Cohen-Macaulayness of trivial extensions

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by  A. Mahdikhani

COHEN-MACAULAYNESS OF TRIVIAL EXTENSIONS
7 A.MAHDIKHANI,P.SAHANDIANDN.SHIRMOHAMMADI
1
0
2 Abstract. Our goal is to determine when the trivial extensions of commu-
tative rings by modules are Cohen-Macaulay in the sense of Hamilton and
n
Marley. Forthispurpose,weprovideageneralizationoftheconceptofCohen-
a
Macaulayness ofringstomodules.
J

0
3

]
C 1. Introduction
A Throughoutthis paperallringsarecommutative,withidentity,andallmodules
. are unital. The theory of Cohen-Macaulay rings admits a rich theory in commu-
h
tative Noetherian rings. No attempts have been made to develop the concept of
t
a Cohen-Macaulayness to non-Noetherian rings until 1992. Then, Glaz [8] begun an
m
investigation on the notion of Cohen-Macaulayness for non-Noetherian rings and,
[ sheaskedhowonecandefineanon-NoetheriannotionofCohen-Macaulaynesssuch

1 that the definition coincides with the originalone in the Noetherian case,and that
v coherent regular rings are Cohen-Macaulay, see [9, p. 220].
9 More recently, Hamilton and Marley [10] established a definition of Cohen-
8
Macaulaynessfornon-Noetherianrings. Moreprecisely,employingSchenzel’sweakly
4 proregular sequences [14], they used the tool of Cˇech cohomology modules to de-
8
0 fine the notion of parameter sequences. A parameter sequence such that every
. truncation on the right is also a parameter sequence is called a strong parameter
1
sequence. In some sense, this is a generalization of the system of parameters to
0
7 the non-Noetherian case. They then called a ring Cohen-Macaulay if every strong
1 parameter sequence is a regular sequence. They showed that their definition coin-
: cides with the original one in the Noetherian case and that coherent regular rings
v
i are Cohen-Macaulay (in the sense of new definition).
X
Let R be a ring and M be an R-module. In 1955, Nagata construct a ring
r extensionofRcalledthetrivial extension ofRbyM,denotedherebyR⋉M. This
a
ring is of particular importance in commutative algebra (cf. [2] and [5, Theorem
3.3.6]). One of the properties of trivial extension is as follows: if R is Noetherian
local and M is finitely generated, then R⋉M is Cohen-Macaulay if and only if
R is Cohen-Macaulay and M is maximal Cohen-Macaulay, see [2, Corollary 4.14].
Motivatedby this and[10,Example4.3],wewishto investigatewhether the trivial
extension R⋉M is Cohen-Macaulay with the definition of Hamilton and Marley.
The main ingredient in this investigation is to formulate a definition for a module
to be Cohen-Macaulay, which was left unaddressed in [10]. So we find ourselves
forced to extend the notion of Cohen-Macaulay to modules.
2010 Mathematics Subject Classification. Primary13C14,13C15.
Key words and phrases. Cohen-Macaulay module,Non-Noetherianring,parameter sequence,
trivialextension.
1
2 A.MAHDIKHANI,P.SAHANDIANDN.SHIRMOHAMMADI
The outline of the paper is as follows. In Section 2, we recall some essential
definitions and results on which we base our approach. In Section 3, after defining
weakly proregular sequences on modules, we give a characterization of such se-
quences using the vanishing of suitable Cˇech cohomologymodules. We then define
(strong)parametersequencesonmodules. Continually,aftercitingsomeelementary
properties of such sequences, we relate these sequences to system of parameters of
finitelygeneratedmodulesinNoetherianlocalrings. InSection4,wedefineCohen-
Macaulayness of modules over non-Noetherian rings. Among other things, we are
able to establish our main result which says when the trivial extension R⋉M is
Cohen-Macaulay.
2. Preliminaries
Let R be a ring, I be an ideal of R and M be an R-module. Following [5], a
sequence x := x ,...,x R is called a weak M-regular sequence if x is a non-
1 ℓ i
∈
zero-divisor on M/(x1,...,xi−1)M for i = 1,...,ℓ. If, in addition, M = xM, we
6
callx anM-regular sequence. The classical grade ofI onM,denoted grade(I,M),
is defined to be the supremum of the lengths of all weak M-regular sequences
contained in I, see [11].
