, and the associativity formula for multiplicity of local cohomology modules given by M Brodmann and R Y Sharp in 2002 In addition to the introduction, conclusions and references, the thesis consists of chapters Chapter recalls some fundamental knowlege to serve the next chapters, including characterizations of Cohen-Macaulay modules and generalized Cohen-Macaulay modules; set of attached primes, dimension and multiplicity of Artinian modules; canonical modules and deficiency modules 4 In Chapter 2, we introduce the notion of canonical system of parameters of finitely generated module M , establish the relation between the standard system of parameters and canonical system of parameters of M We give some the characterizations of canonical generalized Cohen-Macaulay modules through canonical system of parameters and improve the previous results about structure of canonical generalized Cohen-Macaulay modules In Chapter 3, we give the relation between the dimension of non CohenMacaulay locus of M and that of KM Specially, we show that, except the inclusion relation nCM(KM ) ⊆ nCM(M ), these two loci are almost independent of each other In Chapter 4, we clarify the change of the set of attached primes, dimension and multiplicity of Artinian local cohomology modules via certain flat extension Rp → RP , where P ∈ Spec(R) and p = P∩R Based on these results, we establish the formula of computing the dimension of non Cohen-Macaulay in dimension > s locus 5 Chapter Preliminaries Chapter provides some basis knowledge for to proving the results in the sequel chapters including the following content 1.1 Cohen-Macaulay modules and generalized Cohen-Macaulay modules In this section, we recall some properties and well-known characterizations of Cohen-Macaulay modules and generalized Cohen-Macaulay modules 1.2 Artinian modules We present some results on the set of attached primes, dimension and multiplicity of Artinian modules 1.3 Canonical modules and deficiency modules We recall the notions and some properties of canonical modules and deficiency modules that will be used in the sequel 6 Chapter Canonical generalized Cohen-Macaulay modules Throughout this chapter, let (R, m) be a Noetherian local ring and a quotient of a Gorenstein local ring Let M be a finitely generated R-module of dimension d Following P Schenzel, M is said to be a canonical Cohen-Macaulay module if the canonical module KM of M is Cohen-Macaulay Naturally, N T H Loan and L T Nhan introduced the class of canonical generalized CohenMacaulay modules, which are modules M such that KM are generalized CohenMacaulay One of the well-known characterizations of Cohen-Macaulay modules is I(x; M ) = 0, for some (for all) system of parametes x of M In 2012, M Brodmann and L T Nhan gave an analogue version for canonical CohenMacaulay modules as follows: M is canonical Cohen-Macaulay if and only if Rl Hm2 (M/(x1 , , xd−3 )M ) = for some (for all) strict f-sequence system of parameters (x1 , , xd ) of M , where Rl(A) is the residual length of Artinian R-module A was defined by R Y Sharp and M Hamieh in 1985, and the notion of strict f-sequence was introduced by N T Cuong, M Morales and L T Nhan in 2004 The purpose of Chapter is to establish a version for canonical generalized Cohen-Macaulay modules, that is similar to the well-known parametric characterizations of generalized Cohen-Macaulay modules which is proved by N T Cuong, P Schenzel, N V Trung (1978) and N V Trung (1986), where the role of the number I(x; M ) is replaced by that of the number Rl Hm2 (M/(x1 , , xd−3 )M ) and the standard system of parameters is substituted by the canonical system of parameters 2.1 Canonical system of parameter Firstly, we recall the notion and some properties of strict f-sequence Definition 2.1.2 A sequence (x1 , , xt ) of elements in m is said to be a strict f-sequence of M if xj+1 ∈ / p for all prime ideals d−j Att(Hmi (M/(x1 , , xj )M )) \ {m}, p∈ i=1 for