LOCAL COHOMOLOGY ANNIHILATORS AND MACAULAYFICATION 1. Introduction Let X be a ...

24 Pages·2015·0.53 MB·English

LOCAL COHOMOLOGY ANNIHILATORS AND
MACAULAYFICATION
NGUYENTUCUONGAND DOANTRUNGCUONG
Abstract. The aim of this paper is to study a deep connection between lo-
cal cohomology annihilators and Macaulayfication and arithmetic Macaulay-
fication over a local ring. Local cohomology annihilators appear through the
notionofp-standardsystemofparameters. Foralocalring,weproveanequiv-
alenceoftheexistenceofMacaulayfications;theexistenceofap-standardsys-
temofparameters; beingaquotientofaCohen-Macaulaylocalring; andthe
verificationofFaltings’Annihilatortheorem. Forafinitelygeneratedmodule
whichisunmixedandfaithful,weproveanequivalenceoftheexistenceofan
arithmetic Macaulayfication and the existence of a p-standard system of pa-
rameters;andbothareprovedtobeequivalenttotheexistenceofanarithmetic
Macaulayficationonthegroundring. AconnectionbetweenMacaulayfication
anduniversalcatenaricityisalsodiscussed.
1. Introduction
LetX beaNoetherianscheme. AMacaulayficationofX isapair(Y,π)consist-
ingofaCohen-MacaulayschemeY andofabirationalpropermorphismπ :Y →X.
This analogous notion of desingularization is due to Faltings and he constructed
a Macaulayfication for any quasi-projective scheme with 0 or 1-dimensional non-
Cohen-Macaulay locus over a Noetherian ring admitting a dualizing complex [?].
ThekeypointinFaltings’constructionofMacaulayficationisaprofoundconnection
between Macaulayfication of a local ring and the annihilators of local cohomology
modules of the ring. Using this idea, Kawasaki [?] constructed a Macaulayfication
for any quasi-projective scheme over a Noetherian ring provided the ground ring
admits a dualizing complex. Local cohomology annihilators appears in Kawasaki’s
construction through the notion of p-standard system of parameters which is de-
fined first in [?, ?]. The Macaulayfication constructed by Kawasaki in fact is a
blowing up of the scheme with center formulated by a product of parts of a p-
standard system of parameters. While desingularization of Hironaka [?] depends
on characteristic of the ground field, the constructed Macaulayfication is signifi-
cantly characteristic-independent.
2010 Mathematics Subject Classification. Primary13H10,14M05;Secondary13D45,14B05.
Keywordsandphrases. arithmeticMacaulayfication,Macaulayfication,localcohomologyan-
nihilator,p-standardsystemofparameters,quotientofCohen-Macaulayring.
Nguyen Tu Cuong is supported by the NAFOSTED of Vietnam under grant number 101.01-
2011.49.
Doan Trung Cuong is partially supported by the SFB/TR 45 "Periods, moduli spaces and
arithmeticofalgebraicvarieties"andbytheNAFOSTEDofVietnamundergrantnumber101.01-
2012.05.
1
2 NGUYENTUCUONGAND DOANTRUNGCUONG
Beside Faltings and Kawasaki’s Macaulayfication, there are other construction
in some special cases by Brodmann [?], Goto [?], Schenzel [?]. There is also a
formulation of Macaulayfication of modules and sheaves [?], [?].
Goingfurther,Kawasaki[?,?]characterizedtheexistenceofarithmeticMacaulay-
fication of a Noetherian ring. Let R be a commutative Noetherian ring and I be a
proper ideal of positive height of R. The Rees algebra R(I)=(cid:76)∞ Im gives rise
m=0
to a blowing up Y =ProjR(I)→π Spec(R). The morphism π is a Macaulayfication
ifY isaCohen-Macaulayscheme. IftheReesalgebraR(I)itselfisCohen-Macaulay
thenitiscalledanarithmeticMacaulayficationoftheringR. Obviouslytheprojof
an arithmetic Macaulayfication is a Macaulayfication of the spectrum of the ring.
Kawasaki showed that a Noetherian local ring has an arithmetic Macaulayfication
if and only if it is unmixed and all its formal fibers are Cohen-Macaulay. It is
also remarkable that the ideal defining the Cohen-Macaulay Rees algebra is also a
product of parts of a p-standard system of parameters.