The polynomial grade of I on M is defined by
p.grade (I,M):= lim grade(IR[t ,...,t ],R[t ,...,t ] M).
R m→∞ 1 m 1 m ⊗R
It follows from [1] and [11] that
p.grade (I,M)=sup grade(IS,S M)S is a faithfully flat R-algebra .
R { ⊗R | }
Let x be an element of R. Let C(x) denote the complex 0 R R
x
→ → →
0 where the differential is the natural localization map. For a sequence x :=
x ,...,x of elements of R, the Cˇech complex C(x) is inductively defined by
1 ℓ
C(x) := C(x1,...,xℓ−1) R C(xℓ). Then we set C(x;M) := C(x) R M. The
ith Cˇech cohomology Hi(M⊗) of M with respect to the sequence x is d⊗efined to be
x
the ith cohomology of C(x;M).
The following summarizes some essential properties of Cˇech cohomology mod-
ules.
In the sequel, for a finite sequence x of elements of the ring R, ℓ(x) denotes the
length of x.
Proposition 2.1. (see [10, Proposition 2.1]) Let R be a ring, x a finite sequence
of elements of R and M an R-module.
(1) If y is a finite sequence of elements of R such that √yR = √xR, then
Hi(M)=Hi(M) for all i.
y ∼ x
(2) (Change of rings) Let f : R S be a ring homomorphism and N an
→
S-module. Then Hi(N)=Hi (N) for all i.
x ∼ f(x)
(3) (Flat base change) Let f : R S be a flat ring homomorphism. Then
→
Hxi(M)⊗RS ∼=Hxi(M ⊗RS)∼=Hfi(x)(M ⊗RS) for all i.
(4) Hxℓ(x)(M)∼=Hxℓ(x)(R)⊗RM.
Applying parts (2) and (4) of Proposition 2.1 to the natural homomorphism
R R/AnnM together with [10, Proposition 2.7] yields the next corollary.
→
Corollary 2.2. Let R be a ring and M be a finitely generated R-module of finite
dimension. Then Hi(M)=0 for all i>dimM.
x
COHEN-MACAULAYNESS OF TRIVIAL EXTENSIONS 3
The ith local cohomology Hi(M) of M with respect to I is defined by
I
Hi(M):=limExti (R/In,M).
I R
−→
InthecasethatRisNoetherian,onehasHi(M)=Hi(M)foralli,whereI :=xR,
I x
see [6, Theorem 5.1.20].
Let p a prime ideal of R. By ht p, we mean the Krull dimension of the R -
M p
module M . Also, for an ideal I of R
p
ht I :=inf ht pp SuppM V(I) .
M M
{ | ∈ ∩ }
3. Weakly proregular and parameter sequences
3.1. Weakly proregular sequences. It is mentioned in [10] that local cohomol-
ogyandCˇechcohomologyarenotingeneralisomorphicovernon-Noetherianrings.
In [14], Schenzel gave necessary and sufficient conditions on a sequence x of a ring
R such that the isomorphism Hi(M) = Hi(M) holds for all i and R-modules M,
I ∼ x
where I := xR. Such sequences are called R-weakly proregular sequences. In the
following, we provide the module theoretic version of this notion.
Let R be a ring and M be an R-module. For x R, we use K•(x,M) to denote
∈
the Koszul complex
x
0 M M 0.
−→ −→ −→
For a sequence x = x1,...,xℓ the Koszul complex K•(x,M) is defined to be the
complexK•(x1,M) R RK•(xℓ,M). TheithhomologyofK•(x,M)isdenoted
⊗ ···⊗
by H (x,M) and called the Koszul homology of the sequence x with coefficients
i
in M. For m ≥ n, there exists a chain map φmn(x,M) : K•(xm,M) → K•(xn,M)
given by
φm(x,M)=φm(x ,M) φm(x ,M),
n n 1 ⊗R···⊗R n l
where, for all x R, φm(x,M) is the chain map
∈ n
// xm // //
0 M M 0
xm−n =
(cid:15)(cid:15) (cid:15)(cid:15)
// xn // //
0 M M 0.
Hence,{K•(xm,M),φmn(x,M)}isaninversesystemofcomplexes. Note,foreachi,
the map φm(x,M) induces a homomorphism of homology modules H (xm,M)
n i i →
H (xn,M). We also denote this induced homomorphism by φm(x,M) . The se-
i n i
quence x = x ,...,x is called M-weakly proregular if, for each n, there exists an
1 ℓ
m nsuchthatthemapφm(x,M) :H (xm,M) H (xn,M)iszeroforalli 1
≥ n i i → i ≥
(see [14]). Note that an element x R is M-weakly proregular if and only if there
∈
exists an n 1 such that (0: xn)=(0: xn+1).
