Through out this chapter, let (R; m) be a Noetherian local ring, I be an
ideal of R and M be a finitely generated R-module with dim M = d: Let
Var(I) denote the set of all prime ideals of R containing I. Denote by Rb and
Mc the m-adic completions of R and M respectively.
The set of attached primes of the top local cohomology modules with
respect to the maximal ideal Hmd (M) over R and Rb was described clearly by
I. G. Macdonald and R. Y. Sharp. By the Lichtenbaum-Hartshorne Vanishing
Theorem, R. Y. Sharp clarified the set of attached primes of HIdim R(R)
over R: b After that, K. Divaani-Aazar and P. Schenzel extended this result
for module. The purpose of this chapter is to describe this set for HId(M)
over R in the relation with the prime saturation, co-support and associative
formula for multiplicity of HId(M). On the study of the set of attached primes
of Hd
I (M), we characterized the prime saturation for this module via the
catenary of base ring and tranfered to the case of the top local cohomology
modules with respect to the maximal ideal of a quotient module of M

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Mp). Víi gi¶ thiÕt vµnh R lµ catenary phæ dông
vµ mäi thí h×nh thøc lµ Cohen-Macaulay, M. Brodmann vµ R. Y. Sharp ®·
xem vai trß cña Rp-m«®unH
i−dim(R/p)
pRp
(Mp) nh lµ "®èi ®Þa ph¬ng hãa" cña
m«®un ®èi ®ång ®iÒu ®Þa ph¬ng Artin H im(M) nh»m x©y dùng thµnh c«ng
c«ng thøc liªn kÕt cho sè béi cña H im(M). V× thÕ vÊn ®Ò ë ®©y lµ t×m ®iÒu
kiÖn ®Ó tån t¹i mét ®èi ®Þa ph¬ng hãa t¬ng thÝch víi mäi m«®un ®èi ®ång
®iÒu ®Þa ph¬ng Artin H im(M). Trong phÇn nµy chóng t«i ®a ra ®iÒu kiÖn
cÇn ®Ó tån t¹i mét ®èi ®Þa ph¬ng hãa nh vËy cho H im(M).
§Þnh lý 2.3.11. Gi¶ sö tån t¹i, víi mçi p ∈ Spec(R), mét hµm tö khíp, tuyÕn
tÝnh Fp :MR →MRp trªn ph¹m trï c¸c R-m«®un, Fp(H im(M)) lµ Artin vµ
Fp(H
i
m(M)) 6= 0 víi bÊt k× p ⊇ AnnRH im(M), víi mäi sè nguyªn i vµ víi
mäi R-m«®un h÷u h¹n sinhM . Khi ®ã vµnh R lµ catenary phæ dông vµ mäi
thí h×nh thøc lµ Cohen-Macaulay.
KÕt luËn ch¬ng II
Tãm l¹i, trong ch¬ng nµy chóng t«i ®· thu ®îc c¸c kÕt qu¶ sau ®©y:
- §Æc trng vµnh c¬ së ®Ó c«ng thøc chuyÓn tËp i®ªan nguyªn tè g¾n kÕt
qua ®Çy ®ñ tháa m·n víi mäi R-m«®un Artin.
- Chøng minh c«ng thøc chuyÓn tËp i®ªan nguyªn tè g¾n kÕt cña H im(M)
qua ®Çy ®ñ khi vµnh c¬ së lµ th¬ng cña vµnh Gorenstein.
- §a ra mét sè ®Æc trng cña vµnh catenary phæ dông víi c¸c thí h×nh
thøc Cohen-Macaulay th«ng qua mèi quan hÖ gi÷a c¸c tËp AttR̂H
i
m(M) vµ
AttRH
i
m(M); mèi quan hÖ gi÷a Psupp
i(M) vµ Psuppi(M̂).
- ChØ ra mét ®iÒu kiÖn cÇn trªn vµnh c¬ së R ®Ó tån t¹i ®èi ®Þa ph¬ng
hãa t¬ng thÝch víi mäi R-m«®un Artin.
- §a ra mét tiªu chuÈn cña vµnh c¬ së ®Ó tån t¹i ®èi ®Þa ph¬ng hãa
t¬ng thÝch víi mäi m«®un ®èi ®ång ®iÒu ®Þa ph¬ng H im(M).
17
Ch¬ng 3
M«®un ®èi ®ång ®iÒu ®Þa ph¬ng cÊp cao nhÊt
víi gi¸ tïy ý
Trong suèt ch¬ng nµy, lu«n gi¶ thiÕt (R,m) lµ vµnh Noether ®Þa ph¬ng
víi i®ªan cùc ®¹i duy nhÊt m, M lµ R-m«®un h÷u h¹n sinh víi dimM = d
vµ A lµ R-m«®un Artin. Víi mçi i®ªan I cña R, kÝ hiÖu Var(I) lµ tËp c¸c
i®ªan nguyªn tè cña R chøa I . KÝ hiÖu R̂ vµ M̂ lÇn lît lµ ®Çy ®ñ m-adic
cña R vµ M .
TËp i®ªan nguyªn tè g¾n kÕt cña m«®un ®èi ®ång ®iÒu cÊp cao nhÊt víi
gi¸ cùc ®¹i Hdm(M) trªn R vµ R̂ ®· ®îc I. G. Macdonald vµ R. Y. Sharp
m« t¶ râ rµng. Tõ §Þnh lý triÖt tiªu Lichtenbaum-Hartshorne, R. Y. Sharp
tiÕp tôc m« t¶ tËp i®ªan nguyªn tè g¾n kÕt cña HdimRI (R) trªn vµnh R̂. Sau
®ã, K. Divaani-Aazar vµ P. Schenzel ®· më réng kÕt qu¶ nµy cho m«®un.
Môc tiªu cña ch¬ng nµy lµ m« t¶ tËp i®ªan nguyªn tè g¾n kÕt cña m«®un
HdI (M) trªn vµnh R trong mèi liªn hÖ víi tÝnh b·o hßa nguyªn tè, ®èi ®Þa
ph¬ng hãa vµ c«ng thøc béi liªn kÕt cña m«®un HdI (M). §Ó nghiªn cøu
tËp i®ªan nguyªn tè g¾n kÕt cña HdI (M), chóng t«i ®Æc trng tÝnh b·o hßa
nguyªn tè cho m«®un nµy th«ng qua tÝnh catenary cña vµnh råi chuyÓn nã
vÒ m«®un ®èi ®ång ®iÒu ®Þa ph¬ng cÊp cao nhÊt víi gi¸ cùc ®¹i cña mét
m«®un th¬ng cña M.
3.1 TÝnh b·o hßa nguyªn tè
Theo N. T. Cêng vµ L. T. Nhµn, mét R-m«®un A tháa m·n tÝnh b·o hßa
nguyªn tè nÕu AnnR(0 :A p) = p víi mäi i®ªan nguyªn tè p ⊇ AnnRA. Khi
R lµ vµnh ®Çy ®ñ, mäi m«®un ®èi ®ång ®iÒu ®Þa ph¬ng Artin ®Òu tháa m·n
tÝnh b·o hßa nguyªn tè. Tuy nhiªn, nh×n chung c¸c m«®un ®èi ®ång ®iÒu ®Þa
ph¬ng Artin kh«ng cã tÝnh chÊt nµy. TÝnh b·o hßa nguyªn tè ®· ®îc ®Æc
trng cho c¸c m«®un ®èi ®ång ®iÒu ®Þa ph¬ng víi gi¸ cùc ®¹i. Trong tiÕt
nµy, chóng t«i tiÕp tôc ®Æc trng tÝnh b·o hßa nguyªn tè cho líp m«®un ®èi
18
®ång ®iÒu ®Þa ph¬ng Artin cÊp cao nhÊt øng víi gi¸ bÊt k×HdI (M) th«ng qua
tÝnh catenary cña vµnh råi chuyÓn nã vÒ m«®un ®èi ®ång ®iÒu ®Þa ph¬ng
cÊp cao nhÊt víi gi¸ cùc ®¹i cña mét m«®un th¬ng cñaM. Tríc hÕt, chóng
t«i ®a ra mét sè kÝ hiÖu liªn quan ®Õn ph©n tÝch nguyªn s¬ cña m«®un con
0 cña M. KÝ hiÖu nµy ®îc sö dông trong toµn bé Ch¬ng 3.