ArithmeticMacaulayficationhasbeenstudiedfromotherperspectivebyKurano
[?], Aberbach [?], Aberbach-Huneke-Smith [?], Cutkosky-Tai [?], Tai-Trung [?].
Analogously, an arithmetic Macaulayfication of a finitely generated R-module M
is defined to be a Cohen-Macaulay Rees module R(M,I):=(cid:76) ImM for some
m≥0
ideal I of R.
The aim of this paper is to study more extensively the relationship between
Macaulayfication and local cohomology annihilators. We address ourself to two
problems on local rings: Existence of arithmetic Macaulayfication of modules; re-
lationship between existence of p-standard system of parameters and of Macaulay-
fication of the spectrum of a local ring.
Let (R,m) be a Noetherian local ring and M be a finitely generated R-module.
The i-th local cohomology module Hi (M) has the annihilator ideal denoted by
m
a (M). Put a(M) := a (M)a (M)...a (M), where d is the Krull dimension of
i 0 1 d−1
M. A system of parameters x ,...,x ∈ m of M is called p-standard if for any
1 d
i = 1,...,d, we have x ∈ a(M/(x ,...,x )M). These systems of parameters
i i+1 d
have very rich properties. Beside applications in constructing Macaulayfication,
theyareveryusefulinthestudyofstructureoflocalringandmodules, seeCuong-
Schenzel-Trung [?], Schenzel [?], Trung [?], Cuong-Cuong [?, ?]. The concept of
p-standard system of parameters will play an important role in our study of two
problems above.
FortheproblemonarithmeticMacaulayficationofmodules,wegetthefirstmain
result.
Theorem 1.1. Let (R,m) be a Noetherian local ring and M be a finitely generated
R-module. Suppose M is unmixed. The following statements are equivalent:
(a) M has an arithmetic Macaulayfication.
(b) R/p has an arithmetic Macaulayfication for all associated prime ideals p ∈
Ass(M).
(c) M admits a p-standard system of parameters.
HereM isunmixedifdimRˆ/P =dimM foranyassociatedprimeidealP ofthe
m-adic completion of M.
As a direct consequence of Theorem ??, a finitely generated R-module M has
anarithmeticMacaulayficationifandonlyifsodoestheringR/Ann (M). Sothe
R
questiononexistenceofarithmeticMacaulayficationofmodulereducestothesame
LOCAL COHOMOLOGY ANNIHILATORS AND MACAULAYFICATION 3
question on the corresponding ring. In order to prove Theorem ??, we first prove
the equivalence of (a) and (c). Then we make use of the following result which
relates the existence of p-standard system of parameters on a module to that on
the ring.
Theorem 1.2. Let (R,m) be a Noetherian local ring and M be a finitely generated
R-module. The following statements are equivalent:
(a) M admits a p-standard system of parameters.
(b) R/Ann M admits a p-standard system of parameters.
R
(c) Any finitely generated R-module N with Supp(N) ⊆ Supp(M) admits a p-
standard system of parameters.
AninterestingconsequenceofTheorems??and??isthatifM hasanarithmetic
Macaulayfication and N is another finitely generated R-module such that N is
unmixedandSupp (N)⊆Supp (M),thenN hasanarithmeticMacaulayfication.
R R
In the second part of the paper, we study the existence of p-standard system of
parameters on a local ring. It turns out that the existence of p-standard system of
parameters relates closely to various subjects such as Macaulayfication, Faltings’
Annihilator theorem and being quotient of Cohen-Macaulay rings. They are put
all together in the following second main result of the paper.
Theorem 1.3. Let (R,m) be a Noetherian local ring. The following statements are
equivalent:
(a) R admits a p-standard system of parameters.
(b) R is a quotient of a Cohen-Macaulay local ring.
(c) RisuniversallycatenaryandforanyquotientS ofR,Spec(S)hasaMacaulay-
fication.
(d) All essentially of finite type R-algebras verify Faltings’ Annihilator Theorem.
That means, if we let A be an essentially of finite type R-algebra and N be a
finitely generated A-module, then for any pair of ideals b⊆a of A, we have
(cid:112)
inf{i:b(cid:54)⊆ Ann Hi(N)}=inf{depthN +height(a+p)/p:p∈/ V(b)}.
A a p
(e) Any of the above statements holds for all essentially of finite type R-algebras.