M M
≥
Remark 3.1. Let x be a finite sequence of elements of R.
(1) If x is an M-weakly proregular sequence, then so is any permutation of x.
(2) Any M-regular sequence is M-weakly proregular.
Thefollowingresultprovidesanotherdescriptionofweaklyproregularsequences
usingCˇechcohomology. Itsproofisinspiredbytheproofof[14,Lemma2.4]. Here,
forasequencex=x ,...,x ,weuseHi(x,M)todenotetheithcohomologyofthe
1 ℓ
complexHomR(K•(x),M), whereK•(x):=K•(x,R), andwecallitthe ithKoszul
4 A.MAHDIKHANI,P.SAHANDIANDN.SHIRMOHAMMADI
cohomology of the sequence x with coefficients in M. It follows from [6, Theorem
5.2.5] that Hi(M)=limHi(xn,M).
x ∼
−→
Theorem 3.2. Let x be a finite sequence of elements of R. Then the following
conditions are equivalent:
(1) x is M-weakly proregular.
(2) Hi(Hom (M,E))=0 for all injective R-modules E and i=0.
x R 6
Proof. AssumethatxisM-weaklyproregularandthatE isaninjectiveR-module.
Then HomR(K•(xn),HomR(M,E))∼=HomR(K•(xn)⊗RM,E). So
Hi(xn,HomR(M,E))=Hi(HomR(K•(xn),HomR(M,E)))
∼=Hi(HomR(K•(xn)⊗RM,E))
∼=HomR(Hi(xn,M),E)
for all i. Hence
limHi(xn,HomR(M,E))∼=limHomR(Hi(xn,M),E).
By assumption−,→for all n N, the homomo−r→phism
∈
Hom (H (xn,M),E) Hom (H (xm,M),E)
R i R i
−→
is zero, for some m n. Therefore
≥
limHi(xn,Hom (M,E))=limHom (H (xn,M),E)=0.
R R i
−→ −→
So that Hi(Hom (M,E))=0 for i=0.
x R 6
Conversely,assumethatHi(Hom (M,E))=0forallinjectiveR-modulesEand
x R
i = 0. Let f : H (xn,M) E denote an injection of H (xn,M) into an injective
i i
6 →
R-module E. Then f ∈HomR(Hi(xn,M),E)∼=Hi(xn,HomR(M,E)). Since
limHi(xn,Hom (M,E))=0,
R
then there exists an m −→n such that Hom (φm(x,M),1 ) = 0 which means
1 fφm(x,M)=0. So φm≥(x,M)=0, becauseRf isninjective.E (cid:3)
E n n
The abovetheoremtogetherwith Proposition2.1immediately yields the follow-
ing corollary.
Corollary 3.3. Assume that x and y are finite sequences of R such that √xR =
√yR. Then x is M-weakly proregular if and only if y is M-weakly proregular.
As a consequence of the following theorem one obtains that any finite sequence
of elements in a Noetherian ring is weakly proregular on any finitely generated
module.
Theorem 3.4. Let R be a Noetherian ring, I be an ideal of R and M be a finitely
generated R-module. Then Hi(Hom (M,E)) = 0 for all injective R-modules E
I R
and i=0.
6
Proof. Assume that E is an injective R-module. Since the exact sequence 0
→
H0(E) E E/H0(E) 0issplitby[6,Corollary2.1.5],onehasE =H0(E)
I → → I → ∼ I ⊕
E/H0(E). So that, for all i, we have
I
HIi(HomR(M,E))∼=HIi(HomR(M,HI0(E)))⊕HIi(HomR(M,E/HI0(E))).
