LUU PHUONG THAOON CANONICAL GENERALIZED COHEN-MACAULAYMODULES AND SOME NON-COHEN-MACAULAY LOCI OVER NOETHERIAN LOCAL RINGS SUMMARY OF MATHEMATICS DOCTOR THESIS THAI NGUYEN - 2020... THAI
Trang 1LUU PHUONG THAO
ON CANONICAL GENERALIZED COHEN-MACAULAYMODULES AND SOME NON-COHEN-MACAULAY LOCI OVER
NOETHERIAN LOCAL RINGS
SUMMARY OF MATHEMATICS DOCTOR THESIS
THAI NGUYEN - 2020
Trang 2THAI NGUYEN UNIVERSITY
LUU PHUONG THAO
ON CANONICAL GENERALIZED COHEN-MACAULAYMODULES AND SOME NON-COHEN-MACAULAY LOCI OVER
NOETHERIAN LOCAL RINGS
Major: Algebra and number theory
Trang 3a Cohen-Macaulay ring The class of Cohen-Macaulay modules and its alizations have attracted many researchers all over the world The structure
gener-of these modules is characterized via most well-known theories gener-of commutativealgebra such as multiplicity, local cohomology, localization, completion, etc.These modules are studied in many different branches of mathematics includ-ing homology algebra, invariant theory, combinatory and algebraic geometry
The thesis related to two directions of generalization of the class of Macaulay modules as follows The first generalization is based on the differenceI(x; M ) between the length `(M/xM ) and the multiplicity e(x; M ) with respect
Cohen-to a system of parameters x of M Note that M is Cohen-Macaulay if and only ifI(x; M ) = 0 for a (for all) system of parameters x Therefore, a conjecture wasgiven 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 negativeanswer for this conjecture was given by W Vogel and J St¨uckrad in 1973,and they studied the class of rings and modules satisfying the conditions ofthis 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 ) < ∞, wherethe supremum runs over every parameters system x of M , and they calledthem generalized Cohen-Macaulay modules Nowadays, the notion Buchsbaummodule and generalized Cohen-Macaulay module have become well-known inCommutative Algebra Continue on generalizing in this direction, we gain aclass 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
Trang 4of 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 generalizedCohen-Macaulay if and only if M is Cohen-Macaulay in dimension > 0
The second generalization of the class of Cohen-Macaulay modules isbased on structure of the canonical modules, in the case R is a homomorphicimage of a Gorenstein local ring (R0, m0) of dimension n0 For each integer i ≥ 0,set KMi := ExtnR00−i(M, R0) Then KMi is finitely generated R-module and it iscalled i-th deficiency module of M When i = d, we denote KMd by KM andcall 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 modulethen so is KM Thus, the class of canonical Cohen-Macaulay modules is ageneralization of the class of Cohen-Macaulay modules
The notions of canonical Cohen-Macaulay rings and modules was nated from following problem: Let (R, m) be an integral domain and Q(R) thequotient ring of R A natural question is whether exists or not an intermedi-ate R ⊆ B ⊆ Q(R) such that B is a finitely generated R-module and B is aCohen-Macaulay ring? A such ring B (if exist) is called a birational Macaulay-fication of R This is an important problem in Commutative Algebra In 2004,
origi-P Schenzel proved that an integral domain local Noetherian R have a tional Macaulayfication if and only if R is canonical Cohen-Macaulay ring In
bira-2006, L T Nhan gave a characterization for canonical Cohen-Macaulay ules through the vanishing of residual lengths of local cohomology modules withrespect to strict f-sequence systems of parameters, which was introduced by N
mod-T Cuong, M Morales, L mod-T Nhan In 2012, M Brodmann and L mod-T Nhanshowed that, in the case d ≥ 4 and x is a strict f-element of parameter, M iscanonical 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 canonicalgeneralized Cohen-Macaulay modules, which are modules M such that theircanonical modules are generalized Cohen-Macaulay They characterized thisclass of modules through the existence of a uniform bound for residual lengths
of local cohomology modules with respect to strict f-sequence systems of
Trang 5pa-generalized Cohen-Macaulay.