Theorem ?? shows the existence of Macaulayfication of the spectrum of a local
ring provided this local ring is the quotient of a Cohen-Macaulay ring. This result
is,infact,ageneralizationofKawasaki’sTheoremonMacaulayfication[?,Theorem
1.1]forlocalrings. Furthermore, byTheorems??and??, alocalringisaquotient
of a Cohen-Macaulay ring if and only if there is a finitely generated and faithful
module over the ring which has a p-standard system of parameters.
The key point in the proof of Theorem ?? is to show that a local ring has a
p-standard system of parameters if and only if it is universally catenary and all its
formal fibers are Cohen-Macaulay. Then we use several times results of Faltings,
GrothendieckandKawasakitorelatethelaterpropertieswiththeotherequivalent
conditions in the theorem.
To summarize the content of this paper, in Section ?? we study the local coho-
mology supported by an ideal generated by a dd-sequence, a slightly generalized
notion of p-standard system of parameters (see Lemma ??). p-Standard system of
parameters is defined via the local cohomology annihilators while dd-sequence is
defined by using Huneke’s notion of d-sequence. We will see in Section ?? that in
4 NGUYENTUCUONGAND DOANTRUNGCUONG
certain cases, dd-sequences as sequences of elements in m is easier to handle than
p-standardsytemofparameters. Themainresultofthissectionisasplittingshort
exact sequence for local cohomology modules in Theorem ??.
The main technical results for the proofs of Theorems ?? and ?? are presented
inSection??. UsingresultsinSection??,weshowthattheexistenceofp-standard
systemofparametersisalocalproperty. Thatmeans,ifamodulehasap-standard
system of parameters then so does its localization at any prime ideal (Proposition
??).
We prove Theorem ?? in Section ??. We first prove Theorem ?? where the
technical results in the former sections find their applications.
Theorem ?? will be proved in both Sections ?? and ??. In Section ?? the
equivalence of the conditions (a) and (b) in Theorem ?? is proved by Theorem ??.
Various consequences of this theorem characterizing quotients of Cohen-Macaulay
local rings are presented afterward. The last section is devoted to proving the
rest of Theorem ?? (see Theorem ??) and to discussing a conjectural relationship
between Macaulayfication (originally, desingularization) and universal catenaricity
of a Noetherian ring (Theorem ??). This was motivated by Grothendieck’s works
on connection between desingularization and excellent rings in [?].
2. Local cohomology supported by dd-sequences
Throughout this paper (R,m) denotes a commutative Noetherian local ring.
In this section we study the local cohomology with support generated by a dd-
sequenceandtheirannihilatorideals. LetM beafinitelygeneratedR-moduleofdi-
mensiond. Denotebya (M)=Ann Hi (M)theannihilatoridealofthei-thlocal
i R m
cohomologymoduleHi (M)fori=0,1,...,d,andseta(M)=a (M)...a (M).
m 0 d−1
DuetoHuneke[?],asequencex ,...,x ∈misad-sequenceonM if(x ,...,x )M :
1 s 1 i
x =(x ,...,x )M :x x for all 0≤i<j ≤s.
j 1 i i+1 j
Definition 2.1. A system of parameters x ,...,x of M is called p-standard if
1 d
x ∈ a(M) and x ∈ a(M/(x ,...,x )M) for i = d − 1,...,1 (cf. [?]). In
d i i+1 d
[?], the authors extend this notion for sequence of elements. We say that a se-
quence x ,...,x in m is a dd-sequence on M if xn1,...,xni is a d-sequence on
1 s 1 i
M/(xni+i+11,...,xnss)M for all n1,...,ns >0 and i=1,...,s.
The two notions above are very close in the case of systems of parameters, as
shown by the following lemma.
Lemma 2.2. [?, Corollary 3.9] Let M be a finitely generated R-module with a
system of parameters x ,...,x , d = dimM. If x ,...,x is a p-standard system
1 d 1 d
of parameters then it is a dd-sequence on M. Vice versa, if x ,...,x is a dd-
1 d
sequencethenxn1,...,xnd isap-standardsystemofparametersofM foralln ≥i,
1 d i
i=1,...,d.
Belows we recall some basic properties of dd-sequences.
Lemma 2.3. [?, Proposition 3.4] Let M be a finitely generated R-module and
x ,...,x be a dd-sequence on M.
1 s
(i) x ,...,x is a dd-sequence on M for any 1≤i <...<i ≤s.
i1 ir 1 r
(ii) x ,...,x ,x ,...,x is a dd-sequence on M/x M for i=1,...,s.
