✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❈❍➑ ❚❹▼ ❚➑◆❍ ❈❖❍❊◆✲▼❆❈❆❯▲❆❨ ❉❶❨ ❈Õ❆ ✣❸■ ❙➮ ❘❊❊❙ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼ ✷✵✶✽ ✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆ ❚❘×❮◆● ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❈❍➑ ❚❹▼ ❚➑◆❍ ❈❖❍❊◆✲▼❆❈❆❯▲❆❨ ❉❶❨ ❈Õ❆ ✣❸■ ❙➮ ❘❊❊❙ số ỵ tt số số ✹✻ ✵✶ ✵✹ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❈→♥ ữợ P ❚❙✳ ❚r➛♥ ◆❣✉②➯♥ ❆♥ ❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼ ✷✵✶✽ ✐ ▲❮■ ❈❆▼ ✣❖❆◆ ❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ r➡♥❣ ❝→❝ ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr✉♥❣ t❤ü❝ ✈➔ ❦❤ỉ♥❣ trò♥❣ ❧➦♣ ✈ỵ✐ ❝→❝ ✤➲ t➔✐ ❦❤→❝✳ ❚ỉ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ♠å✐ sü ❣✐ó♣ ✤ï ❝❤♦ ✈✐➺❝ t❤ü❝ ❤✐➺♥ ❧✉➟♥ ✈➠♥ ♥➔② ✤➣ ✤÷đ❝ ❝↔♠ ì♥ ✈➔ ❝→❝ t❤ỉ♥❣ t✐♥ tr➼❝❤ ❞➝♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤➣ ✤÷đ❝ ❝❤➾ rã ỗ ố t ❚→❝ ❣✐↔ ◆❣✉②➵♥ ❈❤➼ ❚➙♠ ❳→❝ ♥❤➟♥ ❳→❝ ♥❤➟♥ ❝õ❛ trữ ổ t t ữợ ❤å❝ ✐✐ ▲❮■ ❈❷▼ ❒◆ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ ữợ sỹ ữợ P trữớ ❤å❝ ❲❛s❡❞❛✱ ❚♦❦②♦✱ ◆❤➟t ❇↔♥ ✈➔ ❚❙✳ ❚r➛♥ ◆❣✉②➯♥ ❆♥✱ tr÷í♥❣ ✣↕✐ ❤å❝ ❙÷ ♣❤↕♠ ❚❤→✐ ◆❣✉②➯♥✳ ❚ỉ✐ ①✐♥ ✤÷đ❝ ❜➔② tä ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝ t ỳ qỵ tứ tr ❣✐➜② ✈➔ ❝↔ ♥❤ú♥❣ ❜➔✐ ❤å❝ tr♦♥❣ ❝✉ë❝ sè♥❣ t❤➛② ❞↕② ❣✐ó♣ tỉ✐ tü t✐♥ ❤ì♥ ✈➔ tr÷ð♥❣ t❤➔♥❤ ❤ì♥✳ ❚ỉ✐ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ tỵ✐ t➜t ❝↔ ❝→❝ t❤➛② ❝æ ð ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥✱ ❝→❝ t❤➛② ð ❱✐➺♥ ❚♦→♥ ✈➔ ❝→❝ t❤➛② ❝æ ✤➳♥ tø ◆❤➟t ❇↔♥ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tæ✐ t❤❛♠ ❣✐❛ ❝→❝ ợ ữỡ tr tổ õ t tự qỵ ổ ữủ ỷ ỡ tợ tt t tr ❣✐❛ ✤➻♥❤ ✤➣ t↕♦ ✤✐➲✉ ❦✐➺♥ ❝❤♦ tỉ✐ ✤÷đ❝ ❤å❝ t➟♣✱ ♥❣❤✐➯♥ ❝ù✉ ✈➔ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥✳ ✐✐✐ ▼ö❝ ❧ư❝ ▲í✐ ❝❛♠ ✤♦❛♥ ✐✐ ▲❮■ ❈❷▼ ❒◆ ✐✐✐ ▼Ð ✣❺❯ ✶ ❈❤÷ì♥❣ ✶ ❱➔♥❤ ❧å❝ ✈➔ t➼♥❤ ◆♦❡t❤❡r ❝õ❛ ✈➔♥❤ ❧å❝ ✸ ✶✳✶ ❱➔♥❤ ❧å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✶✳✷ ❚➼♥❤ ◆♦❡t❤❡r ❝õ❛ ✈➔♥❤ ❧å❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ❈❤÷ì♥❣ ✷ ❚➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ✤↕✐ sè ❘❡❡s ✶✽ ✷✳✶ ▲å❝ ❝❤✐➲✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ▼æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ ▼æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✸ ❚➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ✤↕✐ sè ❘❡❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ❑➌❚ ▲❯❾◆ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✹✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✐✈ ✹✷ ▼Ð ✣❺❯ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ◆♦❡t❤❡r ✈➔ ❝❤♦ F = {Fn }n ❧➔ ♠ët ❤å ❝→❝ ✐✤➯❛♥ tr♦♥❣ R✳ ❑❤✐ ✤â t❛ ♥â✐ F ❧➔ ♠ët ❧å❝ ❝õ❛ R ♥➳✉ ✭✐✮ F0 = R❀ Fn+1 ⊆ Fn ✈ỵ✐ ♠å✐ n ∈ Z; ✭✐✐✮ Fn Fm ⊆ Fn+m ✈ỵ✐ ♠å✐ m, n ∈ Z✳ ❱➼ ❞ư ✈➲ ❝→❝ ❧♦↕✐ ❧å❝ ♠➔ ❝❤ó♥❣ t❛ t❤÷í♥❣ ♥❣❤✐➯♥ ❝ù✉ ✤â ❧➔ ❧å❝ I ✲❛❞✐❝ Fn = I n , n ∈ N ✈ỵ✐ I ❧➔ ✐✤➯❛♥ ❝õ❛ R❀ ❧å❝ Fn = p(n) , n ∈ N ❧➔ ❧å❝ ❧ô② t❤ø❛ ❤➻♥❤ t❤ù❝ ❝õ❛ ✐✤➯❛♥ ♥❣✉②➯♥ tè p tr♦♥❣ R❀ ❧å❝ Fn = I n , n ∈ N ❧➔ ❧å❝ ❝→❝ ❜❛♦ ✤â♥❣ ♥❣✉②➯♥ ❝õ❛ I n ❀ ❧å❝ Fn = i≥n Ri tr♦♥❣ ✤â R = i≥0 Ri ❧➔ ♠ët ✈➔♥❤ ♣❤➙♥ ❜➟❝✳ ❱ỵ✐ t ❧➔ ♠ët ❜✐➳♥ tr➯♥ R ✈➔ ✈ỵ✐ ♠é✐ ❧å❝ F ❝õ❛ R t❛ ❝â ❜❛ ✤↕✐ sè ♣❤➙♥ ❜➟❝ ❧✐➯♥ ❦➳t ❧➔ Fn tn ⊆ R[t], R(F) = n≥0 Fn tn = R(F)[t−1 ] ⊆ R[t, t−1 ] ✈➔ R (F) = n∈Z G(F) = R(F)/t−1 R(F) ✈➔ t❛ ❣å✐ t÷ì♥❣ ù♥❣ ❧➔ ✤↕✐ sè ❘❡❡s✱ ✤↕✐ sè ❘❡❡s ♠ð rë♥❣ ✈➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ❧✐➯♥ ❦➳t ❝õ❛ ❧å❝ F ✳ ❑❤✐ F ❧➔ I ✲❛❞✐❝ t❛ t❤÷í♥❣ ❦➼ ❤✐➺✉ ❝→❝ ✤↕✐ sè ❜ð✐ R(I), R (I) ✈➔ G(I) t÷ì♥❣ ù♥❣✳ ❑➳t q✉↔ ✤➛✉ t✐➯♥ ✈➲ ①➨t t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝õ❛ ✈➔♥❤ ❘❡❡s ù♥❣ ✈ỵ✐ ❧å❝ m✲❛❞✐❝ ❧➔ ❝õ❛ ❙✳ ●♦t♦✲❨✳ ❙❤✐♠♦❞❛ ❬✶✸❪ ❤å ✤➣ ①➨t tr♦♥❣ tr÷í♥❣ ❤đ♣ R ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ữỡ ợ tố t m é ✤â ❤å ✤➣ ❦❤➥♥❣ ✤à♥❤ ♥➳✉ dim R ≥ t❤➻ ✈➔♥❤ ❘❡❡s R(m) ✈ỵ✐ m ✐✤➯❛♥ tè✐ ✤↕✐ ❝õ❛ R ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ G(m) ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ a(G(m)) < tr♦♥❣ ✤â a(G(m)) ❧➔ a✲❜➜t ❜✐➳♥ ❝õ❛ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ✭t❤❡♦ ❬✶✹❪✮✳ ❙✳ ■❦❡❞❛ ❬✶✽❪ ♠ð rë♥❣ ❦➳t q✉↔ tr➯♥ ❝❤♦ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ❜➜t ❦ý ❝â ❝❤✐➲✉ dim R ≥ 1✳ ❙❛✉ ✤â ◆✳ ❱✳ ❚r✉♥❣ ✈➔ ❙✳ ■❦❡❞❛ ❬✷✽❪ t➻♠ ❤✐➸✉ ❝❤♦ tr÷í♥❣ ❤đ♣ tê♥❣ q✉→t ❤ì♥✳ ❈ư t❤➸ ❝❤♦ I ❧➔ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ ◆♦t❤❡r ✤à❛ ♣❤÷ì♥❣ R✱ M ❧➔ ✐✤➯❛♥ tè✐ ✤↕✐ ♣❤➙♥ ❜➟❝ ❞✉② ♥❤➜t ❝õ❛ R(I)✳ ❑❤✐ ✤â ♥➳✉ dim R(I) = dim R + t❤➻ R(I) ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✶ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ [Hmi (G(I))]n = (0) ✈ỵ✐ ♠å✐ i, n ∈ Z, i = dim R, n = −1 ✈➔ a(G(I)) < 0✳ ❚➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝õ❛ ❝→❝ ✤↕✐ sè ù♥❣ ✈ỵ✐ ❝→❝ ❧å❝ ❦❤→❝ ❝ơ♥❣ ✤÷đ❝ ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ q✉❛♥ t➙♠ ♥❣❤✐➯♥ ❝ù✉✳ ▼ët ❝➙✉ ❤ä✐ tü ♥❤✐➯♥ ✤➦t r❛ ❧➔ t➻♠ ❤✐➸✉ t➼♥❤ số tr ú ỵ r t t ữủ ợ t ❘✳ P✳ ❙t❛♥❧❡② ❬✷✺❪ ❝❤♦ ❝→❝ ♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤✳ ❙❛✉ ✤â ◆✳ ❚✳ ❈÷í♥❣✱ ▲✳ ❚✳ ◆❤➔♥ ❬✽❪ ✈➔ P✳ ❙❝❤❡❧③❡❧ ❬✷✹❪ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ❧ỵ♣ ♠ỉ✤✉♥ ♥➔② tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣✳ ❚➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ ❝❤♦ ♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ✈➔♥❤ ◆♦❡t❤❡r ❜➜t ❦ý ❜ð✐ ❙✳ ●♦t♦✱ ❨✳ ❍♦r✐✉❝❤✐ ✈➔ ❍✳ ❙❛❦✉r❛✐ ❬✶✶❪✳ ▲ỵ♣ ♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❧➔ ♠ð rë♥❣ tü ♥❤✐➯♥ ❝õ❛ ❧ỵ♣ ♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❱✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ❧ỵ♣ ♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✤â♥❣ ✈❛✐ trá r➜t q✉❛♥ trå♥❣ tr♦♥❣ ✣↕✐ sè ❣✐❛♦ ❤♦→♥✱ ❍➻♥❤ ❤å❝ ✤↕✐ sè✱ ✣↕✐ sè tê ❤ñ♣✱ ✤➦❝ ❜✐➺t tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ✈➔♥❤ ❙t❛♥❧❡②✲❘❡✐♥❡r✳ ❈➜✉ tró❝ ❝õ❛ ♠ỉ✤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ❞➣② ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ ❦❤→ rã t❤ỉ♥❣ q m ữỡ trữ ỗ ✤✐➲✉ ❬✷✺✱ ✽✱ ✷✹✱ ✶✻❪ ✈➔ ❤➺ t❤❛♠ sè tèt✱ ❤➺ t❤❛♠ sè dd✲❞➣② ❬✻❪✳ ❚➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ✤↕✐ sè ❘❡❡s ù♥❣ ✈ỵ✐ ❧å❝ I ✲❛❞✐❝ tr➯♥ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ (R, m) ✤÷đ❝ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❬✼❪✱ tr♦♥❣ ✤â I ❧➔ ✐✤➯❛♥ m✲♥❣✉②➯♥ sì✳ ❚r♦♥❣ ❬✷✻❪ ❝→❝ t→❝ ❣✐↔ ♠ð rë♥❣ ♥❣❤✐➯♥ ❝ù✉ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ❝→❝ ✤↕✐ sè ❘❡❡s ù♥❣ ✈ỵ✐ ❧å❝ tê♥❣ q✉→t ❤ì♥✳ ▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ❤✐➸✉ ✈➔♥❤ ❧å❝✱ ❝→❝ ✤↕✐ sè ❘❡❡s ✈➔ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ❝→❝ ✤↕✐ sè ❘❡❡s✳ ❱✐➺❝ t➻♠ ❤✐➸✉ ❝❤✐ t✐➳t ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝ơ♥❣ ❧➔ ♠ët ♠ư❝ ✤➼❝❤ ❦❤→❝ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ▲✉➟♥ ✈➠♥ t❤❛♠ ❦❤↔♦ ❝❤➼♥❤ t❤❡♦ ❝→❝ t➔✐ ❧✐➺✉ ❬✷✻❪✱ ❬✷✼❪✱ ❬✶✶❪✱ ❬✺❪✱ ❬✽❪✱ ❬✶✾❪✳ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❜è ❝ư❝ ❧➔♠ ❤❛✐ ❝❤÷ì♥❣✳ ❈❤÷ì♥❣ ✶ tr➻♥❤ ❜➔② ✈➲ ✈➔♥❤ ❧å❝ ✈➔ t➼♥❤ ◆♦❡t❤❡r ❝õ❛ ✈➔♥❤ ❧å❝✳ ❈❤÷ì♥❣ ✷ tr➻♥❤ ❜➔② ✈➲ ❧å❝ ❝❤✐➲✉✱ ♠æ✤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ✈➔ ♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✱ t➼♥❤ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ✤↕✐ sè ❘❡❡s✳ ✷ ❈❤÷ì♥❣ ✶ ❱➔♥❤ ❧å❝ ✈➔ t➼♥❤ ◆♦❡t❤❡r ❝õ❛ ✈➔♥❤ ❧å❝ Ð ❝❤÷ì♥❣ ♥➔② t❛ ❧✉ỉ♥ ❣✐↔ t❤✐➳t R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥ ❝â ✤ì♥ ✈à ✈➔ M ❧➔ R✲ ♠ỉ✤✉♥✳ ▼ët sè ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t ❝❤÷❛ ✤÷đ❝ ♥➯✉ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝â t❤➸ t❤❛♠ ❦❤↔♦ tr♦♥❣ ❬✶✻❪✱❬✷✵❪✱ ❬✷✶❪✳ ❈❤÷ì♥❣ ♥➔② t❤❛♠ ❦❤↔♦ t❤❡♦ ❬✷❪✱ ❬✶✼❪✱ ❬✶✾❪✳ ✶✳✶ ❱➔♥❤ ❧å❝ ❚r♦♥❣ ♠ư❝ ♥➔② t❛ s➩ ❣✐ỵ✐ t❤✐➺✉ ✈➲ ✈➔♥❤ ❘❡❡s✱ ✈➔♥❤ ❘❡❡s ♠ð rë♥❣ ✈➔ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ❧✐➯♥ ❦➳t ❝õ❛ ♠ët ✈➔♥❤ ❧å❝✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ✈➔ {Fn}n∈Z ❧➔ ♠ët ❤å ❝→❝ ✐✤➯❛♥ ❝õ❛ R✳ ❉➣② {Fn }n∈Z ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❧å❝ ❝→❝ ✐✤➯❛♥ ❝õ❛ R ♥➳✉ ♥â t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿ ✭✐✮ F0 = R✱ Fn+1 ⊆ Fn ✈ỵ✐ ♠å✐ n ∈ Z❀ ✭✐✐✮ Fn Fm ⊆ Fn+m ✈ỵ✐ ♠å✐ m, n ∈ Z✳ ▼ët ✈➔♥❤ ❧å❝ ❧➔ ❝➦♣ (R, F) tr♦♥❣ ✤â R ❧➔ ✈➔♥❤ ✈➔ F ❧➔ ♠ët ❧å❝ tr➯♥ R✳ ❱➼ ❞ö ✶✳✶✳✷✳ ❈❤♦ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ R ✈➔ ✤➦t Fn = I n✳ ❑❤✐ ✤â t❛ ❝â ❧å❝ R = I0 ⊇ I1 ⊇ I2 ⊇ ⊇ In ⊇ ▲å❝ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❧å❝ ❧ơ② t❤ø❛ ❤❛② ❧å❝ I ✲❛❞✐❝✳ ❱➼ ❞ư ✶✳✶✳✸✳ ❈❤♦ R = ❧➔ ❧å❝ ❝→❝ ✐✤➯❛♥ ❝õ❛ R✳ Ri ❧➔ ✈➔♥❤ Z✲♣❤➙♥ ❜➟❝✳ ✣➦t Fn = i≥n i∈Z ✸ Ri t❤➻ {Fn }n∈Z ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ✈➔ p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ R✳ ❑❤✐ ✤â ❧ô② t❤ø❛ ❤➻♥❤ t❤ù❝ ❜➟❝ n ❝õ❛ p✱ ỵ p(n) ữủ pn Rp ∩ R, n ∈ N✳ ❱➼ ❞ö ✶✳✶✳✺✳ ❈❤♦ p ❧➔ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❝õ❛ ✈➔♥❤ R✳ ❑❤✐ ✤â ✈ỵ✐ ♠å✐ m, n ∈ N✱ pn pm ⊆ pm+n ✳ ❚ø ✤â s✉② r❛ pn Rp pm Rp ⊆ pm+n Rp ✳ ❉♦ ✤â (pn Rp ∩ R).