KÝ hiÖu 3.1.1. Cho 0 =
⋂
p∈AssRM
N(p) lµ mét ph©n tÝch nguyªn s¬ thu gän
cña m«®un con 0 cña M . KÝ hiÖu
AssR(I,M) =
{
p ∈ AssRM | dim(R/p) = d,
√
I + p = m
}
.
§Æt N =
⋂
p∈AssR(I,M)
N(p). V× mçi phÇn tö cña AssR(I,M) ®Òu lµ i®ªan
nguyªn tè liªn kÕt tèi thiÓu cñaM nªn R-m«®un N kh«ng phô thuéc vµo sù
lùa chän ph©n tÝch nguyªn s¬ thu gän cña m«®un con 0.
Sau ®©y lµ ®Æc trng tÝnh b·o hßa nguyªn tè cña m«®un ®èi ®ång ®iÒu ®Þa
ph¬ng HdI (M) trong mèi liªn hÖ víi tÝnh catenary cña vµnh c¬ së vµ tËp
c¸c i®ªan nguyªn tè g¾n kÕt cña HdI (M).
§Þnh lý 3.1.2. Cho R-m«®un N x¸c ®Þnh nh trong KÝ hiÖu 3.1.1. Khi ®ã
c¸c mÖnh ®Ò sau lµ t¬ng ®¬ng:
(i) HdI (M) tháa m·n tÝnh b·o hßa nguyªn tè;
(ii) R/AnnRH
d
I (M) lµ vµnh catenary vµ
√
I + p = m víi mäi i®ªan
nguyªn tè g¾n kÕt p cña HdI (M);
(iii) R/AnnRH
d
I (M) lµ vµnh catenary vµ H
d
I (M)
∼= Hdm(M/N).
3.2 TËp i®ªan nguyªn tè g¾n kÕt
Trong tiÕt nµy chóng t«i më réng kÕt qu¶ cña K. Divaani-Aazar vµ P.
Schenzel tõ trêng hîp R ®Çy ®ñ sang trêng hîp m«®un HdI (M) tháa m·n
tÝnh b·o hßa nguyªn tè. Tríc hÕt, tõ §Þnh lý 3.1.2, chóng ta cã hÖ qu¶ sau
vÒ mèi liªn hÖ gi÷a tËp AssR(I,M) vµ AttRH
d
I (M).
19
HÖ qu¶ 3.2.1. Cho AssR(I,M) nh trong KÝ hiÖu 3.1.1. Khi ®ã ta cã
(i) AssR(I,M) ⊆ AttRHdI (M). §Æc biÖt, nÕu AssR(I,M) 6= ∅ th×
HdI (M) 6= 0.
(ii) Gi¶ sö r»ng AssR(I,M) = ∅. Khi ®ã HdI (M) tháa m·n tÝnh b·o hßa
nguyªn tè khi vµ chØ khi HdI (M) = 0.
HÖ qu¶ sau ®©y lµ kÕt qu¶ chÝnh cña tiÕt nµy.
HÖ qu¶ 3.2.2. NÕu HdI (M) tháa m·n tÝnh b·o hßa nguyªn tè th×
AttRH
d
I (M) =
{
p ∈ AssRM | dim(R/p) = d,
√
I + p = m
}
.
MÖnh ®Ò 3.2.3 Cho AssR(I,M) ®îc x¸c ®Þnh nh trong KÝ hiÖu 3.1.1. Khi
®ã c¸c mÖnh ®Ò sau lµ t¬ng ®¬ng:
(i) AttR̂H
d
I (M) =
⋃
p∈AttRHdI (M)
AssR̂(R̂/pR̂);
(ii) HdI (M) tháa m·n tÝnh b·o hßa nguyªn tè vµ R/p kh«ng trén lÉn víi
mäi p ∈ AssR(I,M).
3.3 §èi gi¸ vµ sè béi
Cho p ∈ Spec(R). K. E. Smith ®· nghiªn cøu hµm tö "®èi ngÉu víi ®Þa
ph¬ng hãa"
Fp(−) = HomR
(
HomR(−, E(R/m)), E(R/p)
)
tõ ph¹m trï c¸c R-m«®un ®Õn ph¹m trï c¸c Rp-m«®un, trong ®ã E(−) lµ
bao néi x¹ .
MÖnh ®Ò 3.3.1. Cho p ∈ Spec(R) vµ Fp(−) lµ hµm tö ®èi ngÉu víi ®Þa
ph¬ng hãa ë trªn. Cho N x¸c ®Þnh nh trong KÝ hiÖu 3.1.1. Gi¶ sö r»ng R
lµ ®Çy ®ñ. Khi ®ã
Fp(H
d
I (M))
∼= Hd−dim(R/p)pRp (M/N)p.
MÖnh ®Ò 3.3.1 gîi ý chóng t«i ®a ra kh¸i niÖm tËp ®èi gi¸ cña m«®un ®èi
®ång ®iÒu ®Þa ph¬ng HdI (M) nh sau.
20
§Þnh nghÜa 3.3.2. Cho N x¸c ®Þnh nh trong KÝ hiÖu 3.1.1. TËp ®èi gi¸ cña
HdI (M), kÝ hiÖu lµ CosR(H
d
I (M)), ®îc cho bëi c«ng thøc
CosR(H
d
I (M)) =
{
p ∈ Spec(R) | Hd−dim(R/p)pRp (M/N)p 6= 0
}
.
Bæ ®Ò díi ®©y cho ta mèi quan hÖ gi÷a tËp ®èi gi¸ CosR(H
d
I (M)) vµ tËp
Var(AnnRH
d
I (M)).
Bæ ®Ò 3.3.3. CosR(H
d
I (M)) ⊆ Var(AnnRHdI (M)).
§Þnh lý sau ®©y ®Æc trng tÝnh b·o hßa nguyªn tè cña m«®un ®èi ®ång
®iÒu ®Þa ph¬ng HdI (M) th«ng qua tËp ®èi gi¸ CosR(H
d
I (M)).
§Þnh lý 3.3.5. C¸c mÖnh ®Ò sau lµ t¬ng ®¬ng:
(i) HdI (M) tháa m·n tÝnh b·o hßa nguyªn tè;
(ii) CosR(H
d
I (M)) = Var(AnnRH
d
I (M)).
§Þnh lý 3.3.5 cho ta thÊy nÕu HdI (M) tháa m·n tÝnh b·o hßa nguyªn tè
th× ®èi gi¸ cña nã lµ mét tËp con ®ãng cña Spec(R) trong t«p« Zariski.
Theo D. Kirby, nÕu q lµ i®ªan cña R sao cho (0 :A q) cã ®é dµi h÷u h¹n
th× `(0 :A q
n+1) lµ mét ®a thøc bËc N-dimRA víi hÖ sè h÷u tû khi n ®ñ lín.