1 i−1 i+1 s i
As usual we denote a sequence x ,...,x ,x ,...,x by x ,...,x ,...,x .
1 i−1 i+1 s 1 (cid:98)i s
By a standard method for d-sequences and Koszul homology (see Goto-Yamagishi
LOCAL COHOMOLOGY ANNIHILATORS AND MACAULAYFICATION 5
[?, Theorem 1.14]) and using the same argument as in the proof of Theorem 2.3 of
Goto-Yamagishi [?] we have
Lemma 2.4. Let x ,...,x be a dd-sequence on M. It holds
1 s
(i) (x ,...,x )Hi (M)=0, for i<r,1≤r ≤s.
r+1 s (x1,...,xr)
(ii) (xn1+m1,...,xns+ms)M :(xm1...xms)
1 s 1 s
s
(cid:88)
= (xn11,...,x(cid:99)nii,...,xnss)M :xi+(xn11,...,xnss)M.
i=1
Inthenext,wewillusethenotation(0:x) forelementsinN killedbyxinstead
N
of the usual notation 0: x when the module N has a complicated formula.
N
Lemma 2.5. Let x ,...,x be a dd-sequence on M. For i<s we have
1 s
Hi (M)(cid:39)Hi (M)(cid:39)(0:x ) .
(x1,...,xs) (x1,...,xi+1) i+1 H(ix1,...,xi)(M)
Proof. Since x ,...,x , 1 ≤ k ≤ s, is also a dd-sequence on M by Lemma ??, it
1 k
suffices to prove that
(i) Hi (M)(cid:39)Hi (M).
(x1,...,xs) (x1,...,xs−1)
(ii) Hs−1 (M)(cid:39)(0:x ) .
(x1,...,xs) s H(sx−11,...,xs−1)(M)
Denotethei-thCˇechcohomologymoduleofM withrespecttotheideal(x ,...,x )
1 s
by Hˇi(x ,...,x ,M). There is a long exact sequence
1 s
...−→Hˇi(x ,...,x ,M)−→Hˇi(x ,...,x ,M)
1 s 1 s−1
−→Hˇi(x ,...,x ,M )−→Hˇi+1(x ,...,x ,M)−→...,
1 s−1 (xs) 1 s
whereM isthelocalizationofM bythemultiplicativelyclosedset{1,x ,x2,...}.
(xs) s s
Since Hˇi(x ,...,x ,M) (cid:39) Hi (M) and Hˇi(x ,...,x ,M ) (cid:39)
1 s (x1,...,xs) 1 s−1 (xs)
Hi (M) , there is a long exact sequence of local cohomology modules
(x1,...,xs−1) (xs)
...−→Hi (M)−→Hi (M)−→Hi (M)
(x1,...,xs) (x1,...,xs−1) (x1,...,xs−1) (xs)
−→Hi+1 (M)−→...
(x1,...,xs)
Recall that x ,...,x is a dd-sequence on M, so x ∈ Ann Hi (M) for
1 s s R (x1,...,xs−1)
all i < s − 1 by Lemma ??(i), and hence Hi (M) = 0. Therefore,
(x1,...,xs) (xs)
Hi (M)(cid:39)Hi (M) for all i<s−1 and there is an exact sequence
(x1,...,xs) (x1,...,xs−1)
0−→Hs−1 (M)−→Hs−1 (M)−ψ→Hs−1 (M)
(x1,...,xs) (x1,...,xs−1) (x1,...,xs−1) (xs)
−→Hs (M)−→0,
(x1,...,xs)
where ψ is the natural homomorphism of localization. Note that we can identify
Hs−1 (M) = limM/(xt,...,xt )M. Let a¯ ∈ Hs−1 (M), where a ∈
(x1,...,xs−1) −→ 1 s−1 (x1,...,xs−1)
t
M/(xt,...,xt )M for some t > 0. So a¯ ∈ Kerψ if and only if xra¯ = 0 for some
1 s−1 s
r >0 if and only if xra(x ...x )t ∈(xt+t(cid:48),...,xt+t(cid:48))M. Equivalently,
s 1 s−1 1 s−1
a∈(xt+t(cid:48),...,xt+t(cid:48))M :xr(x ...x )t(cid:48) =(xt+t(cid:48),...,xt+t(cid:48))M :x (x ...x )t(cid:48)
1 s−1 s 1 s−1 1 s−1 s 1 s−1
=(xt+1,...,xt+1)M :(x ...x x )
1 s−1 1 s−1 s
6 NGUYENTUCUONGAND DOANTRUNGCUONG
by Lemma ??(ii) and the assumption x ,...,x is a dd-sequence. So x a¯ = 0.
