Vietnam Journal of Mathematics 34:4 (2006) 449–458
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A B low ing-up C haract erizat ion of P seudo
B uchsbaum M odules
N guye n Tu C uon g1andNguyenThiHongLoan
2
1Institute of Mathem atics,18 Hoang Quoc V iet, 10307 Hanoi, V ietnam
2Department of Mathematics, V inh University
182 Le Duan Street, V inh City, V ietnam
Dedicated t o P rofessor Do Long Van on the occasion of his 65t h birt hday
Received J une 22, 2005
A bst r act . Let (A , m)b e a com mut at ive Noet herian local ring and Ma finit ely
generat ed A-module. T he aim of t his paper is t o give a blow-up charact erizat ion of
pseudo Buchsbaum modules defined in [2], which says t hat Mis a pseudo Buchsbaum
module if and only if t he Rees module Rq(M)is pseudo Buchsbaum for all paramet er
ideals qof M. We also show t hat t he associat ed graded m odule Gq(M)is pseudo
Cohen Macaulay (resp. pseudo Buchsbaum) provided Mis pseudo C ohen Macaulay
(resp. pseudo Buchsbaum).
2000 Mat hemat ics Sub ject Classificat ion: 13H10, 13A30.
Keywords: P seudo Cohen-Macaulay module, pseudo Buchsbaum module, Rees module,
associat e graded module.
1. Int ro du ct ion
Let Abe a commut ative Noetherian local ring with t he maximal ideal m,M a
finit ely generat ed A-module with dim M=d> 0.Let x= (x1,... ,xd)bea
syst em of paramet ers of A-module M . We consider t he difference between t he
multiplicity and the length
JM(x)= e(x;M)−ℓ(M / Q M(x)),
where QM(x)=
t > 0
((xt+ 1
1,... ,xt+ 1
d)M:xt
1...xt
d) is a submodule of M . It
should be mentioned t hat JM(x) gives a lot of informations on t he structure of M .
450 Nguyen Tu Cuong and Nguyen Thi Hong Loan
For ex a m p le, if Mis a Cohen–Macaulay module then QM(x)= (x1,... ,xd)M
by [7]. T herefore JM(x) = 0 for all system of parameters xof M.Further,
we have known that ℓ(M / Q M(x)) is just t he lengt h of generalized fract ion (see
[10]). Therefore by [10], sup
x
JM(x)<∞if Mis a generalized Cohen-Macaulay
module. In [1] we also showed t hat if Mis a Buchsbaum module then, JM(x)
t akes a const ant value for every syst em of para met ers xof M . Unfort unately,
the converses of all above statement s are not true in general. The structure
of modules Msat isfying JM(x)= 0orsup
x
JM(x)<∞was st udied in [5] and
such modules were called pseudo Cohen-Macaulay modules or pseudo generalized
Cohen-Macaulay modules, resp ect ively. In [2] we st udied t he st ruct ure of mod-
ules Mhaving JM(x) a constant value for all systems of parameters. We called
it pseudo Buchsbaum modules. Not e t hat pseudo Cohen Macaulay (resp. pseudo
Buchsbaum, pseudo generalized Cohen Macaulay) modules st ill have many nice
properties and t hey are relat ively closed t o Cohen Macaulay (resp. Buchsbaum,
generalized Cohen Macaulay) modules.
For a paramet er ideal qof Mwe set Rq(M)= ⊕
i≥0qiM T it he Rees module and
Gq(M)= ⊕
i≥0qiM / qi+ 1 Mthe associat ed graded module of Mwit h respect t o q.
Let M=m⊕ ⊕
i≥1
qiTibe the unique homogeneous maximal ideal of Rq(A).Then
Rq(M)orGq(M) is called a pseudo Cohen Macaulay (resp. pseudo Buchsbaum)
module if and only if Rq(M)Mor Gq(M)Mis a pseudo Cohen Macaulay (resp.
pseudo Buchsbaum) module. T he purpose of this paper is to prove t he following
result .
T heorem 1. Let Abe a comm utative Noetherian local ring and Ma finitely
generated A-module. T hen the following statements are true.
(i) Mis a pseudo Buchsbaum m odule if and only if Rq(M)isapseudoBuchs-
baum m odule for all parameter ideals qof M .
(ii) Let Mbe a pseudo Cohen Macaulay (resp. pseudo Buchsbaum ) m odule.
T hen Gq(M)is a pseudo Cohen Macaulay (resp. pseudo Buchsbaum )mod-
ule for all parameter ideals qof M .
It should be noted t hat an analogous result of t he first statement in the
above theorem for Buchsbaum modules was only proved under t he assumpt ion
that dept h M > 0 (see [11, T heorem 3.3, Chap. IV]).