ItiseasytoseethatHom (M,H0(E))isI-torsion. HenceHi(Hom (M,H0(E)))=
R I I R I
0 for all i = 0. Since E/H0(E) is an injective R-module and H0(E/H0(E)) = 0,
6 I I I
COHEN-MACAULAYNESS OF TRIVIAL EXTENSIONS 5
then to complete the proof it is enough for us to show that Hi(Hom (M,E))=0
I R
fortheinjectiveR-moduleE withadditionalconditionthatE isI-torsion-free. For
this, let
F F M 0
1 0
···−→ −→ −→ −→
be a free resolution of M. Then
0 Hom (M,E) Hom (F ,E) Hom (F ,E)
R R 0 R 1
−→ −→ −→ −→···
isanaugmentedinjectiveresolutionofHom (M,E)suchthatH0(Hom (F ,E))=
R I R i
0 for all i since H0(E)=0. Therefore Hi(Hom (M,E))=0 for all i=0. (cid:3)
I I R 6
The following lemma will be used later.
Lemma 3.5. Suppose that f :R S is a flat ring homomorphism and that M is
→
an R-module. If x is M-weakly proregular, then f(x) is M S-weakly proregular.
R
⊗
The converse holds if f is faithfully flat.
Proof. This easily follows from [5, Proposition 1.6.7]. (cid:3)
3.2. Parameter sequences. Let (R,m) be a local Noetherian ring and M be a
finitely generated R-module. A sequence of elements x in R is said to be a system
of parameters on M if M/xM has finite length and dimM = ℓ(x). In fact, x is a
system of parameters on M if and only if ht (xR)=ℓ(x)=dimM.
M
Using homological properties of the rings instead of height conditions, the au-
thorsin[10]extendedthenotionofsystemofparametersinNoetherianlocalringsto
sequences in non-Noetherian ones called strong parameter sequences. This subsec-
tion is devoted to generalize the notion of strong parameter sequences to modules.
Definition3.6. AfinitesequencexofelementsofRiscalledaparametersequence
on M provided that the following conditions hold:
(1) x is M-weakly proregular,
(2) xM =M,
6
(3) Hℓ(x)(M) =0 for all prime ideals p Supp ( M ).
x p 6 ∈ R xM
The sequence x is called a strong parameter sequence on M if x ,...,x is a
1 i
parametersequenceonM fori=1,...,ℓ(x). Onemayconsidertheemptysequence
is a parameter sequence of length zero on any R-module. The empty sequence will
also be considered as a regular sequence of length zero on any R-module.
Below,we state some elementary properties ofparameter sequences that will be
used in the course of the paper.
Proposition 3.7. Let R be a ring and M be an R-module. Let x be a finite
sequence of elements of R.
(1) Any permutation of a parameter sequence on M is again a parameter se-
quence on M.
(2) Assumethat√xR=√yR,ℓ(x)=ℓ(y),xM =M yM =M andSupp ( M )=
6 6 R xM
Supp ( M ). Then x is a parameter sequence on M if and only if y is a
R yM
parameter sequence on M.
(3) If p.grade (xR,M)=ℓ(x), then x is a parameter sequence on M.
R
(4) Every M-regular sequence is a strong parameter sequence on M.
(5) Let f :R S be a flat ring homomorphism. If x is a (strong) parameter
sequence −on→M and M⊗RS = 0, then f(x) is a (strong) parameter
f(x)M⊗RS 6
sequence on M S. The converse holds if f is faithfully flat.
R
⊗
6 A.MAHDIKHANI,P.SAHANDIANDN.SHIRMOHAMMADI
Proof. For (1) see Proposition2.1 and Remark 3.1 and for (2) see, again, Proposi-
tion 2.1 together with Theorem 3.2.
To prove (3), we first note that p.grade (xnR,M) = p.grade (xR,M) =
R R
ℓ(x) by [13, Section 5.5, Theorem 12] and that H (xn,M) = 0 for all i 1
i
≥
by [10, Proposition 2.7]. Let p Supp (M/xM). Hence xM = M . Then
∈ R p 6 p
p.grade (xR ,M ) < again by [10, Proposition 2.7]. Since localization does
R p p ∞
not decrease the polynomial grade by [13, Section 5.5, Exercise 10] and the poly-
nomial grade is bounded above by the length of the sequence (see [10, Proposition
2.7]), we see that p.grade (xR ,M ) = ℓ(x) for all p Supp (M/xM). Hence
R p p ∈ R
Hℓ(M) = 0 for all p Supp (M/xM) by [10, Proposition 2.7]. Therefore x is a
x p 6 ∈ R
parameter sequence on M.