1 s s
Therefore,
Hs−1 (M)(cid:39)Kerψ (cid:39)(0:x ) .
(x1,...,xs) s H(sx−11,...,xs−1)(M)
(cid:3)
Corollary 2.6. Let x ,...,x be a dd-sequence on M. We have
1 s
Ann Hi (M)= (cid:92) Ann (xn11,...,xnii)M :xi+1
R (x1,...,xs) (xn1+1,...,xni+1)M :(x ...x )
n1,...,ni>0 1 i 1 i
for i<s.
Proof. From Lemma ??,
Hi (M)(cid:39)(0:x ) =lim(0:x ) .
(x1,...,xs) i+1 H(ix1,...,xi)(M) −→t i+1 M/(xt1,...,xti)M
Hence,a∈Ann Hi (M)ifandonlyifforanyt>0andu∈(xt,...,xt)M :
R (x1,...,xs) 1 i
x , there is t(cid:48) >0 such that au(x ...x )t(cid:48) ∈(xt+t(cid:48),...,xt+t(cid:48))M, or equivalently,
i+1 1 i 1 i
au∈(xt+t(cid:48),...,xt+t(cid:48))M :(x ...x )t(cid:48)
1 i 1 i
i
(cid:88)
= (xt+1,...,x(cid:100)t+1,...,xt+1)M :x +(xt,...,xt)M
1 j i j 1 i
j=1
=(xt+1,...,xt+1)M :(x ...x ),
1 i 1 i
by Lemma ??(ii). In other words, a∈Ann Hi (M) if and only if
R (x1,...,xs)
a[(xt,...,xt)M :x ]⊆(xt+1,...,xt+1)M :(x ...x ),
1 i i+1 1 i 1 i
for any t > 0. Since Hi (M) (cid:39) Hi (M) for any n ,...,n > 0, we
can replace x ,...,x b(yx1x,.n..1,x,s.)..,xni to(oxbn1t1a,.i.n.,xnss) 1 s
1 i 1 i
Ann Hi (M)= (cid:92) Ann (xn11,...,xnii)M :xi+1
R (x1,...,xs) (xn1+1,...,xni+1)M :(x ...x )
n1,...,ni>0 1 i 1 i
(cid:3)
The next theorem gives a remarkable splitting short exact sequence of local
cohomology modules. This exact sequence will be useful later on (see also [?,
Proposition 2.6]).
Theorem 2.7. Let (R,m) be a Noetherian local ring, I ⊆R be an ideal and M be
a finitely generated R-module. Fix a non-negative integer k. Assume that there is
x∈msuchthat0: x=0: x2 andxHi(M)=0foralli≤k. LetN ⊆0: xbe
M M I M
a submodule. There is a splitting short exact sequence of local cohomology modules
0−→Hi(M/N)−→Hi(M/xnM +N)−→Hi+1(M/0: x)−→0,
I I I M
for all i<k and n≥5. In particular,
Hi(M/xnM +N)(cid:39)Hi(M/N)⊕Hi+1(M/0: x).
I I I M
LOCAL COHOMOLOGY ANNIHILATORS AND MACAULAYFICATION 7
Proof. We first note that x2Hi(M/0 : x) = 0 for all i ≤ k, which follows from
I M
the long exact sequence of local cohomology modules
...−→Hi(0: x)−→Hi(M)−→Hi(M/0: x)−→Hi+1(0: x)−→....