all j = 0, , t − A strict f-sequence (x1 , , xt ) of M is said to be an unconditioned strict f-sequence of M if it is a strict f-sequence in any order Lemma 2.1.3 (a) A sequence (x1 , , xt ) of elements in m is a strict fi sequence of M if and only if it is a filter regular sequence of KM for all integers i ≥ (b) If (x1 , , xt ) ∈ m is a strict f-sequence of M , then so is (xn1 , , xnt t ) for all positive integers n1 , , nt (c) For each integer t > 0, there exists an unconditioned strict f-sequence of M of length t Let A be an Artinian R-module Following R Y Sharp and M A Hamieh, stable index of A, denoted by s(A), is the smallest positive integer s such that mn A = ms A fof all n ≥ s Set Rl(A) := s(A) A) R (A/m Then, Rl(A) is finite and it is called the residual length of A Remark 2.1.4 (i) Rl(A) = if and only if m ∈ / AttR A (ii)If x ∈ / p for all p ∈ AttR A \ {m}, then n R (A/x A) (iii) If R (A/xA) ≤ Rl(A) In this case = Rl(A) for all n ≥ s(A) R (A) < ∞, then Rl(A) = R (A) The following lemma gives a property of strict f-sequence that is relating to residual length and deficiency modules Lemma 2.1.6 Let x ∈ m be a strict f-element of M Let i ≥ be an integer 8 The following statements are true (a) There exists an integer n0 ≥ such that Rl(Hmi (M )) = i R (Hm (KM )) = i R (0 :KM xn ) for all n ≥ n0 (b) There is an exact sequence i+1 i+1 i → KM /xKM → KM/xM → (0 :KMi x) → j In particular, Hmj KM /xKM ∼ = Hm KM/xM for any j ≥ Next, we introduce the concept of canonical system of parameters as follow Definition 2.1.9 A strict f-sequence x = (x1 , , xd ) is said to be a canonical s.o.p of M if Rl Hm2 (M/(x1 , , xd−3 )M ) = Rl Hm2 (M/(x21 , , x2d−3 )M ) If x is at the same time a strict f-sequence in any order and a canonical s.o.p of M , then it is said to be an unconditioned canonical s.o.p of M The relationship between standard s.o.p of M and canonical s.o.p of M is given by the following proposition Proposition 2.1.10 If (x1 , , xd ) be a standard s.o.p of M, then it is a canonical s.o.p of M The converse statement of Proposition 2.1.10 is not true in general 2.2 Canonical generalized Cohen-Macaulay modules In 2013, N T H Loan and L T Nhan introduced the notion of canonical generalized Cohen-Macaulay modules They also characterized this class modules through the existence of a uniform bound for residual lengths of Artinian local cohomology modules with respect to strict f-sequence systems of parameters Definition 2.2.1 M is said to be a canonical generalized Cohen-Macaulay module if canonical module KM of M is generalized Cohen-Macaulay module Lemma 2.2.3 The following statements are equivalent: (a) M is canonical generalized Cohen-Macaulay; (b) There exist a number c(M ) such that Rl Hmd−k−1 (M/(x1 , , xk )M ) ≤ c(M ) for all strict f-sequence x = (x1 , , xd ) of M and all k = 1, , d − 3; (c) There exist a strict f-sequence x = (x1 , , xd ) of M and a number c(x, M ) such that Rl Hmd−k−1 (M/(xn1 , , xnk k )M ) ≤ c(x, M ) for all k = 1, , d− and all positive integer n1 , , nk Furthermore, if the conditions (a), (b), (c) satisfy, then k Rl Hmd−k−1 (M/(x1 , , xk )M ) ≤ i=0 k i (Hmi+2 (KM )) for any strict f-sequence x = (x1 , , xd ) of M and any k = 1, , d − The equality holds true whenever x1 , , xk ∈ m2 k−1 q , where q = min{t ∈ N | mt Hmi (KM ) = for all i < d} The following theorem, which is the main result of Chapter and the first result of the thesis as well, gives some characterizations of canonical generalized Cohen-Macaulay modules through strict f-sequence systems of parameters and unconditioned canonical system of parameters This theorem is a significant improvement for the result of N T H Loan and L T Nhan (Lemma 2.2.3) Moreover, it is also a version for canonical generalized Cohen-Macaulay modules similar to the well-known parametric characterizations of generalized CohenMacaulay modules Theorem 2.2.4 Let d ≥ The following four statements are equivalent: (a) M is canonical