Ta kÝ hiÖu ®a thøc nµy lµ ΘqA(n). §Æt N-dimA = s. Ta cã biÓu diÔn
ΘqA(n) = `(0 :A q
n+1) =
e′(q, A)
s!
ns + ®a thøc cã bËc nhá h¬n s
khi n 0, trong ®ã e′(q, A) lµ mét sè nguyªn d¬ng. Ta gäi e′(q, A) lµ sè
béi cña A øng víi q. N¨m 2002, M. Brodmann vµ R. Y. Sharp ®· giíi thiÖu
kh¸i niÖm tËp gi¶ gi¸ PsuppiR(M) vµ gi¶ chiÒu thø i, kÝ hiÖu lµ psd
i(M), ®Ó
x©y dùng thµnh c«ng c«ng thøc béi liªn kÕt cho m«®un H im(M). GÇn ®©y,
L. T. Nhµn vµ T. N. An ®· më réng kÕt qu¶ trªn cña M. Brodmann vµ R. Y.
Sharp cho trêng hîp H im(M) tho¶ m·n tÝnh b·o hßa nguyªn tè. V× thÕ, víi
tËp ®èi gi¸ cña HdI (M) võa ®Þnh nghÜa ë trªn, chóng t«i tiÕp tôc x©y dùng
c«ng thøc béi liªn kÕt cho m«®un ®èi ®ång ®iÒu ®Þa ph¬ng nµy khi nã tháa
m·n tÝnh b·o hßa nguyªn tè.
21
HÖ qu¶ 3.3.8. Gi¶ sö q lµ i®ªan m-nguyªn s¬. Cho AssR(I,M) vµ N x¸c
®Þnh nh trong KÝ hiÖu 3.1.1. NÕu HdI (M) tháa m·n tÝnh b·o hßa nguyªn tè
th×
e′(q, HdI (M)) =
∑
p∈CosHdI (M)
dim(R/p)=d
`Rp
(
H0pRp(M/N)p
)
e(q, R/p).
Trong trêng hîp nµy,
e′(q, HdI (M)) = e(q,M/N) =
∑
p∈AssR(I,M)
`Rp(Mp)e(q, R/p).
KÕt luËn ch¬ng III
Tãm l¹i, trong ch¬ng nµy chóng t«i ®· thu ®îc c¸c kÕt qu¶ sau ®©y:
- §a ra mét sè ®Æc trng vÒ tÝnh b·o hßa nguyªn tè cho m«®un ®èi ®ång
®iÒu ®Þa ph¬ng cÊp cao nhÊt víi gi¸ tïy ý th«ng qua tÝnh catenary cña vµnh
c¬ së, tÝnh chÊt cña tËp i®ªan nguyªn tè g¾n kÕt cña HdI (M) vµ mèi liªn hÖ
víi m«®un ®èi ®ång ®iÒu ®Þa ph¬ng cÊp cao nhÊt víi gi¸ cùc ®¹i.
- M« t¶ ®îc tËp i®ªan nguyªn tè g¾n kÕt cña HdI (M) khi m«®un Artin
nµy tháa m·n tÝnh b·o hßa nguyªn tè.
- §a ra kh¸i niÖm tËp ®èi gi¸ cho m«®un ®èi ®ång ®iÒu ®Þa ph¬ng cÊp
cao nhÊt víi gi¸ tïy ý vµ ®Æc trng tÝnh b·o hßa nguyªn tè cña HdI (M) th«ng
qua tËp ®èi gi¸ nµy.
- Th«ng qua tËp ®èi gi¸ ®Ó m« t¶ thµnh c«ng c«ng thøc béi liªn kÕt cho
m«®un HdI (M) khi m«®un Artin nµy tháa m·n tÝnh b·o hßa nguyªn tè.
22
KÕt luËn vµ kiÕn nghÞ
1. KÕt luËn
Trong luËn ¸n nµy, chóng t«i ®· thu ®îc mét sè kÕt qu¶ vÒ tËp i®ªan
nguyªn tè g¾n kÕt cña m«®un ®èi ®ång ®iÒu ®Þa ph¬ng cÊp tïy ý víi gi¸
cùc ®¹i vµ m«®un ®èi ®ång ®iÒu ®Þa ph¬ng cÊp cao nhÊt víi gi¸ bÊt k×.
§èi víi m«®un ®èi ®ång ®iÒu ®¹i ph¬ng cÊp tïy ý víi gi¸ cùc ®¹i, chóng
t«i chøng minh c«ng thøc chuyÓn tËp i®ªan nguyªn tè g¾n kÕt cña m«®un
nµy qua ®Çy ®ñ khi R lµ th¬ng cña vµnh Gorenstein ®Þa ph¬ng (MÖnh ®Ò
2.1.7), ®ång thêi ®Æc trng vµnh catenary phæ dông víi mäi thí h×nh thøc lµ
Cohen-Macaulay th«ng qua c«ng thøc chuyÓn tËp i®ªan nguyªn tè g¾n kÕt
nµy (§Þnh lý 2.2.5). Chóng t«i cßn ®a ra mét ®iÒu kiÖn cÇn cña vµnh c¬
së cho sù tån t¹i hµm tö ®èi ®Þa ph¬ng hãa t¬ng thÝch víi mäi m«®un ®èi
®ång ®iÒu ®¹i ph¬ng víi gi¸ cùc ®¹i (§Þnh lý 2.3.11).
§èi víi m«®un ®èi ®ång ®iÒu ®Þa ph¬ng cÊp cao nhÊt víi gi¸ lµ i®ªan bÊt
k×, chóng t«i m« t¶ tËp i®ªan nguyªn tè g¾n kÕt cña m«®un nµy b»ng c¸ch
®Æc trng tÝnh b·o hßa nguyªn tè th«ng qua tÝnh catenary cña vµnh c¬ së
råi chuyÓn vÒ trêng hîp m«®un ®èi ®ång ®iÒu ®Þa ph¬ng víi gi¸ cùc ®¹i
(§Þnh lý 3.1.2). Chóng t«i cßn ®Æc trng tÝnh b·o hßa nguyªn tè th«ng qua
kh¸i niÖm tËp ®èi gi¸ (§Þnh lý 3.3.5), tõ ®ã x©y dùng c«ng thøc béi liªn kÕt
cho m«®un nµy (HÖ qu¶ 3.3.8).
§Ó lµm s¸ng tá c¸c kÕt qu¶ trong toµn luËn ¸n, chóng t«i ®a ra c¸c vÝ dô
minh häa.
2. KiÕn nghÞ
Trong thêi gian tíi, chóng t«i dù kiÕn sÏ nghiªn cøu c¸c vÊn ®Ò sau:
- §a ra mét sè líp vµnh kh«ng ®Çy ®ñ ®Ó tån t¹i mét hµm tö ®èi ®Þa
23
ph¬ng hãa t¬ng thÝch víi mäi m«®un ®èi ®ång ®iÒu ®Þa ph¬ng cÊp bÊt k×
víi gi¸ cùc ®¹i.
- Më réng c¸c kÕt qu¶ vÒ tËp i®ªan nguyªn tè g¾n kÕt, tÝnh b·o hßa nguyªn
tè, ®èi gi¸ vµ sè béi cña m«®un ®èi ®ång ®iÒu ®Þa ph¬ng víi gi¸ lµ i®ªan
tïy ý t¹i cÊp cao nhÊt sang m«®un ®èi ®ång ®iÒu ®Þa ph¬ng víi gi¸ lµ i®ªan
tïy ý t¹i mét sè cÊp thÊp h¬n.
- Khai th¸c kÜ thuËt gi¶ gi¸ vµ ph©n tÝch nguyªn s¬ trong nghiªn cøu cÊu
tróc cña c¸c m«®un h÷u h¹n sinh trªn vµnh Noether ®Þa ph¬ng.
C¸c c«ng tr×nh liªn quan ®Õn luËn ¸n
1. L. T. Nhan and T. D. M. Chau (2012), On the top local cohomology
modules, J. Algebra, 349, 342-352.
2. L. T. Nhan and T. D. M. Chau (2014), Noetherian dimension and co-
localization of Artinian modules over local rings, Algebra Colloquium,
(4)21, 663-670.
3. T. D. M. Chau and L. T. Nhan (2014), Attached primes of local cohomology
modules and structure of Noetherian local rings, J. Algebra, 403, 459-469.
C¸c kÕt qu¶ trong luËn ¸n ®· ®îc b¸o
c¸o vµ th¶o luËn t¹i
- Xemina §¹i sè - §¹i häc Th¸i Nguyªn.