I M I I M I M
Since 0: xn =0: x, there is a commutative diagram
M M
0 −−−−→ M/0: x −−.−x−n→ M/N −−−−→ M/xnM +N −−−−→ 0
M
 (cid:13) 
.x2(cid:121) (cid:13)(cid:13) p(cid:121)
0 −−−−→ M/0: x −−.x−n−−→2 M/N −−−−→ M/xn−2M +N −−−−→ 0,
M
where p is the natural projection. The above diagram derives the following com-
mutative diagram
···−→Hi(M/0: x) −−−ψ−i→ Hi(M/N) −−−−→ Hi(M/xnM +N)−→···
I M I I
 (cid:13) 
 (cid:13) 
.x2(cid:121) (cid:13) (cid:121)
···−→Hi(M/0: x) −−−ϕ−i→ Hi(M/N) −−−−→ Hi(M/xn−2M +N)−→··· ,
I M I I
.xn
where ψ ,ϕ are homomorphisms derived from the homomorphisms M/0: x−→
i i M
M/N and M/0: x.−xn→−2 M/N respectively. If n≥3 then ψ =ϕ ◦(.x2)=0. So
M i i
if n ≥ 5 then ψ = 0 and ϕ = 0 for all i < k. We obtain a commutative diagram
i i
with exact horizontal lines
0−→Hi(M/N) −−−f1−→ Hi(M/xnM +N) −−−g1−→ Hi+1(M/0: x)−→0
I I I M
(cid:13)  
(cid:13)  
(cid:13) (cid:121)h (cid:121).x2
0−→Hi(M/N) −−−f2−→ Hi(M/xn−2M +N) −−−g2−→ Hi+1(M/0: x)−→0.
I I I M
Notethatg ◦h=(.x2)◦g =0,thusIm(h)⊆Im(f ). Wethengetahomomorphism
2 1 2
f−1◦h:Hi(M/xnM +N)−→Hi(M/N),
2 I I
which satisfies f−1 ◦ h ◦ f = id . Therefore, the following short exact
2 1 Hi(M/N)
I
sequence is splitting
0−→Hi(M/N)−→Hi(M/xnM +N)−→Hi+1(M/0: x)−→0,
I I I M
for all i<k and n≥5. (cid:3)
Corollary 2.8. Let x ,...,x be a dd-sequence on M. There is a splitting short
1 s
exact sequence
0−→Hi (M)−→Hi (M/xnM)−→Hi+1 (M/0: x )−→0,
(x1,...,xs) (x1,...,xs) j (x1,...,xs) M j
for 0≤i<j−1<s and n≥5. In particular, if in addition s=d and x ,...,x
1 d
is a system of parameter of M then there is a splitting short exact sequence
0−→Hi (M)−→Hi (M/xnM)−→Hi+1(M/0: x )−→0,
m m j m M j
for all 0≤i<j−1<d and n≥5.
Proof. We have Hi (M) (cid:39) Hi (M) from Lemma ??. From Lemma
(x1,...,xs) (x1,...,xi+1)
??(i), we have x Hi (M) = 0 for all j > i+1. The conclusion then follows
from Theorem ??j. (x1,...,xs) (cid:3)
8 NGUYENTUCUONGAND DOANTRUNGCUONG
Recallthata (M):=Ann Hi (M)anda(M)=a (M)...a (M),d=dimM.
i R m 0 d−1
The next lemma is a simple but very useful property of the ideal a(M).
Lemma 2.9. Let M be a finitely generated R-module. For any submodule N ⊂M
such that dimN < dimM, a(M) ⊆ Ann N. In particular, a(M) ⊆ p for all
R
p∈Ass(M) with dimR/p<dimM.
Proof. Since dimN < dimM, there is a parameter element x of M such that
x∈Ann N. Hence, N ⊆0: x. Note that by a result of Schenzel [?, Satz 2.4.5],
R M
a(M)⊆Ann (0: x), this implies the first conclusion.
R M
Let p ∈ Ass(M) with dimR/p < dimM. There is a submodule N ⊂ M such
that Ass(N) = {p}. Thus dimN = dimR/p < dimM and by the first conclusion,
a(M)⊆Ann N ⊆p. (cid:3)
R
We end this section with the following corollary of Theorem ??.
Corollary 2.10. Letx,y ∈a(M)betwoparameterelementsofM anddimM =d.
For any n,m≥5 and i=0,1,...,d−2 we have
Hi (M/xnM)(cid:39)Hi (M/ymM).
m m
Proof. Since x,y ∈ a(M), we have x ∈ Ann (0 : y) and y ∈ Ann (0 : x)
R M R M
by Lemma ??. Thus 0 : y = 0 : x, in particular, 0 : xt = 0 : x and
M M M M
0 : yt = 0 : y for any t > 0. Using Theorem ?? we obtain for n,m ≥ 5,
M M
i=0,1,...,d−2,
Hi (M/xnM)(cid:39)Hi (M)⊕Hi+1(M/0: x)
m m m M
(cid:39)Hi (M)⊕Hi+1(M/0: y)(cid:39)Hi (M/ymM).