generalized Cohen-Macaulay; 10 (b) There exists an integer number cM such that Rl Hm2 (M/(x1 , , xd−3 )M ) ≤ cM for all strict f-sequence (x1 , , xd ) of M ; (c) There exist a strict f-sequence (x1 , , xd ) of M such that n d−3 Rl Hm2 (M/(xn1 , , xd−3 )M ) < ∞; sup n1 , ,nd−3 ∈N (d) There is an unconditioned canonical s.o.p of M Furthermore, if (x1 , , xd ) is an unconditioned canonical s.o.p of M , then d−3 Rl Hm2 (M/(x1 , , xd−3 )M ) = i=0 d−3 i (Hmi+2 (KM )) To prove Theorem 2.2.4, we need some auxiliary lemmas Lemma 2.2.5 Let d ≥ and x ∈ m be a strict f-element of M Then (K ) R (0 :Hm M x) < ∞ and Rl(Hmd−2 (M/xM )) = 2 R (Hm (KM )/xHm (KM )) + (K ) R (0 :Hm M x) The following results on the increasing properties of the residual length function plays an important role in proving Theorem 2.2.4 Lemma 2.2.6 Let d ≥ and (x1 , , xd ) be an unconditioned strict f-sequence of M Then mk Rl(Hmd−k−1 (M/(xn1 , , xnk k )M )) ≤ Rl Hmd−k−1 (M/(xm , , xk )M ) for all integer ≤ k ≤ d − and all positive integer ni ≤ mi for i = 1, , k Let A be an Artinian R-module Set dimR A = t A system (x1 , , xt ) of elements in m is called a system of parameters of A if R (0 :A (x1 , , xt )) < ∞ The following property of Artinian modules will be used in the proof of some next results of this section Lemma 2.2.7 Let A be an Artinian R-module If dimR A > and x is a parameter of A, then for all positive integer n we have (0 :A xn ) = (0 :A xn+1 ) 11 Here is a corollary used in the first induction step to prove Theorem 2.2.4 Corollary 2.2.8 Let d ≥ and x ∈ m be a strict f-element of M such that Rl Hmd−2 (M/xM ) = Rl Hmd−2 (M/x2 M ) Then R (Hm (KM )) < ∞, xHmi (KM ) = for all i ≤ 3, and Rl(Hmd−2 (M/xM )) = Rl(Hmd−2 (M/xn M )) = R (Hm (KM )) + R (Hm (KM )) for all n > In particular, if d = 4, then M is canonical generalized CohenMacaulay Specially, the following lemma is an important step in the proof of Theorem 2.2.4 Lemma 2.2.9 Suppose that d ≥ Let x = (x1 , , xk ) be a strict f-sequence of M , where ≤ k ≤ d − Then, there exists a positive integer m(x) such that m(x) Rl Hmd−k (M/(x1 , , xk−1 )M ) ≤ Rl Hmd−k−1 (M/(x1 , , xk−1 , xk )M ) 12 Chapter Non Cohen-Macaulay locus of canonical modules Non Cohen-Macaulay locus of M , denoted by nCM(M ), is the set of all prime ideals p of R satisfying the condition that Mp is not Cohen-Macaulay In general, nCM(M ) is not necessarily a closed subset of Spec(R) under Zariski topology However, nCM(M ) is closed if R is a quotient of a Gorenstein local ring When nCM(M ) is closed, we can define its dimension, dim nCM(M ) If M is Cohen-Macaulay, then nCM(M ) = ∅, and in this case we stipulate that dim nCM(M ) = −1 Throughout the Chapter 3, let (R, m) be a Noetherian local ring and a quotient of a Gorenstein local ring Let M be a finitely generated R-module of dimension d The purpose of this chapter is to study the dimension of non Cohen-Macaulay locus of module M , the dimension of non Cohen-Macaulay locus of canonical module KM and the relation between them 3.1 Some properties via flat extensions The following proposition will be applied to the proof of the main result of Chapter 3, in which we show the dimensional properties of local cohomology modules and the dimension of the non Cohen-Macaulay locus of the module under action of flat extension Proposition 3.1.4 Let f : (R, m) → (S, n) be a flat local homomorphism between Noetherian local rings such that S/mS is Cohen-Macaulay of dimension 13 t If M is not Cohen-Macaulay, then max dimS Hni (M ⊗R S) = dim(S/mS) + max dimR Hmi (M ) i and h0 , , hn−1 ≥ be integers Then there is a Buchsbaum local ring (R, m) such that dim R = n, dimR/m Hmi (R) = hi for every ≤ i ≤ n − Moreover, if h0 = 0, then R is an integral domain Another important tool needed here is the notion of idealization introduced by M Nagata We can make R × M into a ring with respect to the componentwise addition and the multiplication defined by (r1 , m1 ) + (r2 , m2 ) = (r1 + r2 , m1 + m2 ); (r1 , m1 )(r2 , m2 ) = (r1 r2 , r1 m2 + r2 m1 ) This ring is called the idealization of M over R, and denoted by R Note that R M M is a commutative Noetherian local ring with the identity (1, 0) The unique maximal ideal of R M is m × M Since KR , KM , KR M are equidimensional, we have the following lemma Lemma 3.2.3 The following statements are true (a) If dim M < dim R, then dim nCM(KR (b) If dim M = dim R, then dim nCM(KR M) M) = dim nCM(KR ) = max{dim nCM(KR ), dim nCM(KM )} 15 Chapter Local cohomology modules via certain flat extension rings and non Cohen-Macaulay in dimension > s locus Let ϕ : (S, n) → (S , n ) be a flat local homomorphism of Noetherian local rings For each finitely generated S-module L, we have the relations between the set of associated primes of S -module L ⊗S S and that of S-module L as follows AssS (L ⊗S S ) = Ass(S /sS ); (4.1) s∈AssS L −1 AssS L = {ϕ (S) | S ∈ AssS (L ⊗S S )} (4.2) Moreover, for given set of associated primes of a finitely generated module, we can compute its dimension and multiplicity through the associativity formula for multiplicity On the other hand, for each integer i ≥ and r = dim(S /nS ), the local cohomology modules Hni+r (L ⊗S S ) is Artinian S -module and Hni (L) is Artinian S-module In addition, the set of attached primes defined by I G Macdonald for Artinian modules makes an important role similarly to that of the set of associated primes for finitely generated modules Therefore, it is natural to ask about the relation between the set of attached primes of Hni (L) and Hni+r (L ⊗S S )? May we compute dimension and multiplicity of Hni+r (L ⊗S S ) through that of Hni (L)? The first aim of Chapter is giving the answer for the above question in the case ϕ : Rp → RP , where P ∈ Spec(R), p = P ∩ R and rP = dim(RP /pRP ) Concretely, we construct two formulas of the relation i+r i between the sets of attached primes of HpR (Mp ) and that of HPR P (MP ), with p P ∼ Mp ⊗R RP = MP Then, we give the relation between the dimension, mulp 16 tiplicity of these Artinian local cohomology modules The second aim of this i chapter is applying the results about sets of attached primes of HpR (Mp ) and p i+r that of HPR P (MP ) to study Cohen-Macaulayness, Cohen-Macaulayness in diP mension > s via homomorphism ϕ Then, we give the relative formula between dimensions of non Cohen-Macaulay in dimension > s of Mp and that of MP 4.1 Attached primes of local cohomology modules via certain flat extension The purpose of this section is to construct the formulas of the relation between the two sets of attached primes of local cohomology modules via certain flat local homomorphism ϕ : Rp → RP , which is similar to the formulas (4.1), (4.2) of finitely generated modules Note that if rP = 0, then by Flat Base i+r change Theorem, we have isomorphism H i (Mp ) ⊗R RP ∼ = H P (MP ) Let pRp rP > and i HpR (Mp ) p = Then i HpR (Mp ) p p PRP ⊗Rp RP is not Artinian RP -module because i dim SuppRP HpR (Mp ) ⊗Rp RP = rP > p i+r i Therefore, in this case, HpR (Mp ) ⊗Rp RP ∼ = HPR P (MP ) However, we have the p P following isomorphism Lemma 4.1.1 Let P ∈ Spec(R) with p = P ∩ R If R is a quotient of a Cohen-Macaulay local ring, then r HPi R (MP ) P ∼ = i−r HPPR HpRpP (Mp ) ⊗Rp RP if i ≥ rP if i < rP P i−r Moreover, HPi R (MP ) = if and only if HpRpP (Mp ) = for all i ≥ rP P To accomplish the above goal, we need the following Shifted localization Principle and Shifted completion Principle for attached primes of local cohomology modules, which were proved by L T Nhan and P H Quy in 2014 Lemma 4.1.2 Let p ∈ Spec(R) and i ≥ be an integer Suppose that R is a quotient of a Cohen-Macaulay local ring Then i−dim(R/p) (a) AttRp HpRp (Mp ) = qRp | q ∈ AttR (Hmi (M )), q ⊆ p ; (b) AttR (Hmi (M )) = AssR (R/pR) i (M )) p∈AttR (Hm 17 The following theorem, which is the first main result of Chapter 4, gives i some relations between the set of attached primes of HpR (Mp ) and that of p i+r HPR P (MP ) P Theorem 4.1.3 Let R is a quotient of a Cohen-Macaulay local ring Let P ∈ Spec(R) and p = P ∩ R Set rP = dim RP /pRP Then for any integer i ≥ 0, we have i+r i (a) AttRp HpR (Mp ) = QRP ∩ Rp | QRP ∈ AttRP HPR P (MP ) p P (b) AttRP i+r HPR P (MP ) P = Ass RP /qRP i qRp ∈AttRp HpR (Mp ) p (c) For any Q ∈ Spec(R) with Q ⊆ P and q = Q ∩ R, we have QRP ∈ i+r i AttRP HPR P (MP ) if and only if qRp ∈ AttRp HpR (Mp ) and Q ∈ Var(qR) p P 4.2 Dimension and multiplicity via certain flat extension Following I G Macdonald, for each Artinian R-module A, we have AttR A = Var(AnnR A) So, the dimension of Artinian module can be computed as the maximum of the dimensions of its attached primes Therefore, we can use Theorem 4.1.3 to i+r i compare dimension of HpR (Mp ) and that of HPR P (MP ) The following theorem p P is the second main result of Chapter Theorem 4.2.1 Let R is a quotient of a Cohen-Macaulay local ring Let P ∈ Spec(R) with p = P ∩ R Set rP = dim RP /pRP Then for any integer i ≥ 0, we have i+r i dimRP HPR P (MP ) = dimRp HpR (Mp ) + rP p P Next, we use Theorem 4.1.3 and the associativity formula of multiplicity for Artinian modules of M Brodmann and R Y Sharp to give the relation i+r i between multiplicity of HpR (Mp ) and that of HPR P (MP ) p P Theorem 4.2.3 Let R is a quotient of a Cohen-Macaulay local ring Let aRp be an pRp -primary ideal of Rp Let ARP be an ideal of RP such that 18 RP /(ARP + pRP ) is of finite length Set JRP = aRP + ARP Then JRP is an PRP -primary ideal of RP and i+r i e (JRP , HPR P (MP )) = e (aRp , HpR (Mp )).e(ARP , RP /pRP ) p P Note that the statements in Theorems 4.1.3, 4.2.1, 4.2.3 are not valid if we remove the assumption that R is a quotient of a Cohen-Macaulay local ring 4.3 Non Cohen-Macaulay in dimension > s via certain flat extension Let s ≥ −1 be an integer The notion of M -sequence in dimension > s, which was introduced by M Brodmann and L T Nhan is a generalization of the well-known concept of M -sequence and the notion of Cohen-Macaulay module in dimension > s defined in an obvious way by N Zamani is a generalization of the concept of Cohen-Macaulay module The purpose of this section is studying Cohen-Macaulayness, Cohen-Macaulayness in dimension > s and non Cohen-Macaulay loci, non Cohen-Macaulay in dimension > s loci via certain flat homomorphism ϕ : Rp → RP Firstly, we recall the notions of regular sequence in dimension > s and Cohen-Macaulay modules in dimension > s Definition 4.3.1 An element x ∈ m is called M -regular in dimension > s if x∈ / p for all p ∈ AssR M satisfying dim(R/p) > s A sequence x1 , , xt ∈ m is said to be an M -sequence in dimension > s if xi is M/(x1 , , xi−1 )M -regular in dimension > s for all i = 1, , t We say that M is a Cohen-Macaulay module in dimension > s if every system of parameters of M is an M -sequence in dimension > s It is clear that a regular M -sequence in dimension > −1 is a regular M -sequence and a regular M -sequence in dimension > is a filter regular sequence of M So, Cohen-Macaulay modules in dimension > −1 are CohenMacaulay modules If R is a quotient of a Cohen-Macaulay local ring, then Cohen-Macaulay modules in dimension > is generalized Cohen-Macaulay modules Moreover, if M is Cohen-Macaulay in dimension > s then so is M/xM for any parameter x of M 19 The following lemma gives a homological characterization for CohenMacaulay modules in dimension > s Lemma 4.3.3 Let R be a quotient of a Cohen-Macaulay local ring Then M is Cohen-Macaulay in dimension > s if and only if dimR Hmi (M ) ≤ s for all integers i < dimR M The following theorem is the next result of Chapter 4, where we show Cohen-Macaulayness and Cohen-Macaulayness in dimension > s under the impact of ϕ : Rp → RP Theorem 4.3.4 Let R be a quotient of a Cohen-Macaulay local ring Let P ∈ Spec(R) and p = P ∩ R Set rP = dim(RP /pRP ) Let s ≥ be an integer Then (a) Mp is Cohen-Macaulay if and only if so is MP (b) Mp is