For (4), notice that ℓ(x) p.grade (xR,M) ℓ(x) since xM = M. Thus
≥ R ≥ 6
p.grade (xR,M) = ℓ(x). Therefore x is a strong parameter sequence on M by
R
(3).
Finally for (5), assume that x is a parameter sequence on M and M⊗RS =
f(x)M⊗RS 6
0. Then, by Lemma 3.5, f(x) is M S-weakly proregular sequence. Now, let
R
q Supp ( M⊗RS ) and set p:=f−⊗1(q). Then p Supp ( M ) and one has the
∈ S f(x)M⊗RS ∈ R xM
isomorphism
(1) Hfℓ((xx))(M ⊗RS)q ∼=Hxℓ(x)(M)p⊗Rp Sq.
Since S is a faithfully flat R -module, one obtains that f(x) is a parameter
q p
sequence on M S. To prove the converse, assume that f is faithfully flat.
R
Again using Lem⊗ma 3.5, it is enough for us to show that Hℓ(x)(M) = 0 for all
x p
6
p Supp ( M ). Assumethatp Supp ( M ). Sincef isflat,thereexistsaprime
∈ R xM ∈ R xM
ideal q of S such that p=f−1(q). The isomorphism
M S M
R
(f(x)M⊗ S)q ∼=(xM)p⊗Rp Sq
R
⊗
shows that q Supp ( M⊗RS ). The isomorphism (1) now completes the proof.
∈ S f(x)M⊗RS
(cid:3)
Next we provide a description of parameter sequences using height condition.
Proposition 3.8. Let R be a ring, M be a finitely generated R-module and x be a
finite sequence of elements of R.
(1) If x is a parameter sequence on M, then ht (xR) ℓ(x).
M
≥
(2) Further assume that R is Noetherian. Then x is a parameter sequence on
M if and only if ht (xR)=ℓ(x).
M
Proof. (1) If ht (xR)= , there is nothing to prove. So assume that ht (xR)<
M M
∞
. Then there exists p Supp(M) V(xR) such that ht (xR) = ht p =
M M
∞ ∈ ∩
dimM . Since x is a parameter sequence on M, then Hℓ(x)(M) = 0. Therefore
p x p
6
ℓ(x) dimM =ht (xR) by Corollary 2.2.
p M
≤
(2) Assume that x is a parameter sequence on M. Since xM = M, then
6
ht (xR) < . By part (1), we have ht (xR) ℓ(x) and by Krull’s Gen-
M M
∞ ≥
eralized Principal Ideal Theorem, we have ht (xR) ht (xR) ℓ(x). Then
M R
≤ ≤
ht (xR)=ℓ(x).
M
Conversely assume that ht (xR)=ℓ(x). By Theorem 3.4, any sequence of ele-
M
ments in R is M-weaklyproregular. Since ht (xR)=ℓ(x)< , then(x)M =M.
M
∞ 6
COHEN-MACAULAYNESS OF TRIVIAL EXTENSIONS 7
LetpbeaminimalelementofSupp (M/xM). Thenpisaminimalprimeidealover
R
xR+Ann(M). Hence p/Ann(M) is a minimal prime ideal over xR(R/Ann(M))
which is generated by ℓ(x) elements. Then
p R
dimM =ht p=ht =htxR =ℓ(x).
p M
Ann(M) Ann(M)
On the other hand, one has
xR +Ann(M)R = (xR+Ann(M))R =pR .
q p p q p p
Hence
Hℓ(x)(M )=Hℓ(x) (M )=Hℓ(x)(M )=0.
xRp p xRp+Ann(M)Rp p pRp p 6
Therefore x is a parameter sequence on M. (cid:3)
The Noetherian assumption in Proposition 3.8(2) is crucial. In fact, in every
valuationdomainofdimension2,onecanchooseaweaklyproregularsequencex,y
such that ht(x,y)=2, but x,y is not a parameter sequence, see [10, Example 3.7].
4. Cohen-Macaulay Modules
4.1. Definition and basic properties. In [8] and [9], Glaz raised the question
thatwhetherthereexistsageneralizationofthenotionofCohen-Macaulaynesswith
certain desirable properties to non-Noetherian rings. One of those is that every
coherent regular ring is Cohen-Macaulay. In this direction, in [10], it is defined a
notion of Cohen-Macaulaynessfor arbitrary commutative rings. This subsection is
devoted to extend the definition of Cohen-Macaulayness for commutative rings in
the sense of [10] to modules.