m m M m
(cid:3)
3. Localization
Ingeneralafinitelygeneratedmoduleoveralocalringdoesnotnecessarilyhave
a p-standard system of parameters. A typical example is the two dimensional local
domain given by Ferrand-Raynaud [?, Proposition 3.3]. There are only some suf-
ficient conditions for the existence of such a system of parameters, for examples,
when the ground ring admits a dualizing complex (cf. [?]). Other examples are
Cohen-Macaulay modules and generalized Cohen-Macaulay modules, namely, reg-
ular and standard systems of parameters respectively. Recently, the authors have
shown that this is also the case for sequentially Cohen-Macaulay modules [?] and
sequentiallygeneralizedCohen-Macaulaymodules[?]. Inthissection,wewillstudy
theexistenceofp-standardsystemofparameterswhenpassingtolocalization. This
is a key forthe proofs of Theorems ?? and ?? in next sections. Todo this, we need
first some lemmas.
Lemma 3.1. Let M be a finitely generated R-module of dimension d and let x be
a parameter element of M. Then (a(M))d−1 ⊆a(M/xM).
Proof. Set b(M) = (cid:84)d Ann(0 : x ) where x ,...,x runs over the
i=1 i M/(x1,...,xi−1)M 1 d
set of all systems of parameters of M. By Schenzel [?, Satz 2.4.5], we have
a(M)⊆b(M)⊆a (M)∩...∩a (M).
0 d−1
Hence, (a(M))d−1 ⊆(b(M))d−1 ⊆(b(M/xM))d−1 ⊆a(M/xM). (cid:3)
LOCAL COHOMOLOGY ANNIHILATORS AND MACAULAYFICATION 9
Lemma 3.2. Let M be a finitely generated R-module. Assume that M admits a
p-standard system of parameters x ,...,x . Let y ∈a(M) be a parameter element
1 d d
of M. There are y ,...,y ∈ m such that yn,...,yn is a p-standard system of
1 d−1 1 d
parameters of M for all n(cid:29)0.
Proof. Take n≥5·d!. We choose y by induction on r =d−k. The case r =0 is
k
trivial. Assume that r >0 and we have chosen y ,...,y such that
k+1 d
(i) y ,...,y ,x ,...,x isapartofasystemofparametersfori=k+1,...,d.
k+1 i i+1 d
(ii) y ∈a(M/(yn ,...,yn)M) for i=k+1,...,d.
i i+1 d
FromLemma??,xn,yn ∈a(M/(yn,...,yn )M),i=k+1,...,d−1. Thisimplies
d d i d−1
by Corollary ?? that
Hj(M/(yn,...,yn ,yn)M)(cid:39)Hj(M/(yn,...,yn ,xn)M),
m i d−1 d m i d−1 d
for j =0,1,...,i−2. The module M/xnM admits a p-standard system of param-
d
eters x ,...,x and y ,...,y is a part of a system of parameters of it such
1 d−1 k+1 d−1
that
y ∈a(M/(yn ,...,yn ,xn)M),
i i+1 d−1 d
for i=k+1,...,d−1. Using the induction hypothesis, we can choose
y ∈a(M/(yn ,...,yn ,xn)M)=a(M/(yn ,...,yn ,yn)M),
k k+1 d−1 d k+1 d−1 d
such thaty (cid:54)∈p forall p∈AssM/(y ,...,y ,x ,...,x )M with dimR/p=k,
k k+1 i i+1 d
i=k+1,...,d. (cid:3)
In the next proposition we will show that the existence of p-standard system
of parameters is preserved when passing to localization. A technical remark is, in
some cases dd-sequences have more advantage than p-standard systems of param-
eters, one can see this in the proofs of the previous lemmas and of the following
proposition.
Proposition 3.3. Let R be a Noetherian local ring and M be a finitely gen-
erated R-module. Let p ∈ Supp(M) and set r = dimR/p. Assume in addi-
tion that M admits a p-standard system of parameters. There is a p-standard
system of parameters x ,...,x of M such that x ,...,x ∈ p and if we put
1 d r+1 d
s =dimM ,s =depthM , then
1 p 2 p
(i) x ,...,x is a p-standard system of parameters of M .
r+1 r+s1 p
(ii) x ,...,x is a maximal regular sequence on M .
r+1 r+s2 p
In particular, M admits a p-standard system of parameters.
p
Proof. If r < d then we choose first a parameter element x ∈ p ∩ a(M). Let
d
n>5·d!. By Lemma ??, M/xnM also admits a p-standard system of parameters.