Cohen-Macaulay in dimension > s if and only if MP is CohenMacaulay in dimension > s + rP Next, we study the dimension of non Cohen-Macaulay in dimension > s via certain flat local homomorphism Definition 4.3.6 Let s ≥ −1 be an integer The non Cohen-Macaulay locus in dimension > s of M , which is denoted by nCM>s (M ), is defined as follows nCM>s (M ) = {p ∈ Spec(R) | Mp is not Cohen-Macaulay in dimension > s} Note that if s = −1 then nCM>−1 (M ) = nCM(M ) is the non CohenMacaulay locus of M If R is a quotient of a Cohen-Macaulay local ring, then nCM(M ) is closed under Zariski topology In case where s ≥ 0, the locus nCM>s (M ) is not closed in general even R is complete However, nCM>s (M ) is always stable under specialization So we can define its dimension dim nCM>s (M ) = max{dim R/p | p ∈ nCM>s (M )} The folowing theorem is the last main result of the thesis We give the relation between dim nCM>s (Mp ) and dim nCM>s (MP ) Theorem 4.3.7 Let s ≥ −1 be an integer, R be a quotient of a CohenMacaulay local ring Let P ∈ Spec(R) and p = P ∩ R Set rP = dim(RP /pRP ) 20 Then (a) nCM>s (Mp ) = ∅ if and only if dim nCM>s (MP ) ≥ rP (b) If nCM>s (Mp ) = ∅, then dim nCM>s (MP ) = dim nCM>s (Mp ) + rP 21 CONCLUSIONS OF THESIS In this thesis, we have obtained the following results: We introduce the notion of canonical system of parameter of finitely generated module M , and show the relation between the standard system of parameter and canonical system of parameter of M We establish the characterizations of canonical generalized Cohen-Macaulay modules through the strict f-sequence system of parameter and unconditioned canonical system of parameter of M We give the relation between the dimension of non Cohen-Macaulay locus of M and that of KM Specially, we show that, except the inclusion relation nCM(KM ) ⊆ nCM(M ), these two loci are almost independent of each other We clarify the relations between attached primes, dimension and multiplicity i+r i of Artinian local cohomology modules HpR (Mp ) and that of HPR P (MP ) p P We show a new result on passing the Cohen-Macaulayness and CohenMacaulayness in dimension > s via certain flat local homomorphism Finally, we give a connection between the dimension of non Cohen-Macaulay locus in dimension > s of MP and that of Mp 22 The results of the thesis were reported at the Seminars of Algebra and Number Theory group, Thai Nguyˆen University, and at many Conferences such as: Vietnam-Japan Association Conference, Thai Nguyen (01/2017); International Commutative Algebra Conference, Ho Chi Minh city (9/2017); Workshop ”Some selected problems in local algebra”, Ha Long - Quang Ninh (12/2017); 9th National Mathematical Conference, Nha Trang - Khanh Hoa (8/2018); PhD.student of Major Algebra and Number Theory Conference, College of Science - Thai Nguyˆen University (01/2019); Workshop ”Modules on commutative rings and applications”, Tuan Chau - Quang Ninh (5/2019) The published papers related to the thesis T N An, L T Nhan and L P Thao, ”Non Cohen-Macaulay locus of canonical modules” J Algebra, 525 (2019), 435-453 L T Nhan, L P Thao and T N An, ”Local cohomology modules via certain flat extension rings”, J Algebra, 503 (2018), 340-355 L P Thao, ”Non Cohen-Macaulay in dimension more than s locus”, Journal of Science and Technololy - TNU, 192(16) (2018), 23-28  ... is Cohen- Macaulay module if and only if it is Cohen- Macaulay in dimension > −1 In the case R is a quotient of a Cohen- Macaulay ring, M is generalized Cohen- Macaulay if and only if M is Cohen- Macaulay. .. So, Cohen- Macaulay modules in dimension > −1 are CohenMacaulay modules If R is a quotient of a Cohen- Macaulay local ring, then Cohen- Macaulay modules in dimension > is generalized Cohen- Macaulay. .. generalized Cohen- Macaulay, then M is canonical generalized Cohen- Macaulay The thesis studies the class of canonical generalized Cohen- Macaulay modules and some non Cohen- Macaulay loci on Noetherian