The paper is divided into 4 sections. In Sec. 2, we outline some properties
of pseudo Cohen Macaulay (resp. pseudo Buchsbaum) modules over local ring
which will b e needed lat er. T he proof of T heorem 1 is given in Sec. 3. As
consequences of T heorem 1 we will show in t he last sect ion t hat t he R ees module
Rq(M) and t he associat ed graded module Gq(M) are always locally pseudo
Cohen-Macaylay if Mis a pseudo Buchsbaum module.
2. P relim in aries
Let (A , m) be a commutat ive Noetherian local ring and Ma finitely gener-
Blowing-up Characterization of Pseudo Buchsbaum Modules 451
at ed module with dim M=d> 0.Let x= (x1,... ,xd) be a syst em of
paramet ers of Mand n= (n1,... ,nd)ad-t uple of positive integers. Set
x(n)= (xn1
1,... ,xnd
d). T hen t he difference between multiplicities and lengths
JM(x(n)) = n1...nde(x;M)−ℓ(M / Q M(x(n)))
can be considered as a funct ion in n. Not e t hat t his function is non-negat ive
([1, Lemma 3.1]) and ascending, i.e., for n= (n1,... ,nd),m = (m1,... ,md)
wit h ni≥mi,i = 1,... ,d, JM(x(n)) ≥JM(x(m)) ([1, Corollary 4.3]). More-
over, we know t hat ℓ(M / Q M(x(n))) is just the length of generalized fraction
M(1/(xn1
1,... ,xnd
d,1)) defined by Sharp and Hamieh [10]. Therefore, we can
describe Question 1.2 of [10] as follows: is JM(x(n)) a polynomial for large
enough n(n≫0 for short)? A negative answer for t his question is given in [4].
But , t he funct ion JM(x(n)) is bounded above by the polynomial n1...ndJM(x),
and more general, we have the following result.
T heorem 2. [3, Theorem 3.2] T he least degree of all polynomials in nbound-
ing above the function JM(x(n)) is independent of the choice of a system of
param et er s x .
The numerical invariant of Mgiven in the above theorem is called t he
polynom ial type of fractions of Mand denot ed by pf (M) [3, Definit ion 3.3].
For convenience, we stipulat e that t he degree of the zero-polynomial is equal
to − ∞ .
D efi nit ion 1.
(i) [5, Definition 2.2] Mis said to be a pseudo Cohen Macaulay module if
pf (M)= − ∞ .
(ii) [2, Definit ion 3.1] A n A-module Mis called a pseudo B uchsbaum m odule
if there exists a constant Ksuch that JM(x)= Kfor every system of
param et er s xof M .
Ais called a pseudo Cohen Macaulay (resp. pseudo Buchsbaum) ring if it is
a pseudo Cohen Macaulay (resp. pseudo Buchsbaum) module as a module over
itself.
It should be mentioned that every Cohen Macaulay module is pseudo Cohen
Macaulay and t he class of pseudo Buchsbaum modules cont ains t he class of
pseudo Cohen Macaulay modules. In [1] and [2], we showed t hat the class of
pseudo Buchsbaum modules strictly contains the class of Buchsbaum modules,
but it does not cont ain t he class of generalized Cohen Macaulay modules.
Next, we recall characterizations of these modules from [5] and [2].
P rop o sit io n 1. Mis a pseudo Cohen Macaulay (resp. pseudo Buchsbaum ) A-
module if and only if
Mis a pseudo Cohen Macaulay (resp. pseudo Buchsbaum)
A-m odule.
Not e that for an A-module M(Ais not necessarily a local ring) we usually
452 Nguyen Tu Cuong and Nguyen Thi Hong Loan
use in this paper the following not at ions
Assh M={p∈Ass M|dim A / p= dimM}.
Let 0 = ∩
pi∈AssM
N(pi) be a reduced primary decomposit ion of the submodule 0
of M . We put
UM(0) = ∩
pj∈AsshM
N(pj)andM=M / UM(0).
Then UM(0) does not depend on the choice of a primary decomposition of t he
zero-submodule of M . Notice that UM(0) is t he largest submodule of Mof
dimension less than dim Mand Ass M= Assh M,dimM= dimM .
T heorem 3. ([5, Theorem 3.1], [2, Lemma 4.4]) Suppose that Aadm its a
dualizing complex. T hen the following statem ents are true.
(i) M is a pseudo Cohen Macaulay module if and only if Mis a Cohen Macaulay
module.
(ii) Mis a pseudo B uchsbaum A-module if and only if Mis a Buchsbaum A-
module. Moreover, in this case we have
JM(x)=
d−1
i= 1
d−1
i−1
ℓ(Hi
m(M)),
for every system of parameters x= (x1,... ,xd)of M , where Hi
m(M)stands
for the it h local cohomology module of Mwith respect to the m axim al ideal
m.