Definition 4.1. An R-module M is called a Cohen-Macaulay R-module if every
strong parameter sequence on M is an M-regular sequence.
This definition agrees with the usual definition of Cohen-Macaulay finitely gen-
eratedmodulesoverNoetherianrings. Indeed,letRbeaNoetherianringandM be
a finitely generated R-module. Assume that M is Cohen-Macaulay in the sense of
Definition4.1. ToshowthatM is Cohen-Macaulaywiththe usualdefinitioninthe
Noetheriancase,itisenoughtoshowthatgrade(I,M)=ht I forallproperideals
M
I ofR. Toprovethis,assumethatI isaproperidealofRandsetht I =ℓ. Since
M
ht I =htI(R/Ann(M)),employing[5,TheoremA.2]totheringR/Ann(M)one
M
finds the elements x ,...,x in I such that ht (x ,...,x )=i for all i=0,...,ℓ.
1 ℓ M 1 i
ItfollowsfromProposition3.8thatx ,...,x isastrongparametersequenceonM.
1 ℓ
Hence it is an M-regular sequence. This yields that ℓ grade(I,M) ht I =ℓ.
M
≤ ≤
Therefore grade(I,M) = ht I. The converse is true by Theorem 4.2 and Propo-
M
sition 4.6 below and [5, Corollary 1.6.19].
Let R be a ring and M be an R-module. If dimM = 0, then M is Cohen-
Macaulay. Indeed, in this situation, M has not any parameter sequences.
Thanks to polynomial grade, Koszul homology, and Cˇech cohomology of strong
parametersequences,ourfirstresultpresentssomeequivalentstatementsofCohen-
Macaulayness. It generalizes[10, Proposition4.2]for modules. Its proofis mutatis
mutandis the same as that of [10, Proposition 4.2]. But, for the reader’s conve-
nience, we reprove it in the case of modules.
8 A.MAHDIKHANI,P.SAHANDIANDN.SHIRMOHAMMADI
Theorem 4.2. Let R be a ring and M be an R-module. The following conditions
are equivalent:
(1) M is Cohen-Macaulay.
(2) grade(xR,M)=ℓ(x) for every strong parameter sequence x of M.
(3) p.grade (xR,M)=ℓ(x) for every strong parameter sequence x of M.
R
(4) H (x,M) = 0 for all i 1 and for every strong parameter sequence x of
i
≥
M.
(5) Hi(M) = 0 for all i < ℓ(x) and for every strong parameter sequence x of
x
M.
Proof. (1) (2) Assume that x is a strong parameter sequence on M; so, by as-
⇒
sumption, x is M-regular sequence. Hence ℓ(x) grade(xR,M). One also notices
≤
thatgrade(xR,M) p.grade (xR,M)by[13,Page149]andthatp.grade (xR,M)
≤ R R ≤
ℓ(x) by [13, Section 5.5, Theorem 13]. Therefore one has grade(xR,M)=ℓ(x).
(2) (3) Assume that x is a strong parameter sequence on M; so that ℓ(x) =
⇒
grade(xR,M). Asinthe(1) (2),againusing[13,Page149]and[13,Section5.5,
⇒
Theorem 13], one obtains that p.grade (xR,M)=ℓ(x).
R
(3) (1) Assume that x is a strong parameter sequence on M. We proceed
⇒
by induction on ℓ = ℓ(x) to show that x is M-regular sequence. If x = x , then
1
p.grade (x R,M)=1 and 0=H (x ,M)=(0: x ). Hence x is an M-regular
R 1 1 1 M 1 1
element. Suppose that every strong parameter sequence on M of length at most
ℓ 1 is M-regular sequence and that x is a strong parameter sequence on M of
le−ngthℓ. Setx′ =x1,...,xℓ−1. Sincex′ isastrongparametersequenceonM,then
by hypothesis p.grade (x′,M) = ℓ 1. Thus, by induction, x′ is an M-regular
sequence. Let M′ =MR/x′M. Since p−.gradeR(x,M)=ℓ, then Hℓ−i(x,M)=0 for
all i<ℓ by [10, Proposition 2.7]. Hence
(0:M′ xℓ)=H1(xℓ,M′)=H1(xℓ,M/x′M)∼=H1(x,M)=0
by[5,Proposition1.6.13]. Thisimpliesthatx isanM′-regularelement. Therefore
ℓ
x is an M-regular sequence. Finally, notice that (3), (4) and (5) are equivalent by
[10, Proposition 2.7]. (cid:3)
The Cohen-Macaulay property descends along faithfully flat extensions:
Proposition4.3. Letf :R S bea faithfully flat ringhomomorphism. Let M be
→
anR-module. IfM S isCohen-MacaulayS-module,thenM isCohen-Macaulay.