- Xemina cña Tæ §¹i sè - Khoa To¸n - Trêng §¹i häc S Ph¹m HuÕ.
- Héi th¶o phèi hîp NhËt B¶n - ViÖt Nam lÇn thø 7 vÒ §¹i sè giao ho¸n, Quy
Nh¬n, 12/2011.
- Héi nghÞ To¸n häc phèi hîp ViÖt - Ph¸p, HuÕ, 08/2012.
- §¹i héi To¸n häc toµn quèc lÇn thø 8, Nha Trang, 08/2013.
Hue university
University of Education
TRAN DO MINH CHAU
On the attached primes of
local cohomology modules
Speciality: Algebra and Number Theory
Code: 62.46.01.04
A summary of
mathematics doctoral thesis
HUE - 2014
This work is finished at University of Education- Hue University
Supervisors :
1. Assoc. Prof. Dr. Le Thi Thanh Nhan
2. Prof. Dr. Le Van Thuyet
Reviewer 1: ...................................................................
Reviewer 2: ...................................................................
Reviewer 3: ...................................................................
This thesis will be defended at the University-level Council which is
celebrated at Hue University, Number 04, Le Loi Road, Thua Thien Hue
city on .......................
This thesis can be found at
-The Viet Nam National Library
-The Library of University of Education - Hue University
1Introduction
In 1960s, A. Grothendieck introduced the local cohomology theory
based on J. P. Serre's works on algebaric sheaves in 1955 . After that, this
theory developed rapidly and was interested in by many mathematicians all
over the world. Nowadays, local cohomology theory have become an indis-
pensable tool in many different fields of mathematics such as Commutative
Algebra, Algebaric Geometry, Combinatory Algebra ...
One of the important results of local cohomology modules was the van-
ishing. Let M be a module over commutative Noetherian ring R. In1967,
A. Grothendieck proved that H iI(M) always vanishes at all level i >
dim SuppM and if (R,m) is a local ring, M is a finitely generated mod-
ule then Hdm(M) 6= 0, where d = dimM. Then, he also showed that the
depth of M is the least integer i for which H im(M) 6= 0. The Lichtenbaum-
Hartshorne Vanishing Theorem also states that if I is an ideal of local ring
(R,m) with dimR = n thenHnI (R) = 0 if and only if dim R̂/(IR̂+P) ≥ 1
for every associated primes of greastest dimension P of m-adic R̂. The sec-
ond result which were interested in is finitely generated property. In general,
even when M is finitely generated, H iI(M) is not. So it is natural to ask in
which conditionH iI(M) is finitely generated. In 1978, G. Faltings character-
ized the least integer i for which H iI(M) is not finitely generated. Specially,
he also provided the Local-global Principle for finiteness dimension of local
cohomology modules.
Artinian property of local cohomology also have attracted many mathe-
maticians. Let (R,m) be a Noetherian commutative local ring and M be a
finitely generated R-module with dimM = d. In 1971, by a short proof us-
ing minimal injective resolution ofM and Artinian property of injective hull
2E(R/m), I. G. Macdonald and R. Y. Sharp showed that H im(M) is Artinian
for every i ≥ 0. Then, by Lichtenbaum-Hartshorne Theorem, R. Y. Sharp
proved that local cohomology modules HdI (M) is also Artinian. Afterward,
L. Melkersson showed these results by a primary method. Many information
of two classes of Artinian local cohomology modules H im(M) and H
d
I (M)
were shown in works of R. Y. Sharp, M. Brodmann-Sharp, M. Hochster and
C. Huneke , K. E. Smith, K. Divaani-Aazar and P. Schenzel, H. Zo¨schinger
and N. T. Cuong with his students.
By I. G. Macdonald, the set of attached primes of Artinian R-module A,
denoted by AttRA, is in some sense similar to that of associated primes
of finitely generated modules and has an important role in the study of the
structure ofA. The purpose of this thesis is to study the set of attached primes
of local cohomology modules with respect to the maximal ideal H im(M) and
that of the top local cohomology modules with respect to an arbitrary ideal
HdI (M), then clarify the structure ofM and the base ring R.Moreover, these
sets are studied in the relation with multiplcity, the prime saturation and the
co-localization of H im(M) and H
d
I (M). Recall that an Artinian R-module A
is called to satisfy the prime saturation if AnnR(0 :A p) = p for every prime
ideal p containing AnnRA. The prime saturation was introduced by N. T.
Cuong and L. T. Nhan in order to study the structure of Artinian module.
Note that local cohomology modules H im(M) has a structure as an R̂-
module, so the set of attached primes of H im(M) over R̂ is always defined.
A natural question is that how is the relation between AttRH
i
m(M) and
AttR̂H
i
m(M). In1975, R. Y. Sharp showed that when the prime ideal
P of R̂ runs through AttR̂H
i
m(M) then the set of prime ideal P ∩ R
is AttRH
i
m(M). He also provided some information on the dimension of
attached primes of H im(M) over R. However, for given AttRH
i
m(M), how
to define AttR̂H
i
m(M)? This contrary question hasn't been solved. In this
thesis, we will provide the answer.
3When R is a quotient of a Gorenstein local ring, in1975, R. Y. Sharp
proved the Shifted localization principle to pass the set of attached primes
of H im(M) through localization H
i−dimR/p
pRp
(Mp).Then R. Y. Sharp and M.
Brodmann used this result on the study of the dimension and multiplicity for
local cohomology modules H im(M) and extended this result to universally
catenary rings whose all formal fibers are Cohen-Macaulay. Note that this
Shifted Localization Principle is not true in general. So the second question,
which will be solved in this thesis, is to find a neccessary condition of base
ring R such that there exists a co-localization compatible with every local
cohomology modules H im(M).
The set of attached primes of Hdm(M) over R and R̂ was clarified by I. G.
Macdonald and R. Y. Sharp in 1971. Then by the Lichtenbaum-Hartshorne
Vanishing Theorem, R. Y. Sharp described the set of attached primes of
HdimRI (R) over R̂.After that, K. Divaani-Aazar and P. Schenzel extended this
result for module. However, defining this set of HdI (M) over R is open. The
third question, that is proved in this thesis, is describing the set of attached
primes forHdI (M) overR in the relation with the prime saturation, co-support
and associative formula for multiplicity of this module.
For method of approach, we use local duality and known properties of
H im(M) on the study of local cohomology modules with respect to the
maximal ideal when R is a quotient of Gorenstein ring. However, on an
arbitrary ring, we have to exploit the pseudo-support defined byM. Brodmann
and R. Y. Sharp and special property of Artinian modules. In oder to study
HdI (M), we need some deepth tools on the primary decomposition, cofinite
and some known results of these modules.
The thesis includes 3 chapters. Chapter 1 presents some basic knowledge
of secondary representation and the set of attached primes, Artinian local
cohomology modules, the dimension and the prime saturation of Artinian
modules, universally catenary rings. Chapter 2 provides some results of the
4thesis on the set of attached primes for local cohomology modules with
respect to the maximal ideal. We spend Chapter 3 on presenting somes
findings for local cohomology modules with respect to an arbitrary ideal.
Through out the thesis, let (R,m) be a Noetherian local ring, A be an
Artinian R-module, I be an ideal of R and M be a finitely generated R-
module with Krull dimension d.
In Chapter 2, we present some researches on the passing attached primes
of local cohomology modules with respect to the maximal ideal through the
m-adic completions. Concretely, we characterized the universally catenary
rings whose all formal fibers are Cohen-Macaulay via the realation between
AttRH
i
m(M) and AttR̂H
i
m(M). We also provide a neccessary condition of
the base ring for the existence of a co-localization compatible with every
Artinian module H im(M).
For each finitely generated R-moduleM , the relations between the set of
associated primes of M and that of M̂ is given by the following formula
AssR̂ M̂ =
⋃
p∈AssM
AssR̂(R̂/pR̂).