The thesis studies the class of canonical generalized Cohen-Macaulaymodules and some non Cohen-Macaulay loci on Noetherian local ring Thefirst purpose of the thesis is characterizing structure of class of canonical gener-alized Cohen-Macaulay modules when R is a homomorphic image of a Goren-stein local ring The second purpose is clarifying the relative between the nonCohen-Macaulay locus of KM and that of M The third purpose is studying set
of attached primes, dimension and multiplicity of Artinian local cohomologymodules via certain flat Rp → bRP, where P ∈ Spec( bR), p = P ∩ R and R isarbitrary, not necessarily a factor of a Gorenstein ring, upon which we give theformula to compute dimension of non Cohen-Macaulay in dimension > s
About the research method, to characterize the class of canonical eralized Cohen-Macaulay modules, we exploit specific properties of Artinianlocal cohomology modules and use the strict f-sequence parameters systemflexibly On the relation between two non Cohen-Macaulay loci nCM(KM)and nCM(M ), we need the Structure of Buchsbaum ring Theorem proved by
gen-S Goto in 1980, the Theorem of Structure of canonical modules via flat momorphism proved by Y Aoyama and S Goto in 1985, and the formula ofdimension of deficiency module under the action of the formal power series ex-tension To study local cohomology modules under the impact of certain flathomomorphism Rp → bRP, we apply effectively the Shifted localization Princi-ple and Shifted completion Principle for attached primes of local cohomologymodules given by L T Nhan and P H Quy in 2014, and the associativityformula for multiplicity of local cohomology modules given by M Brodmannand R Y Sharp in 2002
ho-In addition to the introduction, conclusions and references, the thesisconsists of 4 chapters
Chapter 1 recalls some fundamental knowlege to serve the next ters, including characterizations of Cohen-Macaulay modules and generalizedCohen-Macaulay modules; set of attached primes, dimension and multiplicity
chap-of Artinian modules; canonical modules and deficiency modules
Trang 6In Chapter 2, we introduce the notion of canonical system of ters of finitely generated module M , establish the relation between the stan-dard system of parameters and canonical system of parameters of M We givesome the characterizations of canonical generalized Cohen-Macaulay modulesthrough canonical system of parameters and improve the previous results aboutstructure of canonical generalized Cohen-Macaulay modules.
parame-In Chapter 3, we give the relation between the dimension of non Macaulay locus of M and that of KM Specially, we show that, except theinclusion relation nCM(KM) ⊆ nCM(M ), these two loci are almost independent
Cohen-of each other
In Chapter 4, we clarify the change of the set of attached primes, mension and multiplicity of Artinian local cohomology modules via certain flatextension Rp → bRP, where P ∈ Spec( bR) and p = P∩R Based on these results,
di-we establish the formula of computing the dimension of non Cohen-Macaulay
in dimension > s locus
Trang 7mod-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 andmultiplicity of Artinian modules
1.3 Canonical modules and deficiency modules
We recall the notions and some properties of canonical modules and ficiency modules that will be used in the sequel
Trang 8de-Chapter 2
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
Cohen-Macaulay modules, which are modules M such that KM are generalized
Cohen-Macaulay
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
Cohen-Macaulay modules as follows: M is canonical Cohen-Cohen-Macaulay if and only if
Rl Hm2(M/(x1, , xd−3)M ) = 0 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
no-tion of strict f-sequence was introduced by N T Cuong, M Morales and L T
Nhan in 2004
The purpose of Chapter 2 is to establish a version for canonical generalized
Cohen-Macaulay modules, that is similar to the well-known parametric
char-acterizations 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
Trang 92.1 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 strictf-sequence of M if xj+1 ∈ p for all prime ideals/
i ≥ 0
(b) If (x1, , xt) ∈ m is a strict f-sequence of M , then so is (xn1
1 , , xnt
t ) forall 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 mnA = msA fof all n ≥ s Set Rl(A) := `R(A/ms(A)A) Then, Rl(A)
is finite and it is called the residual length of A
Remark 2.1.4 (i) Rl(A) = 0 if and only if m /∈ AttRA
(ii)If x /∈ p for all p ∈ AttRA \ {m}, then `R(A/xA) ≤ Rl(A) In this case
`R(A/xnA) = Rl(A) for all n ≥ s(A)
(iii) If `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 ≥ 0 be an integer
Trang 10The following statements are true.