R
⊗
Proof. Assume that M S is Cohen-Macaulay S-module. Let x be a strong
R
⊗
parameter sequence on M. Then, by Proposition 3.7(5), f(x) is strong parameter
sequence on M S. Hence, the assumption together with Theorem 4.2, [13,
R
⊗
Section 5.5, Theorem 19] and [10, Proposition 2.7] yields that
ℓ(x)=ℓ(f(x))
=p.grade (f(x)S,M S)
R ⊗R
=p.grade (xR,M S)
R ⊗R
=sup k 0 Hi(M S)=0 for all i<k
{ ≥ | x ⊗R }
=sup k 0 Hi(M) S =0 for all i<k
{ ≥ | x ⊗R }
=sup k 0 Hi(M)=0 for all i<k
{ ≥ | x }
=p.grade (xR,M).
R
COHEN-MACAULAYNESS OF TRIVIAL EXTENSIONS 9
Therefore M is Cohen-Macaulay. (cid:3)
One can immediately obtain the following corollaries.
Corollary 4.4. Let (R,m) be a quasi-local ring and M an R-module. If M R
R
⊗
is Cohen-Macaulay R-module where R is the m-adic completion of R, then M ibs
Cohen-Macaulay. b b
Corollary 4.5. Let R be a ring, M an R-module and t an indeterminate over R.
If M R[t] is Cohen-Macaulay R[t]-module, then M is Cohen-Macaulay.
R
⊗
Proposition 4.6. Let R be a ring and M be a finitely generated R-module. If
M is Cohen-Macaulay R -module for all maximal ideals m of R, then M is a
m m
Cohen-Macaulay R-module.
Proof. Assume that x=x ,...,x is a strong parameter sequence on M and that
1 ℓ
m is a maximal ideal containing xR. Since M is finitely generated, then M =
m
(x)M ; so that x = x1,...,xℓ is a strong parameter sequence on M . Hence 6x
1 m 1 1 1 m 1
is M -regular sequence. Thus x is M-regular sequence. Therefore M is Cohen-
m
Macaulay. (cid:3)
Asmentionedintheintroduction,everycoherentregularringisCohen-Macaulay.
Inparticular,everyvaluationdomainisCohen-Macaulay. Inthefollowing,weshow
thateverytorsion-freemoduleoversuchdomainisCohen-Macaulay. Infact,wedo
thisfortorsion-freemodulesoveralmostvaluationdomains. Recallthatanintegral
domain R with quotient field K is called an almost valuation domain if for every
nonzero x K, there exists an integer n 1 such that either xn R or x−n R
∈ ≥ ∈ ∈
[4].
Proposition 4.7. Every torsion-free module over an almost valuation domain is
Cohen-Macaulay.
Proof. Suppose that (R,m) is an almost valuation domain and M is a torsionfree
R-module. Assume that x := x ,x is a sequence in R of length 2. Assume that
1 2
xnR xnR for some positive integer n. Then
1 ⊆ 2
Hx2(M)=Hx2n1,xn2(M)=Hx2n2(M)=0.
Hencex=x ,x cannotbeaparametersequenceonM. Then,foreachparameter
1 2
sequence x of M, ℓ(x) 1. Therefore M is Cohen-Macaulay. To this end, one
notices that M is torsion≤-free. (cid:3)
Recallthatthe module M is calledCohen-Macaulay in the sense of ideals (resp.
finitely generated ideals) if ht (I) = p.grade (I,M) for all ideals (resp. finitely
M R
generatedideals)I,see[3,Definition3.1]. The followingpropositiongeneralizes[3,
Theorem 3.4] to finitely generated modules.