However, the dual formula for Artinian modules A
AttR̂A =
⋃
p∈AttRA
AssR̂(R̂/pR̂) (1)
is not true in general, even when A = H im(M). We show that the formula
(1) is satisfied for every Artinian modules A if and only if the induced
map fa : Spec(R̂) → Spec(R) is bijective (Proposition 2.1.3). When R
is a quotient of a Gorenstein local ring (R′,m′) of dimension n, we prove
the following relation between AttRH
i
m(M)) and AttR̂H
i
m(M) (Proposition
2.1.7)
AttR̂(H
i
m(M)) =
⋃
p∈AttR(Him(M))
AssR̂(R̂/pR̂). (2)
5Note that there exists a Noetherian local ring R which can't be expressed as
a quotient of a Gorenstein local ring but the formula (2) holds true for every
finitely generated R-module M and for every integer i ≥ 0 (see Example
2.1.8). Therefore it is natural to ask about the problem in a more general
situation, where R is universally catenary and all its formal fibers are Cohen-
Macaulay. Theorem 2.2.5, which is the first main result of Chapter 2, gives a
partial to this question. We also provide some characterizations of universally
catenary local rings (R,m) having all Cohen-Macaulay formal fibers via the
relation between attached primes of H im(M) over R and that over R̂.
The main tool to prove Theorem 2.2.5 is the notion of pseudo support
given by Brodmann - Sharp. For an integer i ≥ 0, the i-th pseudo support
PsuppiR(M) of M is defined by
PsuppiRM = {p ∈ Spec(R) | H i−dim(R/p)pRp (Mp) 6= 0}.
Note that the role of PsuppiR(M) for the Artinian modules H
i
m(M) is in
some sense similar to that of the support for finitely generated modules. As
a consequence of Theorem 2.2.5, we characterize the structure of the base
ring R via the relation between PsuppiR(M) and Psupp
i
R̂
(M̂)
PsuppiRM = {p ∈ Spec(R) | H i−dim(R/p)pRp (Mp) 6= 0}.
It should be mentioned that, for each p ∈ Spec(R), the localization at p is
a linear exact functor from MR to MRp such that Mp is Noetherian and
Mp 6= 0 whenever p ⊇ AnnRM . However, even when p ⊇ AnnRA, if
p 6= m then Ap = 0. It is of interest to construct for each p ∈ Spec(R)
a ``co-localization" functor Fp : MR −→ MRp such that Fp is compatible
with Artinian R-modules, concretely Fp has the following properties:
(a) Fp is linear and exact on the category of Artinian R-modules;
(b) Fp sends Artinian R-modules to Artinian Rp-modules;
(c) Fp(A) 6= 0 if p ⊇ AnnRA for every Artinian R-module A.
6We will give a necessary condition for the existence of such a co-
localization which states that the natural map R −→ R̂ satisfies going up
property.
Some authors have tried to construct a co-localization Fp, for every p ∈
Spec(R). However, none of the known co-localizations have all the desired
properties (a), (b), (c). Under the assumption that R is universally catenary
and all formal fibers ofR are Cohen-Macaulay, Brodmann - Sharp considered
the role of Rp-module H
i−dim(R/p)
pRp
(Mp) as the ``co-localization" of the
Artinian local cohomology moduleH im(M) in order to establish successfully
an associativity formula for the multiplicity of H im(M). Therefore, it is
of interest to study the existence of a co-localization compatible with all
Artinian local cohomology modules H im(M). Theorem 2.3.11 gives a partial
answer to this question. We will show that a neccessary condition of base
ring for which, for each p ∈ Spec(R), there exists a co-localizationFp :
MR →MRp compatible with all local cohomology modules H im(M) is that
R is universally catenary and all formal fibers of R are Cohen-Macaulay.
In Chapter 3, we study the set of the attached primes for HdI (M) over R
in the relation with the prime saturation, co-support and associative formula
for multiplicity of HdI (M). By N. T. Cuong and L. T. Nhan, an Artinian
R-module A satisfies the prime saturation if AnnR(0 :A p) = p for all
p ∈ Var(AnnRA). This property is not satisfied for all Artinian local
cohomology modules in general. N. T. Cuong and L. T. Nhan - N. T.
Dung characterized the prime saturation for Hdm(M) via the catenary of
the ring R/AnnR(H
d
m(M)) and proved that if R is a domain, non-catenary
then HdimRm (R) doesn't satisfy the prime saturation. At any i-th level, L.
T. Nhan and T. N. An gave a characterization of the prime saturation for
H im(M). They showed that H
i
m(M) satisfies the prime saturation if and only
if PsuppiR(M) = Var
(
AnnRH
i
m(M)
)
. Note that HdI (M) is Artinian and
this module may not satisfy the prime saturation even whenR is a quotient of
7regular ring (see Example 3.3.7). On the study of the set of attached primes
of HdI (M), we characterized the prime saturation for this module via the
catenary of the base ring and reduce to the case of local cohomology modules
with respect to the maximal ideal of a quotient module ofM (Theorem 3.1.2).
By Lichtenbaum-Hartshorne Vanising Theorem, R. Y. Sharp described the
set of attached primes of HdimRI (R) over R̂. This result is extended for
module by K. Divaani-Aazar and P. Schenzel. In this thesis, by Theorem
3.1.2, we will extend R. Y. Sharp's result for HdI (M) when this module
satisfies the prime saturation (Corollary 3.2.2).
In the last section of this chapter, we study the co-support and multiplicity
for HdI (M). The functor "co-localization" defined by K. E. Smith
Fp(−) = HomR
(
HomR(−, E(R/m)), E(R/p)
)
from the category of R-module to the category of Rp-module, where E(−) is
the injective hull, suggested us to provide the concept co-support of HdI (M),
denoted by CosR(H
d
I (M)). Then we present another characterization of the
prime saturation for HdI (M) via the co-support (Theorem 3.3.5).
For each Artinian R-module A, we denote by N-dimRA the Noetherian
dimension of A defined by R. N. Roberts. By D. Kirby, if q is a ideal of
R such that (0 :A q) has finite length then `R(0 :A q
n+1) is a polynomial
of degree N-dimRA with rational coefficients for n 0. We denote this
polynomial by ΘqA(n). Set N-dimA = s. We have
ΘqA(n) = `(0 :A q
n+1) =
e′(q, A)
s!
ns + polynomial of degree less than s
when n 0, in which e′(q, A) is a positive integer. We call e′(q, A) by
multiplicity of A with respect to q. In 2002, M. Brodmann and R. Y. Sharp
introduced the concept pseudo-support PsuppiR(M) to establish successfully
an associativity formula for the multiplicity of H im(M). In the last section
of Chapter 3, we study the co-support CosR(H
d
I (M)) and then give the
8associative formula for multiplicity of HdI (M) when this module satisfies
the prime saturation (Corollary 3.3.8).
Chapter 1
Preliminaries
In this chapter, we recall some basic knowledge of secondary represen-
tation and the set of attached primes, Artinian local cohomology modules,
dimension and the prime saturation of Artinian modules, universally catenary
rings which is needed for next chapters.
Section 1.1 prepares some notion and known result on secondary repre-
sentation.
Section 1.2 includes some concept and properties of local cohomology
modules such as the Independence Theorem, the vanishing and Artinian
property. Specially, we are interested in the set of attached primes of these
modules.
In Secttion 1.3, we recall some notion and known result of universally
catenary rings.
We also spend section 1.4 on reviewing the Noetherian dimension and the
prime saturation of Artinian modules.
Chapter 2
local cohomology modules with respect to
the maximal ideal
Through out this chapter, (R,m) is a Noetherian local ring,A is an Artinian
R-module, I is an ideal of R and M is a finitely generated R-module with
dimM = d. Let Var(I) denote the set of all prime ideals of R containing
I . Denote by R̂ and M̂ the m-adic completions of R and M respectively.
