Tóm tắt Luận án Về tập iđêan nguyên tố gắn kết của môđun đối đồng điều địa phương

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.