(a) There exists an integer n0 ≥ 0 such that
Rl(Hmi (M )) = `R(Hm0(KMi )) = `R(0 :Ki
M xn)for all n ≥ n0
(b) There is an exact sequence
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 acanonical 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 ical generalized Cohen-Macaulay modules They also characterized this classmodules through the existence of a uniform bound for residual lengths of Ar-tinian local cohomology modules with respect to strict f-sequence systems of
Trang 11canon-Definition 2.2.1 M is said to be a canonical generalized Cohen-Macaulaymodule 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
1 , , xnk
k )M ) ≤ c(x, M) for all k = 1, , d−
3 and all positive integer n1, , nk
Furthermore, if the conditions (a), (b), (c) satisfy, then
`(Hmi+2(KM))
for any strict f-sequence x = (x1, , xd) of M and any k = 1, , d − 3 Theequality holds true whenever x1, , xk ∈ m2 k−1 q, where
q = min{t ∈ N | mtHmi(KM) = 0 for all i < d}
The following theorem, which is the main result of Chapter 2 and the firstresult of the thesis as well, gives some characterizations of canonical generalizedCohen-Macaulay modules through strict f-sequence systems of parameters andunconditioned canonical system of parameters This theorem is a significantimprovement 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 modulessimilar to the well-known parametric characterizations of generalized Cohen-Macaulay modules
Theorem 2.2.4 Let d ≥ 3 The following four statements are equivalent:(a) M is canonical generalized Cohen-Macaulay;
Trang 12(b) There exists an integer number cM such that
Rl Hm2(M/(x1, , xd−3)M ) ≤ cMfor all strict f-sequence (x1, , xd) of M ;
(c) There exist a strict f-sequence (x1, , xd) of M such that
(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
`(Hmi+2(KM))
To prove Theorem 2.2.4, we need some auxiliary lemmas
Lemma 2.2.5 Let d ≥ 4 and x ∈ m be a strict f-element of M Then
Let A be an Artinian R-module Set dim
b
RA = t A system (x1, , xt) ofelements 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 somenext results of this section
Lemma 2.2.7 Let A be an Artinian R-module If dim
b
RA > 0 and x is aparameter of A, then for all positive integer n we have
(0 :A xn) 6= (0 :A xn+1)
Trang 13Corollary 2.2.8 Let d ≥ 4 and x ∈ m be a strict f-element of M such that
Rl Hmd−2(M/xM ) = Rl Hd−2
m (M/x2M )
Then `R(Hm3(KM)) < ∞, xHmi (KM) = 0 for all i ≤ 3, and
Rl(Hmd−2(M/xM )) = Rl(Hmd−2(M/xnM )) = `R(Hm2(KM)) + `R(Hm3(KM))for all n > 0 In particular, if d = 4, then M is canonical generalized Cohen-Macaulay
Specially, the following lemma is an important step in the proof of orem 2.2.4
The-Lemma 2.2.9 Suppose that d ≥ 4 Let x = (x1, , xk) be a strict f-sequence
of M , where 1 ≤ k ≤ d − 3 Then, there exists a positive integer m(x) such that
Rl Hmd−k(M/(x1, , xk−1)M ) ≤ Rl Hd−k−1
m (M/(x1, , xk−1, xm(x)k )M )
Trang 14M is Cohen-Macaulay, then nCM(M ) = ∅, and in this case we stipulate thatdim nCM(M ) = −1.
Throughout the Chapter 3, let (R, m) be a Noetherian local ring and aquotient 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 nonCohen-Macaulay locus of module M , the dimension of non Cohen-Macaulaylocus 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 cohomologymodules and the dimension of the non Cohen-Macaulay locus of the moduleunder action of flat extension
Proposition 3.1.4 Let f : (R, m) → (S, n) be a flat local homomorphismbetween Noetherian local rings such that S/mS is Cohen-Macaulay of dimension