The aim of this chapter is to research the passing attached primes of local
9cohomology modules with respect to the maximal ideal through the m-adic
completions. The structure of the base ring is also reflected by the existence
of a co-localization compatible with every Artinian module H im(M) which
is studied in the last part of this chapter.
2.1 Passing attached primes through m−adic completion
For each finitely generated R-moduleM , the relations between the set of
associated primes of M and that of M̂ is given by the following formula.
Lemma 2.1.1. (i) AssR̂ M̂ =
⋃
p∈AssM
AssR̂(R̂/pR̂);
(ii) AssRM =
{
P ∩R | P ∈ AssR̂ M̂
}
.
Let A be an Artinian R-module.Artin. Since A has a natural structure as
an Artinian R̂-module, the set of attached primes of A over R̂ is well defined.
R. Y. Sharp proved that the relation between AttRA and AttR̂A is given by
the following formula
AttRA = {P ∩R | P ∈ AttR̂A}.
This formula makes the role in some sense similar to that of associated primes
for finitely generated modules in Lemma 2.1.1(ii). However, the dual formula
of (i) in Lemma 2.1.1
AttR̂A =
⋃
p∈AttRA
AssR̂(R̂/pR̂) (1)
is not true in general, even whenA is a local cohomology module with respect
to the maximal ideal (Example 2.1.2).
Now we give a characterization for the base ring R such that the formula
(1) holds true for all Artinian R-module A.
Proposition 2.1.3. The following statements are equivalent:
(i) AttR̂A =
⋃
p∈AttRA
AssR̂(R̂/pR̂) for every Artinian R-module A.
10
(ii) The induced map Spec(R̂)→ Spec(R) is bijective.
Note that there exists a Noetherian local ring (R,m) which is not complete
such that
AttR̂A =
⋃
p∈AttRA
AssR̂(R̂/pR̂)
for every Artinian R-module A. For example, let R be an arbitrary discrete
valuation ring which is non-complete. Then R satisfies the conditions in
Proposition 2.1.3 and so this formula is true.
Example 2.1.2 showed that the formula (1) is not true in general for all
local cohomology modules with respect to the maximal ideal. So, we consider
the relation
AttR̂(H
i
m(M)) =
⋃
p∈AttR(Him(M))
AssR̂(R̂/pR̂)
between AttRH
i
m(M) and AttR̂H
i
m(M). The following result gives a suffi-
cient condition for which this formula holds true.
Proposition 2.1.7. Assume that R is a quotient of a Gorenstein local ring.
Then for any finitely generated R-moduleM and any integer i ≥ 0 we have
AttR̂(H
i
m(M)) =
⋃
p∈AttR(Him(M))
AssR̂(R̂/pR̂).
Note that there exists a Noetherian local ring (R,m) such that the formula
(2) is true for every H im(M), but R is not a quotient of a Gorenstein local
ring ( see Example 2.1.8).
2.2 The case of a universally catenary ring all of whose formal fibers are
Cohen-Macaulay
As we know, if R is a quotient of a Gorenstein local ring then R is a
quotient of a Cohen-Macaulay local ring but the contrary is not true, for
example when R is a domain of dimension 1 in Example 2.1.8. In other way,
11
the class of a quotient of a Gorenstein local ring is exactly subset of that
of a quotient of a Cohen-Macaulay local ring. T. Kawasaki proved that R
a quotient of a Cohen-Macaulay local ring if and only if R is universally
catenary with all its formal fibers are Cohen-Macaulay. Recall that, for
each p ∈ Spec(R) and P ∈ Spec(R̂) such that P ∩ R = p, the natural
homomorphism R → R̂ induces a local one f : Rp → R̂P. Then the
fiber ring R̂P ⊗ (Rp/pRp) ∼= R̂P/pR̂P of f over the maximal ideal pRp
of Rp is called formal fiber of R with respect to p and P. Therefore it
is natural to ask about the problem in a more general situation, where R
is universally catenary and all its formal fibers are Cohen-Macaulay? The
purpose of this section gives a partial answer to this question. Concretely,
we show a characterization of universally catenary with all its formal fibers
are Cohen-Macaulay via the relation between the set of minimal elements of
attached primes of H im(M) over R and R̂. The crucial tool to prove the main
result of this section is the notion of pseudo-support defined by Brodmann -
Sharp as follows.
Definition 2.2.1. Let i ≥ 0 be an integer. The i-th pseudo-support of M ,
denoted by Psuppi(M), is defined by
PsuppiR(M) =
{
p ∈ Spec(R) | H i−dim(R/p)pRp (Mp) 6= 0
}
.
The following theorem is the main result of this section.
Theorem 2.2.5. The following statements are equivalent:
(i)R is universally catenary and all its formal fibers are Cohen-Macaulay;
(ii) min AttR̂(H
i
m(M)) = min
⋃
p∈AttR(Him(M))
AssR̂(R̂/pR̂) for any finitely
generated R-module M and any integer i ≥ 0;
(iii) dimR(R/AnnRH
i
m(M)) = dimR̂(R̂/AnnR̂H
i
m(M)) for any finitely
generated R-module M and any integer i ≥ 0.
As a consequence, we characterize the structure of the base ring R via the
12
relation between the pseudo-supports of M and M̂ . In general, we have the
following relation between PsuppiR(M) and Psupp
i
R̂
(M̂).
Lemma 2.2.7. For every integer i ≥ 0, we have
PsuppiRM ⊆ {P ∩R | P ∈ PsuppiR̂(M̂)}.
Now, we present another characterization of R when it is universally
catenary and all formal fibers are Cohen-Macaulay via the i-th pseudo-
support of M.
Corollary 2.2.8. The following statements are equivalent:
(i)R is universally catenary and all its formal fibers are Cohen-Macaulay;
(ii) PsuppiRM = {P ∩ R | P ∈ PsuppiR̂(M̂)} for any finitely generated
R-module M and any integer i ≥ 0.
2.3 Co-localization
When R is a quotient of a Gorenstein local ring, in1975, R. Y. Sharp
proved the Shifted Localization Principle to pass the set of attached primes
of H im(M) through localization H
i−dimR/p
pRp
(Mp). P. Schenzel also considered
H
i−dimR/p
pRp
(Mp) as " the localization" of H
i
m(M) at prime ideal p to study
finitely generated modules over local rings. This idea was used by R. Y.
Sharp and M. Brodmann to research the dimension and multiplicity for local
cohomology modules H im(M). They also extended this result when R is
universally catenary and has property that all its formal fibers are Cohen-
Macaulay. However, this Shifted Localization Principle is not true in general.
The aim of this section is to find a neccessary condition of base ring such that
there exists a co-localization compatible with all local cohomology modules
H im(M).
It should be mentioned that, for each p ∈ Spec(R), the localization at
p is a linear exact functor from MR to MRp such that Mp is Noetherian
and Mp 6= 0 whenever p ⊇ AnnRM . This functor takes an important role
13
on study of Noetherian modules. However, for each Artinian R- module,
even when p ⊇ AnnRA, if p 6= m then Ap = 0 and Supp(A) ⊆ {m}.
So the localization functor is not useful on study of Artinian modules. It is
of interest to construct for each p ∈ Spec(R) a ``co-localization" functor
Fp : MR −→ MRp such that Fp is compatible with Artinian R-modules,
concretely Fp has the following properties:
(a) Fp is linear and exact on the category of Artinian R-modules;
(b) Fp sends Artinian R-modules to Artinian Rp-modules;
(c) Fp(A) 6= 0 if p ⊇ AnnRA for every Artinian R-module A.
Some authors have tried to construct a co-localization Fp, for every
p ∈ Spec(R). However, none of the known co-localizations have all the
desired properties (a), (b), (c). In this section, we give a neccessary condition
for the existence of such a co-localization satisfying properties (a), (b), (c).