Proposition 4.8. Let R be a ring and M be a finitely generated R-module. If M
is Cohen-Macaulay in the sense of ideals (or finitely generated ideals), then M is
Cohen-Macaulay in the sense of Definition 4.1.
Proof. Assume that x is a strongparametersequence onM. Then, by Proposition
3.8 and [13, Section 5.5, Theorem 13], we have
ht (xR) ℓ(x) p.grade (xR,M).
M ≥ ≥ R
However,by assumptionp.grade (xR,M)=ht (xR). Then p.grade (xR,M)=
R M R
ℓ(x). Therefore M is Cohen-Macaulay in the sense of Definition 4.1. (cid:3)
10 A.MAHDIKHANI,P.SAHANDIANDN.SHIRMOHAMMADI
4.2. The Cohen-Macaulayness of some constructions. Let R and S be two
commutative rings with unity, let J be an ideal of S and f : R S be a ring
→
homomorphism. ThesubringR⊲⊳f J := (x,f(x)+j)x R and j J ofR S is
{ | ∈ ∈ } ×
calledtheamalgamation of Rwith Salong J withrespect tof [7]. Thisconstruction
generalizes several classical constructions. Among them is the Nagata’s trivial
extension, see [7, Examples 2.5 and 2.6].
Undermildconditions,thenextpropositionshowsthattheCohen-Macaulayness
of R⊲⊳f J descends to that of R.
Proposition 4.9. Let R and S be commutative rings with unity, let J be an ideal
of S and f : R S be a ring homomorphism. Assume that J is flat as an R-
→
module induced by f. If R ⊲⊳f J is Cohen-Macaulay, then R is Cohen-Macaulay.
Moreover, if any strong parameter sequence on J is a strongparameter sequenceon
R, then J is Cohen-Macaulay.
Proof. Note that as an R-module R ⊲⊳f J = R J. This in conjunction with the
∼ ⊕
assumption implies the natural embedding ι : R R ⊲⊳f J is faithfully flat.
R
−→
Hence, by Proposition 4.3, R is Cohen-Macaulay. The rest of the conclusion is
clear since J is flat. (cid:3)
We do not know whether the Cohen-Macaulay property of R ascends to that of
R ⊲⊳f J. The difficulty lies in linking strong parameter sequences of R ⊲⊳f J to
strong parameter sequences of R and J. However, we solve this difficulty under
certain assumptions. Let M be an R-module. Then R⋉M denotes the trivial
extension ofRbyM. As indicatedin[7,Example2.8],ifS :=R⋉M,J :=0⋉M,
and f :R S be the natural embedding, then R⊲⊳f J =R⋉M.
→ ∼
Assume that R is Noetherian local and that M is finitely generated. It is well
known that the trivial extension R ⋉ M is Cohen-Macaulay if and only if R is
Cohen-Macaulay and M is maximal Cohen-Macaulay, see [2, Corollary 4.14]. Our
maintheoreminthispapergeneralizesthisresult. Toproveitweneedthefollowing
lemma.
Lemma 4.10. Let R be a ring and M be an R-module. Set S := R⋉M. Let
π :S R be the natural projection. Then x S is S-weakly proregular if and only
→ ⊆
if π(x) is R and M-weakly proregular.
Proof. First of all notice that using the structure of prime spectrum of the trivial
extensiononehas√xS = π(x)S. UsingthistogetherwithProposition2.1(2)and
p
Theorem 3.2, one can deduce that x is S-weakly proregular if and only if (π(x),0)
is S-weakly proregular. Thus, it suffices to prove that if y is a finite sequence of
elements from R, then y is R and M-weakly proregular if and only if (y,0) is S-
weaklyproregular. However,asR-modules,Hi((yn,0),S)∼=Hi(yn,R)⊕Hi(yn,M)
for all i and n. Therefore the definition of weakly proregular sequence completes
the proof. (cid:3)
Theorem 4.11. Let R be a ring and M be an R-module such that every R-weakly
proregular sequence is an M-weakly proregular sequence. Then R⋉M is Cohen-
Macaulay if and only if R is Cohen-Macaulay and every R-regular sequence is a
weak M-regular sequence.
Proof. FirstassumethatR⋉M isCohen-Macaulayandthatxisastrongparameter
sequence on R. So, in particular, x is M-weakly proregular. By Lemma 4.10,