d
Ifr <d−1wecanchooseaparameterselementx ∈p∩a(M/xnM)ofM/xnM.
d−1 d d
By the recursive method, we get a system of parameters x ,...,x of M such that
1 d
x ,...,x ∈ p and x ∈ a(M/(xn ,...,xn)M), i = 1,...,d. Then xn,...,xn
r+1 d i i+1 d 1 d
is a p-standard system of parameters of M with x ,...,x ∈ p. Without lost
r+1 d
of generality we assume that x ,...,x is a p-standard system of parameters. By
1 d
Lemma??,x ,...,x isadd-sequenceonM,hencex ,...,x isadd-sequenceon
1 d r+1 d
M by Lemma ??(i). Passing to localization, it can be seen easily that x ,...,x
r+1 d
is also a dd-sequence on M for any prime ideal q ∈ Spec(R). Moreover, we have
q
10 NGUYENTUCUONGAND DOANTRUNGCUONG
(cid:112)
(x ,...,x )R =pR and
r+1 d p p
∞
(cid:91)
0: x = 0: (xt ,...,xt)=H0 (M ).
Mp r+1 Mp r+1 d pRp p
t=1
So if dimM > 0 then x is a parameter element of M . Using Lemma ??(ii),
p r+1 p
x ,...,x(cid:91),...,x is a dd-sequence on M/x M. Induction on the dimension of
1 r+1 d r+1
M shows that x ,...,x is a system of parameters of M which is also a dd-
r+1 r+s1 p
sequence on M . Using Lemma ?? again we see that for t big enough, xt,...,xt is
p 1 d
a p-standard system of parameters of M with x ,...,x ∈p and xt ,...,xt
r+1 d r+1 r+s1
is a p-standard system of parameters of M .
p
The regularity of the sequence x ,...,x on M is proved similarly. (cid:3)
r+1 r+s2 p
Corollary3.4. Keepallassumptionsinthepreviousproposition. Fork ≥dimR/p,
we have
a (M)R ⊆a (M ).
k p k−dimR/p p
Proof. Set r = dimR/p. We have shown in Proposition ?? the existence of a p-
standard system of parameters x ,...,x of M such that x ,...,x ∈ p. By
1 d r+1 d
Corollary ??,
a (M)[(xn1,...,xnk)M :x ]⊆(xn1+1,...,xnk+1)M :(x ...x ),
k 1 k k+1 1 k 1 k
for k =0,1,...,d and x =0. Applying Krull’s Intersection Theorem we get
d+1
a (M)[(xnr+1,...,xnk)M :x ]= (cid:92) a (M)[(xn1,...,xnk)M :x ]
k r+1 k k+1 k 1 k k+1
n1,...,nr
(cid:92)
⊆ (xn1+1,...,xnk+1)M :(x ...x )
1 k 1 k
n1,...,nr
=(xnr+1+1,...,xnk+1)M :(x ...x ).
r+1 k 1 k
Since x ,...,x (cid:54)∈p, the above inclusion becomes
1 r
a (M)R [(xnr+1,...,xnk)M :x ]⊆(xnr+1+1,...,xnk+1)M :(x ...x ).
k p r+1 k p k+1 r+1 k p r+1 k
(cid:112)
Itshouldbenotedthatx ,...,x isadd-sequenceonM and (x ,...,x )R =
r+1 d p r+1 d p
pR . So by Corollary ?? again,
p
a (M )= (cid:92) Ann (xnr+r+11,...,xnkk)Mp :xk+1 .
k−r p (xnr+1+1,...,xnk+1)M :(x ...x )
nr+1,...,nk>0 r+1 k p r+1 k
Therefore, a (M)R ⊆a (M ). (cid:3)
k p k−r p
Intherestofthissection,wewillusetheresultsabovetoinvestigatethebehavior
of the depth of a module when passing to localization. Note that this result has
been known in some special cases (cf. Schenzel [?]). We start with a lemma.
Lemma 3.5. Let M be a finitely generated R-module and x ,...,x be a system
1 d
of parameters of M. Assume that x ,...,x is a dd-sequence on M. For k =
1 d
0,1,...,d−1 and n≥5 we have
k+1
(cid:92) (cid:112)
Ann(0:x ) ⊆ a (M)...a (M).
i M/(xn ,...,xn )M 0 k
i+1 k+1
i=1