It follows that the desired one does not exist in general. Firstly, we present
some lemmas for the proof of main result.
Lemma 2.3.1. If p ∈ AttRA then AnnR(0 :A p) = p.
Lemma 2.3.3. Let A = A1+. . .+An be a minimal secondary representation
of A, where Ai is pi-secondary. Let r be an integer such that 0 < r < n. Set
B = A1 + . . . + Ar. Then
AttR(A/B) = {pr+1, . . . , pn}.
For each finitely generated R-module M and p ∈ Spec(R), we know that
p 6⊇ AnnRM if and only if Mp = 0. If Fp is a linear co-localization functor
of Artinian module then we have the same property as follows.
Lemma 2.3.4. Let p ∈ Spec(R). Suppose that p 6⊇ AnnRA. Then Fp(A) = 0
for every linear functor Fp :MR −→MRp.
Lemma 2.3.6. Let p ∈ Spec(R). Let Fp : MR −→ MRp be a functor.
Assume that Fp is linear and exact on the category of Artinian R-modules.
14
Let A be an Artinian R-module such that Fp(A) is a non-zero Artinian Rp-
module. Then AnnR(0 :A p) = p.
By N. T. Cuong and L. T. Nhan, for each Artinian R-module A, we have
N-dimRA ≤ dim(R/AnnRA). Moreover, there exists R-module Artinian
A such that N-dimRA < dim(R/AnnRA). The following proposition
presents some characterizations for which the equality holds. Note that the
equivalence between (i) and (iii) was shown by H. Zo¨schinger.
Proposition 2.3.7. The following statements are equivalent:
(i) dim(R/AnnRA) = N-dimRA for every Artinian R-module A.
(ii) dim(R/AnnRA) = N-dimRA for every top local cohomology module
A = Hkm̂(R̂/P ), where P ∈ Spec(R̂) and k = dim(R̂/P ).
(iii) dim(R̂/P ) = dim(R/(P ∩R)) for all P ∈ Spec(R̂).
The following Theorem shows a neccessary condition such that there exists
a co-localization compatible with every Artinian modules ( it means that to
satisfy properties (a), (b), (c) as above).
Theorem 2.3.8. Suppose that there exists, for every p ∈ Spec(R), a functor
Fp : MR −→ MRp satisfying the properties (a), (b), (c). Then the natural
map R −→ R̂ satisfies going up property. In particular, all the formal fibers
of R are Artinian.
In general, for Artinian local cohomology modules H im(M), we also
have N-dimR(H
i
m(M)) ≤ dim(R/AnnRH im(M)). Moreover, there exists
local rings (R,m) such that N-dimR(H
1
m(R)) < dim(R/AnnRH
1
m(R)).
By Proposition 1.4.8, at each i-th level, if H im(M) satisfies the prime
saturation then N-dimR(H
i
m(M)) = dim(R/AnnRH
i
m(M)), it means that
the prime saturation is stronger and follows the equality of dimension. The
following result not only gives some characterizations for which the equality
N-dimR(H
i
m(M)) = dim(R/AnnRH
i
m(M)) to be true for every Artinian
15
local cohomology modulesH im(M) but also states that if this equality holds
for any finitely generated R-module M and for any integer i then it is
equivalent to the prime saturation.
Lemma 2.3.10. The following statements are equivalent:
(i) dim(R/AnnRH
i
m(M)) = N-dimR(H
i
m(M)) for every integer i and
every Noetherian R-module M .
(ii)AnnR(0 :Him(M) p) = p for every integer i, every NoetherianR-module
M and every p ∈ Var(AnnRH im(M)).
(iii) R is universally catenary and all its formal fibers are Cohen-
Macaulay.
For each p ∈ Spec(R), let Fp be the co-localization defined by A.
S. Richardson. If R is complete then it follows from local duality that
Fp(H
i
m(M)) = H
i−dim(R/p)
pRp
(Mp). Under the assumption that R is universally
catenary and all formal fibers of R are Cohen-Macaulay, Brodmann - Sharp
considered the role of Rp-module H
i−dim(R/p)
pRp
(Mp) as the ``co-localization"
of the Artinian local cohomology module H im(M) in order to establish suc-
cessfully an associativity formula for the multiplicity of H im(M). Therefore,
it is of interest to study the existence of a co-localization compatible with all
Artinian local cohomology modules H im(M).
Theorem 2.3.11. Suppose that there exists, for every p ∈ Spec(R), a
functor Fp : MR −→ MRp such that Fp is linear exact on the category
of Artinian R-modules, Fp(H
i
m(M)) is Artinian, and Fp(H
i
m(M)) 6= 0
whenever p ⊇ AnnRH im(M) for every integer i and every Noetherian R-
module M . Then R is universally catenary and all formal fibers of R are
Cohen-Macaulay.
Conclusions of chapter 2
In this chapter, we achieved some results as follows:
16
- Characterizing the base ring so that the formula of passing attached
primes through completion is satisfied for all Artinian modules.
- Showing the formula of passing attached primes through completion for
H im(M) when the base ring is a quotient of a Gorenstein local ring.
- Presenting some characterizations of universally catenary rings whose
all formal fibers are Cohen-Macaulay via the relations between AttR̂H
i
m(M)
and AttRH
i
m(M); Psupp
i(M) and Psuppi(M̂).
- Proving a neccessary condition of base ring in order to exist a co-
localization compatible with every Artinian R-module.
- Giving a criterion of base ring such that there exits a co-localization
compatible with every Artinian local cohomology modulesH im(M).
17
Chapter 3
The top local cohomology modules
with respect to an arbitrary ideal
Through out this chapter, let (R,m) be a Noetherian local ring, I be an
ideal of R and M be a finitely generated R-module with dimM = d. Let
Var(I) denote the set of all prime ideals of R containing I . Denote by R̂ and
M̂ the m-adic completions of R and M respectively.
The set of attached primes of the top local cohomology modules with
respect to the maximal ideal Hdm(M) over R and R̂ was described clearly by
I. G. Macdonald and R. Y. Sharp. By the Lichtenbaum-Hartshorne Vanishing
Theorem, R. Y. Sharp clarified the set of attached primes of HdimRI (R)
over R̂. After that, K. Divaani-Aazar and P. Schenzel extended this result
for module. The purpose of this chapter is to describe this set for HdI (M)
over R in the relation with the prime saturation, co-support and associative
formula for multiplicity ofHdI (M). On the study of the set of attached primes
of HdI (M), we characterized the prime saturation for this module via the
catenary of base ring and tranfered to the case of the top local cohomology
modules with respect to the maximal ideal of a quotient module of M.
3.1 The prime saturation
By N. T. Cuong and L. T. Nhan, an Artinian R-module A satisfies the
prime saturation if AnnR(0 :A p) = p for all p ∈ Var(AnnRA). When R is
complete, the prime saturation is satisfied for all Artinian local cohomology
modules. However, this property is not satisfied in general. The prime
saturation is characterized for local cohomology modules with respect to
the maximal ideal. In this section, we give the characterization for the prime
saturation for the top local cohomology modules with respect to an arbitrary
idealHdI (M) via the catenary of base ring and tranfered to the case of the top
local cohomology modules with respect to the maximal ideal of a quotient
18
module of M .
First, we keep the following notations which will be used through this
chapter .
Notation 3.1.1. Let 0 =
⋂
p∈AssRM
N(p) be a reduced primary decomposition
of the submodule 0 of M . Set
AssR(I,M) =
{
p ∈ AssRM | dim(R/p) = d,
√
p + I = m
}
.
Set N =
⋂
p∈AssR(I,M)
N(p). Note that N does not depend on the choice of the
reduced primary decomposition of 0 because AssR(I,M) ⊆ min AssRM .
The following theorem gives a characterization for the prime saturation
of HdI (M) in the relation with the catenary of the base ring and the set of
attached primes of HdI (M).