3. ProofofTheorem 1
Let
ϕ:Rq(M)→Rq(M)andπ:Gq(M)→Gq(M)
be t he canonical epimorphisms, where M=M / UM(0).T hen we have
Ker ϕ=⊕
i≥0
(UM(0) ∩qiM)Tiand Ker π=⊕
i≥0
qiM∩(qi+ 1 M+UM(0))
qi+ 1 M.
To prove T heorem 1 we need some auxiliary lemmata.
Le m m a 1. W ith the sam e notations as above, then we have Ker ϕ=URq(M)(0).
Proof. It is clear t hat Ass Ker ϕ⊆Ass Rq(M).For ea ch p∈Spec Awe
denote p:= ⊕
i≥0(p∩qi)Ti.T a ke a ny P∈Assh Rq(M).T hen t here exist s
p∈Assh Msuch t ha t P=p(see [11, Lemma 1.7 and Lemma 3.1, chap IV]).
Since dim UM(0) <dim M,(UM(0))p= 0. T herefore we have
(Ker ϕ)(p)= (⊕
i≥0(UM(0) ∩qiM)Ti)(p)= 0.
Blowing-up Characterization of Pseudo Buchsbaum Modules 453
Thus (Ker ϕ)(P)= 0.It follows t hat (Ker ϕ)P= 0.T herefore dim Ker ϕ <
dim Rq(M).
Let K=⊕
i≥0
KiTibe a homogeneous submodule of Rq(M)withK⊃Ker ϕ.
Then we have KiTi⊇(UM(0) ∩qiM)Tifor all i≥0andthereexistsj≥0such
that Kj⊃(UM(0) ∩qjM). Since K⊆Rq(M),Kj⊆qjM . Hence Kj⊆UM(0).
Set V=Kj+UM(0). We have V⊃UM(0).T hus dim V= dimM . T herefore
there exist s p∈Assh V∩Assh M . Hence 0 =Vp= (Kj)p⊆K
p,h ⊆K
p.
Thus we get K
p= 0, i.e, p∈Supp K⊆Supp Rq(M).On t he ot her hand
P∈Assh Rq(M) (see [11, Lemma 1.7 and Lemma 3.1, chap IV]). Combining
these facts, we get dim K= dimRq(M) and t herefore Ker ϕis the largest
homogeneous submodule of Rq(M) of dimension less than dim Rq(M).
Moreover, we can choose a reduced primary decomposit ion of the submodule
0inRq(M) such that 0Rq(M)=l
∩
i= 1Qiwit h Qiis the homogeneous primary
submodule of Rq(M) belonging to homogeneous prime Pi(see [9, P rop osit ion
10 B]). T hen URq(M)(0) is a homogeneous submodule of Rq(M).On t he other
hand, URq(M)(0) is t he largest submodule of Rq(M) of dimension less than
dim Rq(M).T herefore Ker ϕ=URq(M)(0).
Let Nbe a submodule of Msuch t hat dim N < dim M . Set M′=M / N . If
qis a pa ramet er idea l of M, then it is clear that qis a paramet er ideal of M′.
But t he converse is not t rue. It mea ns t hat t here exist s a para met er ideal q′of
M′but q′is not a parameter ideal of M . However, we have t he following result.
Le m m a 2. Let q′be a param eter ideal of M′.T hen there exists a parameter
ideal qof Msuch that q+ AnnM′=q′+ AnnM′.In particular we have
Rq(M′)= Rq′(M′)and Gq(M′)= Gq′(M′).
Proof. Let q′b e a pa ramet er ideal of M′and let (x1,... ,xd) b e a syst em of
paramet ers of M′such t ha t q′= (x1,... ,xd)A . T hen t he lemma is proved if we
can show t he exist ence of a syst em of paramet ers (y1,... ,y
d)ofMsuch t hat
(y1,... ,y
d)R+ AnnM′= (x1,... ,xd)R+ AnnM′.
To prove t his we first claim by int roduction on it hat t here exist s a syst em of
paramet ers (y1,... ,y
d)ofMsuch t hat yi=xi+aiwith ai∈(xi+ 1 ,... ,xd)A+
Ann M′for all i= 1,... ,d. In fa ct , since x1is a paramet er element of M′
and Assh M= Assh M′,we have x1is a paramet er elem ent of M . W e ch o ose
y1=x1.Suppose t hat we already have for 1 k < da part of t he syst em of
paramet ers (y1,... ,y
k)ofMas required. We have t o show t hat t here exist s
a paramet er element yk+ 1 of M / (y1,... ,y
k)Msuch t hat yk+ 1 =xk+ 1 +ak+ 1
wit h ak+ 1 ∈(xk+ 2,... ,xd)A+ AnnM′.Let q1= (x1,... ,xd)A+ AnnM′.
Since (x1,... ,xd) is a syst em of paramet ers of M′,we have q1is a m-primary
ideal. T herefore q1⊆pfor all prime ideals pwit h dim A / p>0.It then follows
that
(xk+ 1,... ,xd)A+ AnnM′⊆p