Theorem 3.1.2. Let N be defined as in Notations 3.1.1. The following
statements are equivalent:
(i) HdI (M) satisfies the prime saturation.
(ii) The ring R/AnnRH
d
I (M) is catenary and
√
p + I = m for all
p ∈ AttRHdI (M).
(iii) The ring R/AnnRH
d
I (M) is catenary and H
d
I (M)
∼= Hdm(M/N).
3.2 The set of attached primes
In this section, we extend the result of K. Divaani-Aazar and P. Schenzel
from the case where R is complete to the case thatHdI (M) satisfies the prime
saturation. First, by Theorem 3.1.2, we have the following corollary of the
relation between AssR(I,M) and AttRH
d
I (M).
Corollary 3.2.1. Let AssR(I,M) be defined as in Notations 3.1.1. Then we
have
(i) AssR(I,M) ⊆ AttRHdI (M). In particular, if AssR(I,M) 6= ∅ then
19
HdI (M) 6= 0.
(ii) Suppose that AssR(I,M) = ∅. Then HdI (M) satisfies the prime
saturation if and only if HdI (M) = 0.
The following corollary is the main result of this section.
Corollary 3.2.2. If HdI (M) satisfies the property prime saturation then
AttRH
d
I (M) =
{
p ∈ AssRM | dim(R/p) = d,
√
I + p = m
}
.
Proposition 3.2.3 Let AssR(I,M) be defined as in Notations 3.1.1. Then
the following statements are equivalent:
(i) AttR̂H
d
I (M) =
⋃
p∈AttRHdI (M)
AssR̂(R̂/pR̂).
(ii) HdI (M) satisfies the prime saturation and R/p is unmixed for all
p ∈ AssR(I,M).
3.3 Co-support and multiplicity
Let p ∈ SpecR. K. E. Smith studied a functor called ``dual to localization"
Fp(−) = HomR
(
HomR(−, E(R/m)), E(R/p)
)
from the category of R-modules to the category of Rp-modules, where E(−)
is the injective hull. Note that this functor Fp is linear exact, Fp(A) 6= 0 if
and only if p ⊇ AnnRA, and when R is complete then Fp(A) is Artinian for
any Artinian R-module A.
Proposition 3.3.1. Let p ∈ Spec(R) and let Fp(−) be the above dual to
localization. Let N be defined as in Notations 3.1.1. Suppose that R is
complete. Then
Fp(H
d
I (M))
∼= Hd−dim(R/p)pRp (M/N)p.
Proposition 3.3.1 suggested us giving the following notion of co-support
of HdI (M).
20
Definition 3.3.2. Let N be defined as in Notations 3.1.1. The co-support of
HdI (M), denoted by CosR(H
d
I (M)), is defined as follows
CosR(H
d
I (M)) =
{
p ∈ Spec(R) | Hd−dim(R/p)pRp (M/N)p 6= 0
}
.
The following lemma gives the relation between the co-supportCosR(H
d
I (M))
and Var(AnnRH
d
I (M)).
Lemma 3.3.3. CosR(H
d
I (M)) ⊆ Var(AnnRHdI (M)).
The following theorem characterizes the prime saturation of HdI (M) in
terms of the co-support.
Theorem 3.3.5. The following statements are equivalent:
(i) HdI (M) satisfies the prime saturation;
(ii) CosR(H
d
I (M)) = Var(AnnRH
d
I (M)).
Theorem 3.3.5 asserts that if HdI (M) satisfies the prime saturation then its
co-support is a closed subset of Spec(R) in the Zariski topology.
By D. Kirby, if q is an ideal of R in which (0 :A q) has finite length then
`(0 :A q
n+1) is a polynomial of degree N-dimRA with rational coefficients
for n 0. We denote this polynomial by ΘqA(n). Set N-dimA = s. We
have
ΘqA(n) = `(0 :A q
n+1) =
e′(q, A)
s!
ns + polynomial of degree less than s
when n 0, in which e′(q, A) is a positive integer. We call e′(q, A)
by multiplicity of A with respect to q. In 2002, M. Brodmann and R. Y.
Sharp introduced the concept pseudo-support PsuppiR(M) and ith pseudo-
dimension, denoted by psdi(M), to establish successfully an associativity
formula for the multiplicity of H im(M). Recently, L. T. Nhan and T. N. An
have extended this result of M. Brodmann and R. Y. Sharp for the case of
H im(M) satisfies the prime saturation. So, using the co-support of H
d
I (M),
we establish the associative formula for multiplicity of this module when it
satisfies the prime saturation.
21
Corollary 3.3.8. Let q be an m-primary ideal. Let AssR(I,M) and N be
defined as in Notations 3.1.1. If HdI (M) satisfies the prime saturation then
e′(q, HdI (M)) =
∑
p∈CosHdI (M)
dim(R/p)=d
`Rp
(
H0pRp(M/N)p
)
e(q, R/p).
In this case, e′(q, HdI (M)) = e(q,M/N) =
∑
p∈AssR(I,M)
`Rp(Mp)e(q, R/p).
Conclusions of chapter 3
In this chapter, we obtained the following results:
- Presenting some characterizations of the prime saturation of local coho-
mology modules with respect to an arbitrary ideal in terms of the catenary
of the base ring, the set of the attached primes of HdI (M) and relation with
the top local cohomology modules with respect to the maximal ideal.
- Describing the set of attached primes of HdI (M) in case of this Artinian
module satisfies the prime saturation.
- Introducing the concept of co-support of local cohomology modules with
respect to an arbitrary ideal and characterizing the prime saturation ofHdI (M)
via it.
- Establishing successfully an associative formula for multiplicity of
HdI (M) in terms of the co-support when this Artinian module satisfies the
prime saturation.
22
CONCLUSIONS AND RECOMMENDATIONS
FOR FURTHER STUDY
1. Conclusions
In this thesis, we obtained some results in the set of attached primes of
local cohomology modules with respect to the maximal ideal and that of the
top local cohomology modules with respect to an arbitrary ideal.
For local cohomology modules with respect to the maximal ideal, we
proved the formula passing attached primes through m−adic completion
when R is a quotient of a Gorenstein local ring (Proposition 2.1.7). We also
gave characterizations of universally catenary local rings having all Cohen-
Macaulay formal fibers via this formula (Theorem 2.2.5). Moreover, we
showed a neccesary condition of the base ring for the existence of a co-
localization compatible with every local cohomology modules with respect
to the maximal ideal (Theorem 2.3.11).
For the top local cohomology modules with respect to an arbitrary ideal,
we described the set of attached primes of this module by characterizing
the saturation in terms of the catenary of base ring and then we reduced to
the case of local cohomology modules with respect to the maximal support
(Theorem 3.1.2). We also gaved the characterizations of the prime saturation
via the co-support (Theorem 3.3.5), and then proved the associative formula
for the multiplicity of this module (Corollary 3.3.8).
In order to clarify the results in the thesis, we also gave some examples.
2. Recommendations
We intend to study some topics as follows next time:
- Showing some classes of non-complete rings for the existence of a co-
localization compatible with every local cohomology modules in artribary
23
order with respect to the maximal ideal.
- Extending some results in the set of attached primes, the prime saturation,
co-support and multiplicity of the top local cohomology modules to those at
some lower levels.
- Exploiting the pseudo-support and primary decomposition on the study
of the structure of finitely generated modules over Noetherian local ring.
24
LIST OF POSTGRADUET'S WORKS RELATED
TO THE THESIS
1. L. T. Nhan and T. D. M. Chau (2012), On the top local cohomology
modules, J. Algebra, 349, 342-352.
2. L. T. Nhan and T. D. M. Chau (2014), Noetherian dimension and co-
localization of Artinian modules over local rings, Algebra Colloquium,
(4)21, 663-670.
3. T. D. M. Chau and L. T. Nhan (2014), Attached primes of local cohomology
modules and structure of Noetherian local rings, J. Algebra, 403, 459-469.

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