(pm Rp ∩ R) ⊆ pm+n Rp ∩ R ❤❛② p(n) p(m) ⊆ p(n+m) ✈➔ ♥❤÷ ✈➟② {p(n) }n ❧➔ ♠ët ❧å❝✳ ❚✐➳♣ t❤❡♦ t❛ ①➨t ♠ët ✈➼ ❞ö ✈➲ ❧å❝ ❝→❝ ✐✤➯❛♥ ❝õ❛ ✈➔♥❤ ✤❛ t❤ù❝ k[x] ✈ỵ✐ k ❧➔ ♠ët tr÷í♥❣✳ ❱➼ ❞ư ✶✳✶✳✻✳√❈❤♦ Fn √ = (x n ), n ∈ N ❧➔ ♠ët ❤å ❝→❝ ✐✤➯❛♥ tr♦♥❣ k[x] √ n ❧➔ sè ♥❣✉②➯♥ ♥❤ä ♥❤➜t ❧ỵ♥ ❤ì♥ ❤♦➦❝ n ợ ỵ ự {Fn } ởt k[x] rữợ t t❛ ❝❤ù♥❣ ♠✐♥❤ √ √ √ √ √ √ m+n ≤ m + n ✳ ❘ã r➔♥❣ t❛ ❝â m+n ≤ m+ n ✳ ❑❤✐ ✤â √ √ √ √ √ m+n≤ m+ n≤ m + n √ √ √ √ √ √ m + n ∈ N ♥➯♥ m + n = m + n ✳ ❉♦ ❱➻ √ √ √ ✤â m+n ≤ m + n ✳ ❚❛ ❝â ✤✐➲✉ ❦✐➺♥ (ii) tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ❧å❝ ❧➔ t❤ä❛ ♠➣♥✳ ❍✐➸♥ ♥❤✐➯♥ ✤✐➲✉ ❦✐➺♥ (i) ❝õ❛ ✤à♥❤ ♥❣❤➽❛ ❧✉æ♥ t❤ä❛ ♠➣♥✳ ❉♦ ✈➟② {Fn } ❧➔ ❧å❝ ❝→❝ ✐✤➯❛♥ ❝õ❛ R✳ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✼✳ ❈❤♦ ✈➔♥❤ ❧å❝ (R, F)✱ ✈ỵ✐ ❧å❝ F = {Fn}n∈Z ✈➔ M ❧➔ R✲ ♠æ✤✉♥✳ ▼ët ❧å❝ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ❧➔ ❤å {Mn }n∈Z ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M t❤ä❛ ♠➣♥ M0 = M ✈➔ Mn+1 ⊆ Mn ✳ ▲å❝ {Mn }n∈Z ữủ tữỡ t ợ F F ✲❧å❝ ♥➳✉ Fm Mn ⊆ Mm+n ✱ ✈ỵ✐ ♠å✐ m, n ∈ Z✳ ❱➼ ❞ö ✶✳✶✳✽✳ ❈❤♦ R = Rn ❧➔ ✈➔♥❤ Z✲♣❤➙♥ ❜➟❝ ✈➔ M = n∈Z ♠æ✤✉♥ ♣❤➙♥ ❜➟❝✳ ✣➦t Mn = i≥n Gn ❧➔ R✲ n∈Z Gi ✱ ❦❤✐ ✤â {Mn }n∈Z ❧➔ ♠ët F ✲❧å❝ ❝õ❛ M ợ F = {Fn } ữ tr ✶✳✶✳✸✳ ❱➼ ❞ö ✶✳✶✳✾✳ ❈❤♦ (R, F) ❧➔ ✈➔♥❤ ❧å❝ ✈ỵ✐ ❧å❝ F = {Fn}n∈Z ✈➔ M ❧➔ R✲♠ỉ✤✉♥✳ ✣➦t Mn = Fn M ✳ ❑❤✐ ✤â {Mn }n∈Z ❧➔ F ✲❧å❝✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ❧å❝ ✈ỵ✐ ❧å❝ F = {Fn }n∈Z ✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ ✤↕✐ sè s R tữỡ ự ợ F F n tn R = R(F) = n≥0 ✹ ❑❤✐ ✤â R(F) ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ ✈➔♥❤ R[t]✳ ❚❛ ❝ô♥❣ ✤à♥❤ ♥❣❤➽❛ ✤↕✐ sè ❘❡❡s ♠ð rë♥❣ R tữỡ ự ợ F Fn tn R = R (F) = n∈Z ❑❤✐ ✤â R(F) ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët ✈➔♥❤ ❝♦♥ ❝õ❛ ✈➔♥❤ R[t, t−1 ]✳ ◆❣♦➔✐ r❛✱ t❛ ✤à♥❤ ♥❣❤➽❛ ✈➔♥❤ ♣❤➙♥ ❜➟❝ ❧✐➯♥ t R tữỡ ự ợ F G = G(F) = Fn /Fn+1 n=0 ◆â ❧➔ ♠ët ✈➔♥❤ ♣❤➙♥ ❜➟❝ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ❝↔♠ s✐♥❤ ❜ð✐ ♣❤➨♣ ♥❤➙♥ →♥❤ ①↕ Fm × Fn −→ Fm+n ✳ ✣➦t ❈❤♦ M ❧➔ R✲♠æ✤✉♥✱ M = {Mn }n∈Z ❧➔ F ✲❧å❝ ❝→❝ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ tn ⊗ Mn ⊆ R[t] ⊗R M R(M ) = n≥0 tn ⊗ Mn ⊆ R[t, t−1 ] ⊗R M R (M ) = n∈Z G(M ) = R (M )/t−1 R(M ) ✤÷đ❝ ❣å✐ ❧➔ ♠ỉ✤✉♥ ❘❡❡s✱ ♠ỉ✤✉♥ ❘❡❡s ♠ð rë♥❣ ✈➔ ♠ỉ✤✉♥ ♣❤➙♥ ❜➟❝ ❧✐➯♥ ❦➳t ❝õ❛ M ✳ ❈❤ó þ✱ ✤ỉ✐ ❦❤✐ ✤➸ ✤ì♥ ❣✐↔♥ t❛ ❝ơ♥❣ ✈✐➳t R(M ) = Mn tn ✈➔ n≥0 Mn t ✳ n R (M ) = n∈Z ❚❛ ①➨t tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ❣✐↔ sû ✈➔♥❤ R ❧➔ ✈➔♥❤ ❣✐❛♦ ❤♦→♥✱ I ⊆ R ❧➔ ✐✤➯❛♥ ✈➔ M ❧➔ R✲♠æ✤✉♥✳ ❑❤✐ ✤â t ỵ s s rở ✈➔♥❤ ♣❤➙♥ ❜➟❝ ❧✐➯♥ ❦➳t ù♥❣ ✈ỵ✐ ❧å❝ {I n M }n∈Z ❜ð✐ (I n M )tn R(I, M ) = n∈N ✈➔ (I n M )tn R (I, M ) = n∈Z I n M/I n+1 M G(I, M ) = nN sỷ q ữợ I n = R ♥➳✉ n ≤ ✈➔ ①➨t R(I, M ) ✈➔ R (I, M ) ❧➔ ♥❤â♠ ❝♦♥ ❝õ❛ M [t, t−1 ] = M ⊗R R[t, t−1 ] ợ j < q ữ [M (p)]P = Dj (MP ) [Di ]P = p∈AssR M, dim RP /p RP >nj , p⊆P (2) ❚❛ ❝â [M (p)]P , [Di−1 ]P = p∈AssR M, dim R/p≥di , p⊆P ✈➻ Di−1 = p∈AssR M, dim R/p≥di M (p)✳ ❚❛ ❝â t❤➸ ❣✐↔ sû j > 1✳ ❑❤✐ ✤â di = nj > nj−1 ♥➯♥ [M (p)]P [Di−1 ]P = p∈AssR M, dim R/p≥di , p⊆P = [M (p)]P p∈AssR M, dim RP /p RP >nj−1 = Dj1 (MP ) t tử ự ỵ ❚ø ❦❤➥♥❣ ✤à♥❤ tr➯♥ t❛ ❝â Cj (MP ) = Dj (MP )/Dj−1 (MP ) ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➻ MP ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✱ tr♦♥❣ ✤â [Ci ]P ❧➔ RP ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ổ ỵ t õ ❝❤ù♥❣ ♠✐♥❤✳ ◆➳✉ ✈➔♥❤ ❝ì sð R ❧➔ ♠ët ✤↕✐ sè ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ♠ët tr÷í♥❣ t❤➻ ❣✐↔ t❤✐➳t ỵ tọ t õ ❦❤➥♥❣ ✤à♥❤ s❛✉✳ ❍➺ q✉↔ ✷✳✷✳✷✸✳ ❈❤♦ R ❧➔ ♠ët ✤↕✐ sè ❤ú✉ ❤↕♥ s✐♥❤ tr➯♥ ♠ët tr÷í♥❣✱ M = (0) ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✮ M ❧➔ R✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✭✐✐✮ MP ❧➔ RP ✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✈ỵ✐ ♠é✐ P SuppR M ú ỵ R = ❧➔ ✈➔♥❤ Z✲♣❤➙♥ ❜➟❝ ◆♦❡t❤❡r ✈➔ ❣✐↔ sû r➡♥❣ R ởt H ữỡ ợ H tố ✤↕✐ P ❝õ❛ R t❤❡♦ ♥❣❤➽❛ tr♦♥❣ ❙✳ ●♦t♦ ❛♥❞ ❑✳ ❲❛t❛♥❛❜❡ ✭❳❡♠ ❬✶✺✱ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✺✱ ✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✻❪✮✳ ❈❤♦ M = (0) ❧➔ R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤ ❝â ❝❤✐➲✉ d✳ ❈❤♦ {Di }0≤i≤ ❧➔ ♠ët ❧å❝ ❝❤✐➲✉ ❝õ❛ M ✳ ❚❛ ✤➦t q = dim R/P ✳ ❱ỵ✐ ✐✤➯❛♥ I ❜➜t ❦➻ ❝õ❛ ✈➔♥❤ ♣❤➙♥ ❜➟❝✱ I ∗ ❧➔ ♠ët ✐✤➯❛♥ s✐♥❤ ❜ð✐ ❝→❝ ♣❤➛♥ tû t❤✉➛♥ ♥❤➜t tr♦♥❣ I ✳ n∈Z Rn ❇ê ✤➲ ✷✳✷✳✷✺✳ ợ tt ữ ú ỵ t õ dimR M = dimR MP + q ✳ ❉♦ ✤â dimR M = dimRm Mm ✈ỵ✐ ♠é✐ ✐✤➯❛♥ tè✐ ✤↕✐ m ❝õ❛ R ✈ỵ✐ m ⊇ P ✳ ✸✵ P ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â t❤➸ ❣✐↔ sû q > ✭❞♦ ✤â q = 1✮✳ ❈❤♦ m ❧➔ ♠ët ✐✤➯❛♥ tè✐ ✤↕✐ ❝õ❛ R s❛♦ ❝❤♦ m ⊇ P ✈➔ dim Rm /P Rm = 1✳ ❈❤♦ p ∈ AssR M s❛♦ ❝❤♦ P ⊇ p ✈➔ dim RP / p RP = dimRP MP ✳ ❑❤✐ ✤â t❛ ❝â dimR M ≥ dimRP MP + 1✳ ◆❣÷đ❝ ❧↕✐✱ t❛ ❝❤å♥ p ∈ AssR M ✈➔ ♠ët ✐✤➯❛♥ tè✐ ✤↕✐ m ❝õ❛ R s❛♦ ❝❤♦ p ⊆ m✱ dim Rm / p Rm = d✳ ❱➻ p ❧➔ ♠ët ✐✤➯❛♥ ♣❤➙♥ R p m ữ ỵ m ❦❤æ♥❣ ❧➔ ✐✤➯❛♥ ♣❤➙♥ ❜➟❝ ❝õ❛ R ✈➻ q = 1✳ ❉♦ ✤â t❛ ❝â m∗ ⊆ P ✈➔ d = dim Rm / p Rm = dim Rm∗ / p Rm∗ +1 ≤ dim RP /p RP +1 ≤ dimRP MP +1 ỵ ✤➲ ✷✳✷✳✷✺✱ t❛ ❝â ❤➺ q✉↔ s❛✉✳ ❍➺ q✉↔ ✷✳✷✳✷✻✳ ợ tt ữ ú ỵ õ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② ❧➔ ✤ó♥❣✳ ✭✐✮ [D0 ]m = (0) [D1 ]m [D ]m = Mm ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ Mm ✈ỵ✐ ♠é✐ ✐✤➯❛♥ tè✐ ✤↕✐ m ❝õ❛ R s❛♦ ❝❤♦ m ⊇ P ✳ ✭✐✐✮ [D0 ]P = (0) [D1 ]P [D ]P = MP ❧➔ ♠ët ❧å❝ ❝❤✐➲✉ ❝õ❛ MP s❛♦ ❝❤♦ M ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ MP ❧➔ RP ✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ◆❤➟♥ ①➨t ✷✳✷✳✷✼✳ ❱ỵ✐ tt ữ ú ỵ sỷ P ✐✤➯❛♥ tè✐ ✤↕✐ ❝õ❛ R✳ ❑❤✐ ✤â [D0 ]P = (0) [D1 ]P [D ]P = MP ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ MP s❛♦ ❝❤♦ M ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ MP ❧➔ RP ✲♠æ✤✉♥ ố ũ t õ ỵ s ữỡ õ ỵ ợ tt ữ ú ỵ s tữỡ ✤÷ì♥❣✳ ✭✐✮ M ❧➔ R✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✭✐✐✮ MP ❧➔ RP ✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❑❤✐ ✤â tr♦♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ Mp ❧➔ Rp ✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✈ỵ✐ ♠é✐ p ∈ SuppR M ✳ ❈❤ù♥❣ ♠✐♥❤✳ ❙ü t÷ì♥❣ ✤÷ì♥❣ ❝õ❛ ✭✐✮ ✈➔ ✭✐✐✮ ✤÷đ❝ s✉② r❛ tø ❍➺ q✉↔ ✷✳✷✳✷✻✳ ự ố ú ỵ r p∗ ⊆ P ✈ỵ✐ ❜➜t ❦➻ p ∈ SuppR M ✳ ❚ø ▼➺♥❤ ✤➲ ✷✳✷✳✷✶ t❤➻ Mp∗ ❧➔ Rp∗ ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❱➻ p∗ R(p) ❧➔ ♠ët ✐✤➯❛♥ H ✲tè✐ ✤↕✐ ❝õ❛ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ❤â❛ t❤✉➛♥ ♥❤➜t R(p) ❝õ❛ R✱ M(p) ❧➔ R(p) ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛✉❝❛✉❧❛② ❞➣②✳ ❚❛ ❝â t❤➸ ❣✐↔ sû r➡♥❣ p ❦❤æ♥❣ ❧➔ ✐✤➯❛♥ ✸✶ ♣❤➙♥ ❜➟❝ ❝õ❛ R s❛♦ ❝❤♦ p R(p) ❧➔ ✐✤➯❛♥ tè✐ ✤↕✐ ❝õ❛ R(p) ✳ ❉♦ ✤â Mp ❧➔ Rp ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ◆❤➟♥ ①➨t ✷✳✷✳✷✾✳ ❈❤♦ R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r✱ M = (0) ❧➔ R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ dimR M < ∞✳ ❈❤♦ R[t] ❧➔ ♠ët ✈➔♥❤ ✤❛ t❤ù❝ tr➯♥ R✳ ❚❛ ✤➦t S = R[t] ❤♦➦❝ R[t, t−1 ]✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✮ M ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✭✐✐✮ S ⊗R M ❧➔ S ✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ P❤➛♥ ❝✉è✐ ❝õ❛ ♠ư❝ ♥➔② t❛ t➻♠ ❤✐➸✉ tê♥❣ trü❝ t✐➳♣ ❝õ❛ ❝→❝ ♠æ✤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ❞➣②✳ ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ữỡ ợ tố m Mn ❧➔ R✲♠ỉ✤✉♥ ❝♦♥ ❧ỵ♥ ♥❤➜t ❝õ❛ M ✈ỵ✐ dimR Mn ≤ n ✈ỵ✐ ♠é✐ n ∈ Z✳ ❇ê ✤➲ ✷✳✷✳✸✵✳ ●✐↔ sû (R, m) ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ◆♦❡t❤❡r✱ M ✈➔ N ❧➔ ❝→❝ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â [M ⊕ N ]n = Mn ⊕ Nn ✈ỵ✐ ♠é✐ n ∈ Z✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â [M ⊕ N ]n ⊇ Mn ⊕ Nn ✈➻ dimR (Mn ⊕ Nn ) = max{dimR Mn , dimR Nn } ≤ n ❈❤♦ p : L = M ⊕ N → M, (x, y) → x ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❧➯♥ t❤➔♥❤ ♣❤➛♥ t❤ù ♥❤➜t✳ ❑❤✐ ✤â p(Ln ) ⊆ Mn ✱ ✈➻ dimR p(Ln ) ≤ dimR Ln ≤ n✳ ❚÷ì♥❣ tü t❛ ❝â q(Ln ) ⊆ Nn ✱ tr♦♥❣ ✤â q : M ⊕ N → N, (x, y) → y ❦➼ ❤✐➺✉ ❧➔ ♣❤➨♣ ❝❤✐➳✉ ❧➯♥ t❤➔♥❤ ♣❤➛♥ t❤ù ❤❛✐✳ ❉♦ ✤â [M ⊕ N ]n ⊆ Mn ⊕ Nn ✳ ▼➺♥❤ ✤➲ ✷✳✷✳✸✶✳ ●✐↔ sû (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ M ✈➔ N (M, N = (0)) ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â M ⊕ N ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ❞➣② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ M ✈➔ N ❧➔ ❝→❝ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ✤➦t L = M ⊕ N ✈➔ = S(L)✳ ❑❤✐ ✤â S(L) = S(M ) ∪ S(N ) ❤❛② AssR L = AssR M ∪ AssR N ✳ ❉♦ ✤â ♥➳✉ = t❤➻ S(L) = S(M ) = S(N ) ✈➔ dimR L = dimR M = dimR N ✳ ❉♦ ✈➟② ❦❤✐ = t❤➻ L ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛② ❞➣② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ L ❧➔ R✲♠æ✤✉♥ tự tữỡ ữỡ ợ R✲♠æ✤✉♥ M ✈➔ N ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ♥❣❤➽❛ ❧➔ M ✈➔ N ❧➔ ❝→❝ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ●✐↔ sû > ✈➔ ❦❤➥♥❣ ✤à♥❤ ❝õ❛ t❛ ✤ó♥❣ ✈ỵ✐ − 1✳ ❈❤♦ D0 = (0) D1 D2 ··· D =L ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ L = M ⊕ N ✱ tr♦♥❣ ✤â S(L) = {d1 < d2 < · · · < d }✳ ❑❤✐ ✤â {Di /D1 }1≤i≤ ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ L/D1 ✈➔ ❞♦ ✈➟② L ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✸✷ ❞➣② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ D1 ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✈➔ L/D1 ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❇ð✐ ✈➻ Md1 ⊕ (0)(d1 ∈ S(M ) \ S(N )), D1 = Md1 ⊕ Nd1 (d1 ∈ S(M ) ∩ S(N )), (0) ⊕ Nd1 (d1 ∈ S(N ) \ S(M )) t❤❡♦ ❇ê ✤➲ ✷✳✷✳✸✵✱ ❣✐↔ t❤✐➳t ✈ỵ✐ ❞➵ ❞➔♥❣ ❝❤ù♥❣ tä ❝❤♦ ❦❤➥♥❣ ✤à♥❤ t❤ù ❤❛✐ ♥â✐ r➡♥❣ R✲▼æ✤✉♥ M ✈➔ N ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✷✳✸ ❚➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ✤↕✐ sè ❘❡❡s ▼ö❝ ♥➔② t❤❛♠ ❦❤↔✐ ❝❤➼♥❤ t❤❡♦ ❜➔✐ ❜→♦ ❬✷✻❪✳ ❚r♦♥❣ s✉èt ♠ö❝ ♥➔②✱ t❛ ❣✐↔ sû (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ M = (0) ❧➔ ♠ët R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ❝â ❝❤✐➲✉ d✳ ❈❤♦ F = {Fn }n∈Z ❧➔ ♠ët ❧å❝ ❝→❝ ✐✤➯❛♥ ❝õ❛ R ✈ỵ✐ F1 = R✱ M = {Mn }n∈Z ❧➔ ♠ët F ✲❧å❝ ❝→❝ R✲ ♠æ✤✉♥ ❝♦♥ ❝õ❛ M ✳ ❚❛ ✤➦t a = R(F)+ = n>0 Fn tn ✳ ❈❤♦ ≤ i ≤ ✳ ❚❛ ✤➦t Di = {Mn ∩ Di }n∈Z , Ci = {[(Mn ∩ Di ) + Di−1 ]/Di−1 }n∈Z ❑❤✐ ✤â Di ✭t÷ì♥❣ ù♥❣ Ci ✮ ❧➔ ♠ët F ✲❧å❝ ❝→❝ R✲♠æ✤✉♥ ❝♦♥ ❝õ❛ Di tữỡ ự Ci ứ ợ s → [Di−1 ]n → [Di ]n → [Ci ]n → ❝→❝ R✲♠ỉ✤✉♥ ✈ỵ✐ ♠å✐ n ∈ Z✳ ❑❤✐ ✤â t❛ ❝â ❞➣② ❦❤ỵ♣ s❛✉ → R(Di−1 ) → R(Di ) → R(Ci ) → ✭✷✳✶✮ → R (Di−1 ) → R (Di ) → R (Ci ) → ✭✷✳✷✮ → G(Di−1 ) → G(Di ) → G(Ci ) → ✭✷✳✸✮ ✸✸ ❝õ❛ ❝→❝ ♠æ✤✉♥ ♣❤➙♥ ❜➟❝✳ ❱➻ R(Di ) ❧➔ R✲♠æ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ♥➯♥ R(Ci ) ❝ơ♥❣ ❧➔ R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ rữợ t r❛ ♠ët sè t➼♥❤ ❝❤➜t s❛✉✳ ❇ê ✤➲ ✷✳✸✳✶✳ ❬✷✵✱ ▼➺♥❤ ✤➲ ✾❆❪ ❈❤♦ φ : A → B ❧➔ ởt ỗ tr M ởt B ✲♠ỉ✤✉♥✳ ❚❛ ❝â t❤➸ ①❡♠ M ♥❤÷ A✲♠ỉ✤✉♥ t❤ỉ♥❣ q✉❛ φ✳ ❑❤✐ ✤â AssA M = φ−1 (AssB M ) ❇ê ✤➲ ✷✳✸✳✷✳ ❬✷✵✱ ▼➺♥❤ ✤➲ ✾❇❪ ❈❤♦ φ : A B ởt ỗ tr E ❧➔ ♠ët A✲♠æ✤✉♥ ✈➔ F ❧➔ B ✲♠æ✤✉♥✳ ●✐↔ sû F ❧➔ ♠ët A✲♠ỉ✤✉♥ ♣❤➥♥❣✳ ❑❤✐ ✤â ✭✐✮ ❱ỵ✐ ♠é✐ ✐✤➯❛♥ ♥❣✉②➯♥ tè p ∈ A ❜➜t ❦➻✱ {p} ♥➳✉ F/ p F = −1 φ (AssB (F/ p F )) = AssA (F/ p F ) = ∅ ♥➳✉ F/ p F = ✭✐✐✮ AssB (E ⊗A F ) = AssB (F/ p F )✳ p∈Ass(A) ▼➺♥❤ ✤➲ s❛✉ ✤➙② ❝❤♦ t❛ ❝æ♥❣ t❤ù❝ t➼♥❤ ❝❤✐➲✉ ❝õ❛ ✣↕✐ sè ❘❡❡s✳ ▼➺♥❤ ✤➲ ✷✳✸✳✸✳ ❬✷✻✱ ▼➺♥❤ ✤➲ ✷✳✸❪ ❈❤♦ (R, m) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✤à❛ ♣❤÷ì♥❣✱ M ❧➔ R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤✳ ❑❤✐ ✤â ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ ✤➙② ❧➔ ✤ó♥❣✳ ✭✐✮ ❈❤♦ P ∈ AssR R(M) ✈➔ ✤➦t p = P ∩ R✳ ❑❤✐ ✤â p ∈ AssR M ✱ P = p R[t] ∩ R ✈➔ dim R/p + ♥➳✉ dim R/p < ∞, F1 p, dim R/P = ❝→❝ tr÷í♥❣ ❤đ♣ ❦❤→❝ dim R/p ✭✐✐✮ p R[t] ∩ R ∈ AssR R(M) ✈ỵ✐ ♠é✐ p ∈ AssR M ✳ F1 ✭✐✐✐✮ ●✐↔ sû M = (0)✱ d = dimR M < ∞ tỗ t p AsshR M s p ❑❤✐ dimR R(M) = d + 1✳ ❈❤ù♥❣ ♠✐♥❤✳ ✭✐✮ ❈❤♦ P ∈ AssR R(M)✳ ❑❤✐ ✤â P ∈ AssR R[t] R M ữ tỗ t Q AssR[t] R[t] ⊗R M = AssR[t] R[t]/p R[t], p∈AssR M ✸✹ s❛♦ ❝❤♦ P = Q ∩ R✳ ❉♦ ✤â tỗ t p AssR M s p = Q ∩ R ✈➔ Q = p R[t]✳ ❑➨♦ t❤❡♦ P = p R[t] ∩ R✱ p = P ∩ R✳ ✣➦t R = R/p✳ ❑❤✐ ✤â F = {Fn R}n∈Z ❧➔ ♠ët ❧å❝ ❝→❝ ✐✤➯❛♥ ❝õ❛ R ✈➔ R/P ∼ = R(F) ❧➔ ♠ët R✲✤↕✐ sè ♣❤➙♥ ❜➟❝✳ ❉♦ ✤â ✤✐➲✉ ❦❤➥♥❣ ✤à♥❤ ❧➔ ✤ó♥❣ t❤❡♦ ❬✶✷✱ P❤➛♥ ■■✱ ❇ê ✤➲ ✭✷✳✷✮❪✳ ✭✐✐✮ ❈❤♦ p ∈ AssR M ✳ ❚❛ ✈✐➳t p = (0) :R x ✈ỵ✐ x ∈ M ✳ ❑❤✐ ✤â (0) :R ξ = p R[t] ∩ R tr♦♥❣ ✤â ξ = ⊗ x ∈ [R(M)]0 ✳ ✭✐✐✐✮ ❙✉② r❛ tø ❦❤➥♥❣ ✤à♥❤ ✭✐✮✱ ✭✐✐✮✳ ❍➺ q✉↔ ✷✳✸✳✹✳ ●✐↔ sû R ❧➔ ♠ët ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✈➔ M = (0)✳ ❑❤✐ ✤â dimR R(M) = dimR M + ♥➳✉ tỗ t p AsshR M s F1 dimR M p, ❝→❝ tr÷í♥❣ ❤đ♣ ❝á♥ ❧↕✐ ❚÷ì♥❣ tü t❛ ❝ơ♥❣ ❝â t❤➸ ①➙② ❞ü♥❣ ♠ët ✐✤➯❛♥ ♥❣✉②➯♥ tè ❧✐➯♥ ❦➳t ❝õ❛ ♠æ✤✉♥ ❘❡❡s ♠ð rë♥❣✳ ▼➺♥❤ ✤➲ ✷✳✸✳✺✳ ❈→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤ó♥❣✳ ✭✐✮ ❈❤♦ P ∈ AssR R (M)✳ ❑❤✐ ✤â p ∈ AssR M ✱ P = p R[t, t−1 ] ∩ R ✈➔ dim R/P = dim R/p + 1✱ tr♦♥❣ ✤â p = P ∩ R✳ ✭✐✐✮ p R[t, t−1 ] ∩ R ∈ AssR R (M) ✈ỵ✐ ♠é✐ p ∈ AssR M ✳ ✭✐✐✐✮ ●✐↔ sû r➡♥❣ M = (0)✳ ❑❤✐ ✤â dimR R (M) = dimR M + 1✳ ❙û ❞ö♥❣ ▼➺♥❤ ✤➲ ✷✳✸✳✺✱ t❛ ❝â ❦➳t q✉↔ s❛✉✳ ❍➺ q✉↔ ✷✳✸✳✻✳ ●✐↔ sû R ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ✈➔ M = (0)✳ ❑❤✐ ✤â dimG G(M) = dimR M ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ R ❧➔ ✈➔♥❤ ✤à❛ ♣❤÷ì♥❣ ♥➯♥ R ❧➔ ✈➔♥❤ H ✲✤à❛ ♣❤÷ì♥❣✳ ❝❤♦ N ❧➔ ✐✤➯❛♥ ♣❤➙♥ ❜➟❝ tè✐ ✤↕✐ ❞✉② ♥❤➜t ❝õ❛ R ✳ ❑❤✐ ✤â t❛ ❝â R (M)N = (0) ✈➔ u ∈ N✳ ❚❤❡♦ ❜ê ✤➲ ◆❛❦❛②❛♠❛✱ t❛ ❝â G(M)N = (0)✱ tr♦♥❣ ✤â G(M) = (0)✳ ❉♦ ✈➟② t❛ ❝â dimG G(M) = dimR M t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✸✳✺✳ ❇ê ✤➲ ✷✳✸✳✼✳ ❚❛ ❝â {R (Di)}0≤i≤ ❧➔ ♠ët ❧å❝ ❝❤✐➲✉ ❝õ❛ R (M)✳ ◆➳✉ F1 ✈ỵ✐ ♠é✐ p ∈ AssR M t❤➻ {R(Di )}0≤i≤ ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ R(M)✳ p ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ ≤ i ≤ ✳ ❑❤✐ ✤â dimR R (Di ) = di + 1✱ ✈➻ Di = (0)✳ ❈❤♦ P ∈ AssR R (Ci )✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✸✳✺✱ t❛ ❝â dim R /P = di + = dimR R (Ci ) ỵ ✷✳✶✳✻✱ {R (Di )}0≤i≤ ❧➔ ♠ët ❧å❝ ❝❤✐➲✉ ❝õ❛ R (M) ỵ ỵ M ❧➔ ♠ët R✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✈➔ F1 p ✈ỵ✐ ♠é✐ p ∈ AssR M ✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✮ R (M) ❧➔ ♠ët R ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✭✐✐✮ G(M) ❧➔ ♠ët G ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✱ {G(Di )}0≤i≤ ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ G(M)✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② M ❧➔ R✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❈❤ù♥❣ ♠✐♥❤✳ ✣✐➲✉ ❦✐➺♥ ✭✐✮ ✈➔ ✭✐✐✮ ❧➔ t÷ì♥❣ ✤÷ì♥❣ ✈➻ R (M ) ❧➔ R ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R (Di )/R (Di−1 ) ❧➔ R ✲♠æ✤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✳ ❚❤❡♦ ❞➣② ❦❤ỵ♣ ✭✷✳✷✮ t❛ ❝â R (Di )/R (Di−1 ) ∼ = R (Ci ) ♥➯♥ ✤✐➲✉ ♥➔② t÷ì♥❣ ữỡ ợ R (Ci ) R ổ G(Ci ) ∼ = −1 −1 R (Ci )/t R (Ci ) ✈➔ t ❧➔ R (Ci )✲❝❤➼♥❤ q✉② ♥➯♥ G(Ci ) ❧➔ G ✲♠æ✤✉♥ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✳ ▲↕✐ t❤❡♦ ❞➣② ❦❤ỵ♣ ✭✷✳✸✮✱ t❛ ❝â G(Ci ) ∼ = G(Di )/G(Di−1 ) ♥➯♥ G(Di )/G(Di−1 ) ❧➔ G ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ✣✐➲✉ tữỡ ữỡ ợ G(M) G ổ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❦❤➥♥❣ ✤à♥❤ M ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❚ø ❞➣② ❦❤ỵ♣ ϕ → R (Ci ) → R[t, t−1 ] ⊗R Ci → X = Coker ϕ → ❝→❝ R ✲♠ỉ✤✉♥ ✈ỵ✐ ≤ i ≤ ✳ ❱➻ R (Ci ) ❧➔ ♠ët R ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ X(t−1 ) = (0) ♥➯♥ R[t, t−1 ] ⊗R Ci ∼ = Ci ⊗R R (F)(t−1 ) ∼ = R (F)(t−1 ) ❧➔ ❈♦❤❡♥✲ −1 ▼❛❝❛✉❧❛②✳ ▲↕✐ ❝â R → R[t, t ] = R[t](t−1 ) ❧➔ ♣❤➥♥❣ ♥➯♥ Ci ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❉♦ ✈➟② M ❧➔ ♠ët R✲♠♦❞✉❧❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❙❛✉ ✤➙② t❛ t➻♠ ♠è✐ ❧✐➯♥ ❤➺ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝õ❛ R(M) ✈➔ G(M)✳ ❇ê ✤➲ ✷✳✸✳✾✳ ❈❤♦ P ∈ Spec R s❛♦ ❝❤♦ P a✳ ◆➳✉ G(M)P = (0) (t÷ì♥❣ ù♥❣ R(M)P = (0) ✈➔ P ⊇ ua) t❤➻ R(M)P = (0) (t÷ì♥❣ ù♥❣ G(M)P = (0))✳ ❑❤✐ ✤â✱ ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉ ✤➙② ❧➔ ✤ó♥❣✳ ✭✐✮ R(M)P ❧➔ RP ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ G(M)P ❧➔ GP ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ✭✐✐✮ dimRP R(M)P = dimRP G(R)P + 1✳ ❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ P ∈ Spec R s❛♦ ❝❤♦ P a✱ ♥❤÷♥❣ P ⊇ ua✳ ▲➜② ♠ët n ♣❤➛♥ tû t❤✉➛♥ ♥❤➜t ξ = at ∈ a \ P tr♦♥❣ ✤â n > 0✱ a ∈ Fn ✳ ❑❤✐ ✤â t❛ ❝â x = uξ = atn−1 ∈ P ✱ ✈➻ P ⊇ ua✳ ✸✻ ❑❤➡♥❣ ✤à♥❤✳ ◆➳✉ Q ∈ AssR R(M) s❛♦ ❝❤♦ Q ⊆ P t❤➻ x ∈ / Q ữ x ổ ữợ ổ tr♦♥❣ R(M)P ✳ ❈❤ù♥❣ ♠✐♥❤ ❦❤➥♥❣ ✤à♥❤✳ ❚❛ ❣✐↔ sû tỗ t Q AssR R(M) s Q P ♥❤÷♥❣ x ∈ Q✳ ❱✐➳t Q = (0) :R η tr♦♥❣ ✤â η = t ⊗ m ( ∈ Z, m ∈ M )✳ ❑❤✐ ✤â t❛ ❝â ξ = atn ∈ (0) :R η = Q ⊆ P ✳ ✣✐➲✉ ♥➔② ❧➔ ♠➙✉ t❤✉➝♥✳ ❱➻ P a ♥➯♥ t❛ ❝â RP = RP ✈➔ R(M)P = R (M)P ✳ ❉♦ ✈➟② (ua)RP = (ua)RP = uRP = xRP ❛♥❞ (ua)R(M) ⊆ u[R(M)]+ ❉♦ ✤â [uR(M)+ ]P = xR (M)P = xR(M)P ♥➯♥ R(M)P /xR(M)P ∼ = G(M)P ♥❤÷ RP ✲♠ỉ✤✉♥✳ ▼➦t ❦❤→❝✱ ❝❤♦ P ∈ Spec R s❛♦ ❝❤♦ G(M)P = (0)✳ ❑❤✐ ✤â i ε P ⊇ ua ✈➻ ua = uR ∩ R = Ker(R → R → G)✳ ❉♦ ✤â ❦❤➥♥❣ ✤à♥❤ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ♥❤í ✤➥♥❣ ❝➜✉ tr➯♥✳ ❚❛ ❝â ❦➳t q✉↔ s❛✉ ữủ ữ r ts ỵ I ❧➔ ♠ët ✐✤➯❛♥ ❝õ❛ R ✈➔ t ∈ Z✳ t s ỗ t ởt sè ♥❣✉②➯♥ > s❛♦ ❝❤♦ I ·Him (M ) = (0) ✈ỵ✐ ♠é✐ i = t✳ ✭✐✐✮ Mp ❧➔ Rp ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ t = dimRp Mp + dim R/ p ợ ộ p SuppR M ữ p I ✳ ❑❤✐ ✤â (i) ⇒ (ii) ❧➔ ✤ó♥❣✳ ❑➳t q ữủ ú R ỗ ❝õ❛ ♠ët ✈➔♥❤ ●♦r❡♥st❡✐♥ ✤à❛ ♣❤÷ì♥❣✳ ❈❤♦ M ❧➔ ✐✤➯❛♥ ♣❤➙♥ ❜➟❝ tè✐ ✤↕✐ ❞✉② ♥❤➜t ❝õ❛ R✳ ▼➺♥❤ ✤➲ s ữủ ự ởt tr ú ỵ ợ I t ỵ ❤✐➺✉ I ∗ ❧➔ ✐✤➯❛♥ s✐♥❤ ❜ð✐ ❝→❝ ♣❤➛♥ tû t❤✉➛♥ ♥❤➜t ❝õ❛ I ✳ ❈❤♦ R ❧➔ ✈➔♥❤ Z✲♣❤➙♥ ❜➟❝✱ M ❧➔ R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝✱ I ❧➔ ✐✤➯❛♥ ♣❤➙♥ õ ổ ố ỗ ữỡ HIi (M )(i ∈ N) ❝ơ♥❣ ❝â ❝➜✉ tró❝ ❧➔ R✲♠ỉ✤✉♥ ♣❤➙♥ ❜➟❝✳ ▼➺♥❤ ✤➲ ✷✳✸✳✶✶✳ ❬✷✻✱ ▼➺♥❤ ✤➲ ✸✳✺❪ ●✐↔ sû HiM(G(M)) ❧➔ ❝→❝ R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i = d✳ ❑❤✐ ✤â HiM (R(M)) ❧➔ ♠ët ❤å ❝→❝ R✲♠ỉ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ ♠å✐ i = d + 1✳ ✸✼ ❚❤❡♦ ❬✸✱ ▼➺♥❤ ợ ộ i tỗ t r Z s❛♦ ❝❤♦ [Htm (N )]a = (0) ✈ỵ✐ ♠å✐ n ≥ r✳ ❚❛ ✤➦t a(N ) = max{n ∈ Z | [HtM (N )]n = (0)} ✈ỵ✐ N ❧➔ R✲♠ỉ✤✉♥ ♣❤➙♥ ❜➟❝ ❤ú✉ ❤↕♥ s✐♥❤ ❝â ❝❤✐➲✉ t✳ ❙è a(N ) ✤÷đ❝ ❣å✐ ❧➔ ❛✲❜➜t ❜✐➳♥ ❝õ❛ N ✭①❡♠ ❬✶✹✱ ✣à♥❤ ♥❣❤➽❛ ✸✳✶✳✹❪✮✳ ❱ỵ✐ ❦➼ ❤✐➺✉ ♥➔② t❛ ❝â ❜ê ✤➲ s❛✉✳ ❇ê ✤➲ ✷✳✸✳✶✷✳ ❬✷✻✱ ❇ê ✤➲ ✸✳✼❪ ❈→❝ ❦❤➥♥❣ ✤à♥❤ s❛✉ ✤➙② ❧➔ ✤ó♥❣✳ ✭✐✮ [Hd+1 M (R(M))]n = (0) ✈ỵ✐ ♠å✐ n ≥ 0✳ d+1 ✭✐✐✮ ◆➳✉ [Hd+1 M (R(M))]−1 = (0) t❤➻ HM (R(M)) = (0)✳ ❉♦ ✤â ♥➳✉ dimR R(M) = d + t❤➻ a(R(M)) = −1✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ①➨t ❞➣② ❦❤ỵ♣ s❛✉ → L → R(M) → M → 0 → L(1) → R(M) → G(M) → ❝õ❛ ❝→❝ R✲♠æ✤✉♥ ♣❤➙♥ ❜➟❝✱ tr♦♥❣ ✤â L = R(M)+ ✳ ⑩♣ tỷ ố ỗ ữỡ tr➯♥✱ t❛ ❝â d+1 Hdm (M ) → Hd+1 M (L) → HM (R(M)) → ✈➔ d+1 HdM (G(M)) → Hd+1 M (L)(1) → HM (R(M)) → ❉♦ ✤â ∼ d+1 [Hd+1 M (L)]n = [HM (R(M))]n ✈ỵ✐ n = 0, ✈➔ d+1 [Hd+1 M (L)]n+1 → [HM (R(M))]n ợ n Z d+1 ữ [Hd+1 M (R(M))]n = (0) ✈ỵ✐ n ≥ ✈➻ HM (R(M)) ❧➔ ❆rt✐♥✳ ❍ì♥ ♥ú❛ t❛ ❝â d+1 [HM (R(M))]−1 → [Hd+1 M (R(M))]n → ✈ỵ✐ n < 0✱ ♥❤÷ ✈➟② t❛ ❝ơ♥❣ ❝â ❦❤➥♥❣ ✤à♥❤ ✭✐✐✮✳ ❑➳t q✉↔ s❛✉ ✤➙② ♠ð rë♥❣ ❦➳t q✉↔ ✤➣ ❜✐➳t tø ✈➔♥❤ s❛♥❣ ♠æ✤✉♥✱ ❝→❝ ❦➳t q✉↔ ✤➣ ❜✐➳t ✈➲ ✈➔♥❤ t P ỵ ỵ ỵ ỵ ỵ ỵ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✮ R(M) ❧➔ R✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ dimR R(M) = d + 1✳ ✭✐✐✮ HiM (G(M)) = [HiM (G(M))]−1 ✈ỵ✐ ♠é✐ i < d ✈➔ a(G(M)) < 0✳ ❑❤✐ ✤â tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② [HiM (G(M))]−1 = Him (M ) ữ Rổ ợ i < d✳ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t ❞➣② ❦❤ỵ♣ s❛✉ i+1 (∗) · · · → Him (L) → HiM (R(M)) → Him (M ) → Hi+1 M (L) → HM (R(M)) → · · · (∗∗) · · · → Him (L)(1) → HiM (R(M)) → HiM (G(M)) → i+1 → Hi+1 M (L)(1) → HM (R(M)) → · · · ợ ộ i < d rữợ t t sỷ r➡♥❣ R(M) ❧➔ R✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈ỵ✐ sè ❝❤✐➲✉ d + 1✳ ❑❤✐ ✤â i ∼ i+1 Him (M ) ∼ = Hi+1 M (L) ❛♥❞ HM (G(M)) = HM (L)(1) ✈ỵ✐ i < d✳ ❉♦ ✈➟② t❛ ❝â HiM (G(M)) = [HiM (G(M))]−1 ✈➔ [HiM (G(M))]−1 ∼ = i Hm (M ) ♥❤÷ R✲♠ỉ✤✉♥✳ ❱➻ R(M) ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ t❛ ❝â d+1 → Hdm (M ) → Hd+1 M (L) → HM (R(M)) → 0 → HdM (G(M)) → Hd+1 M (L)(1) ❉♦ ✈➟② a(G(M)) < t❤❡♦ ❇ê ✤➲ ✷✳✸✳✶✷✳ ◆❣÷đ❝ ❧↕✐✱ ❝❤♦ i < d✳ ❚ø ❞➣② (∗)✱ (∗∗) ð tr➯♥ ✈➔ ❣✐↔ t❤✐➳t✱ t❛ ❝â ∼ i+1 [Hi+1 M (L)]n+1 = [HM (R(M))]n ∼ i+1 [Hi+1 M (L)]n+1 = [HM (R(M))]n+1 ✈ỵ✐ ♠é✐ n ≥ 0✳ ❉♦ ✤â [HiM (R(M))]n = (0) ✈ỵ✐ n ≥ ✈➻ Hi+1 M (R(M)) ❧➔ ❆rt✐♥✳ ❍ì♥ ♥ú❛✱ t❛ ❝â i+1 → [Hi+1 M (R(M))]n → [HM (R(M))]n−1 ✈ỵ✐ n < t❤❡♦ ❞➣② (∗) ✈➔ (∗∗) ð tr➯♥ ✈➔ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✸✳✶✶✱ Hi+1 M (R(M)) i+1 ❧➔ R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ ✈ỵ✐ i < d✳ ❑❤✐ ✤â [HM (R(M))]n = (0) s✉② r❛ Hi+1 M (R(M)) = (0) ✈ỵ✐ ♠å✐ i < d✳ ❉♦ ✤â R(M) ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝â sè ❝❤✐➲✉ d + 1✳ ✸✾ ❍➺ q✉↔ ✷✳✸✳✶✹✳ ●✐↔ sû M ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✮ R(M) ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ dimR R(M) = d + 1✳ ✭✐✐✮ G(M) ❧➔ G ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ a(G(M)) < 0✳ ✣à♥❤ ỵ ỵ sỷ M R ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✈➔ F1 p ✈ỵ✐ ♠é✐ p ∈ AssR M ✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✮ R(M) ❧➔ R✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✭✐✐✮ G(M) ❧➔ G ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✱ {G(Di )}0≤i≤ ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ G(M) ✈➔ a(G(Ci )) < ✈ỵ✐ ♠é✐ ≤ i ≤ ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② R (M) ❧➔ R ✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✸✳✼✱ R(M) ❧➔ R✲♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ R(Ci ) ❧➔ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈ỵ✐ ♠é✐ ≤ i ≤ ✳ s tữỡ ữỡ ợ G(Ci ) G ✲♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ a(G(Ci )) < ✈ỵ✐ ♠å✐ ≤ i ≤ t❤❡♦ ❍➺ q✉↔ ✷✳✸✳✶✹✳ ❉♦ ✤â t❛ ❝â sü t÷ì♥❣ ✤÷ì♥❣ ❣✐ú❛ ♠➺♥❤ ✤➲ ✭✐✮ ✈➔ ✭✐✐✮✳ ❚❛ ❦➳t t❤ó❝ ♠ư❝ ♥➔② ❜ð✐ ❝→❝ ♣❤→t ❜✐➸✉ ố ợ ởt ỵ ỵ (R, m) tr ữỡ F = {Fn }n∈Z ❧➔ ♠ët ❧å❝ ❝→❝ ✐✤➯❛♥ ❝õ❛ R s❛♦ ❝❤♦ F1 = R✳ ❚❛ ❣✐↔ sû r➡♥❣ R = R(F) ❧➔ ✈➔♥❤ ◆♦❡t❤❡r✳ ❈❤♦ {Di }0≤i≤ ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ R✳ ❑❤✐ ✤â Di = {Fn ∩ Di }n∈Z ✭t÷ì♥❣ ù♥❣ Ci = {[Fn ∩ Di + Di−1 ]/Di−1 }n∈Z ✮ ❧➔ ♠ët F ✲❧å❝ ❝õ❛ Di tữỡ ự Ci ợ i ỵ ỵ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✮ R ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✭✐✐✮ G ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ✈➔ {G(Di )}0≤i≤ ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ G ✳ ❚r♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② R ỵ ỵ sỷ R F1 p ✈ỵ✐ ♠é✐ p ∈ Ass R✳ ❑❤✐ ✤â ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✳ ✭✐✮ R ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✭✐✐✮ G ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✱ {G(Di )}0≤i≤ ❧➔ ❧å❝ ❝❤✐➲✉ ❝õ❛ G ✈➔ a(G(Ci )) < ợ i r trữớ ❤ñ♣ ♥➔② R ❧➔ ✈➔♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✹✵ ❑➌❚ ▲❯❾◆ ▼ö❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ t➻♠ ❤✐➸✉ ✈➔♥❤ ❧å❝✱ ❝→❝ ✤↕✐ sè ❘❡❡s ✈➔ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝õ❛ ❝→❝ ✤↕✐ sè ❘❡❡s✳ ❱✐➺❝ t➻♠ ❤✐➸✉ ❝❤✐ t✐➳t ♠ët sè t➼♥❤ ❝❤➜t ❝õ❛ ♠ỉ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝ơ♥❣ ❧➔ ♠ët ♠ư❝ ✤➼❝❤ ❦❤→❝ ❝õ❛ ❧✉➟♥ ✈➠♥✳ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à tr♦♥❣ ❧✉➟♥ ✈➠♥ ❝❤õ ②➳✉ ✤÷đ❝ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ tø ❤❛✐ ❝✉è♥ s→❝❤ ✧❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✧ ❝õ❛ ❍✳ ▼❛ts✉♠✉r❛ ✈➔ ✧▲♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✿ ❆♥ ❛❧❣❡❜r❛✐❝ ✐♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ❣❡♦♠❡tr✐❝ ❛♣♣❧✐❝❛t✐♦♥s✧ ❝õ❛ ▼✳ ❇r♦❞♠❛♥♥ ❛♥❞ ❘✳ ❨✳ ❙❤❛r♣✳ ❈ö t❤➸ ❧✉➟♥ ✈➠♥ ✤➣ t❤✉ ✤÷đ❝ ♠ët sè ❦➳t q✉↔ s❛✉✿ ✲ ❚r➻♥❤ ❜➔② ❤➺ t❤è♥❣ ♠ët sè ❦✐➳♥ t❤ù❝ ✈➲ ✈➔♥❤ ❧å❝ ✈➔ t➼♥❤ ◆♦❡t❤❡r ❝õ❛ ✈➔♥❤ ❧å❝✳ ✲ ❚r➻♥❤ ❜➔② ✈➲ ❧å❝ ❝❤✐➲✉ ✈➔ ♠æ t↔ ❝❤✐ t✐➳t ✈➲ ❧å❝ ❝❤✐➲✉✳ ✲ ❚➻♠ ❤✐➸✉ ✈➲ ❧å❝ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ ♠æ t↔ ❝❤✐ t✐➳t ✈➲ ❧å❝ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②✳ ✲ ❚➻♠ ❤✐➸✉ ✈➲ ♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈➔ ♠æ✤✉♥ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣②✳ ✲ ❚➻♠ ❤✐➸✉ ✈➲ t➼♥❤ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞➣② ❝❤♦ ❝→❝ ✤↕✐ sè ❘❡❡s✳ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❇✐s❤♦♣ ❲✳✱ P❡tr♦ ❏✳ ❲✳✱ ❘❛t❧✐❢❢ ▲✳ ❏✳✱ ❘✉s❤ ❉✳ ❊✳ ✭✶✾✽✾✮✱ ✧◆♦t❡ ♦♥ ◆♦❡t❤❡r✐❛♥ ❢✐❧tr❛t✐♦♥s✧✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✶✼✱ ◆♦ ✷✱ ✹✼✶✲✹✽✺✳ ❬✷❪ ❇♦✉r❜❛❦✐ ◆✳ ✭✶✾✽✾✮✱ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❙♣r✐♥❣❡r ♣r❡ss✳ ❬✸❪ ❇r♦❞♠❛♥♥ ▼✳ ❛♥❞ ❙❤❛r♣ ❘✳❨✳ ✭✶✾✾✽✮✱ ▲♦❝❛❧ ❈♦❤♦♠♦❧♦❣②✿ ❆♥ ❆❧❣❡❜r❛✐❝ ■♥tr♦❞✉❝t✐♦♥ ✇✐t❤ ●❡♦♠❡tr✐❝ ❆♣♣❧✐❝❛t✐♦♥s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✹❪ ❇r✉♥s ❲✳ ❛♥❞ ❍❡r③♦❣ ❏✳ ✭✶✾✾✽✮✱ ❈♦❤❡♥✲▼❛❝❛✉❧❛② r✐♥❣s✱ ❈❛♠❜r✐❞❣❡ ❙t✉❞✐❡s ✐♥ ❆❞✈❛♥❝❡❞ ▼❛t❤❡♠❛t✐❝s✱ ✻✵✳ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✱ ❈❛♠❜r✐❞❣❡✳ ❬✺❪ ❈✉♦♥❣ ◆✳ ❚✳ ❛♥❞ ❈✉♦♥❣ ❉✳ ❚✳ ✭✷✵✵✼✮✱ ✧❖♥ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✧✱ ❑♦❞❛✐ ▼❛t❤✳ ❏✳✱ ✭✸✵✮✱ ♣♣✳✹✵✾✲✹✷✽✳ ❬✻❪ ❈✉♦♥❣ ◆✳ ❚✳ ❛♥❞ ❈✉♦♥❣ ❉✳ ❚✳ ✱ ✧❖♥ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✲ ✉❧❡s✧✱ ♣r❡♣r✐♥t✳ ❬✼❪ ❈✉♦♥❣ ◆✳ ❚✳ ✱ ●♦t♦ ❙✳ ❛♥❞ ❚r✉♦♥❣ ❍✳ ▲✳ ✭✷✵✶✸✮✱ ✧❚❤❡ ❡q✉❛❧✐t② I = q I ✐♥ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② r✐♥❣s✧✱ ❏✳ ❆❧❣❡❜r❛✱ ✭✸✼✾✮ ✱ ♣♣✳ ✺✵✲✼✾✳ ❬✽❪ ❈✉♦♥❣ ◆✳ ❚✳ ❛♥❞ ◆❤❛♥ ▲✳ ❚✳ ✭✷✵✵✸✮✱ ✧Ps❡✉❞♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❛♥❞ ♣s❡✉❞♦ ❣❡♥❡r❛❧✐③❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✱✧ ❏✳ ❆❧❣❡❜r❛ ✷✻✼✱ ♣♣✳ ✶✺✻✲✶✼✼✳ ❬✾❪ ❈✉♦♥❣ ◆✳ ❚✳ ❛♥❞ ◆❤❛♥ ▲✳ ❚✳ ✭✷✵✵✷✮✱ ✧❖♥ ◆♦❡t❤❡r✐❛♥ ❞✐♠❡♥t✐♦♥ ♦❢ ❆rt✐♥✐❛♥ ♠♦❞✉❧❡s✧✱ ❱✐❡t♥❛♠ ❏✳ ▼❛t❤✳ ✸✵✱ ♣♣✳ ✶✷✶✲✶✸✵✳ ❬✶✵❪ ❋❛❧t✐♥❣s ●✳✭✶✾✼✽✮✱ ✧❯❜❡r ❞✐❡ ❆♥♥✉❧❛t♦r❡♥ ❧♦❦❛❧❡r ❑♦❤♦♠♦❧♦❣✐❡❣r✉♣♣❡♥✧✱ ❆r❝❤✐✈ ❞❡r ▼❛t❤✳✱ ✸✵✱ ♣♣✳ ✹✼✸✕✹✼✻✳ ✹✷ ❬✶✶❪ ●♦t♦ ❙✳✱ ❍♦r✐✉❝❤✐ ❨✳ ❛♥❞ ❙❛❦✉r❛✐ ❍✳✭✷✵✶✵✮✱ ✧❙❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②♥❡ss ✈❡rs✉s ♣❛r❛♠❡tr✐❝ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ♣♦✇❡rs ♦❢ ♣❛r❛♠❡t❡r ✷ ✐❞❡❛❧s✧✱ ❏✳ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✱ ♣♣✳ ✸✼✲✺✹✳ ❬✶✷❪ ●♦t♦ ❙✳ ❛♥❞ ◆✐s❤✐❞❛ ❑✳✭✶✾✾✹✮✱ ✧❚❤❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❛♥❞ ●♦r❡♥st❡✐♥ ♣r♦♣❡rt✐❡s ♦❢ ❘❡❡s ❛❧❣❡❜r❛s ❛ss♦❝✐❛t❡❞ t♦ ❢❧tr❛t✐♦♥s✧✱ ▼❡♠✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✶✶✵✳ ❬✶✸❪ ●♦t♦ ❙✳ ❛♥❞ ❙❤✐♠♦❞❛ ❨✳✭✶✾✽✷✮✱ ✧❖♥ t❤❡ ❘❡❡s ❛❧❣❡❜r❛ ♦❢ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧♦❝❛❧ r✐♥❣s✧✱ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ▲❡❝t✉r❡ ◆♦t❡ ✐♥ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▼❛t❤❡♠❛t✐❝s✱ ▼❛r❝❡❧ ❉❡❦❦❡r ■♥❝✱ ✻✽✱ ♣♣✳ ✷✵✶✲✷✸✶✳ ❬✶✹❪ ●♦t♦ ❙✳ ❛♥❞ ❲❛t❛♥❛❜❡ ❑✳✭✶✾✼✽✮✱ ✧❖♥ ❣r❛❞❡❞ r✐♥❣s✱ ■✧✱ ❏✳ ▼❛t❤✳ ❙♦❝✳ ❏❛♣❛♥✱ ✸✵✱ ♣♣✳ ✶✼✾✲✷✶✸✳ ❬✶✺❪ ●♦t♦ ❙✳ ❛♥❞ ❲❛t❛♥❛❜❡ ❑✳✭✶✾✼✽✮✱ ❖♥ ❣r❛❞❡❞ r✐♥❣s ■■ ✭Zn ✲❣r❛❞❡❞ r✐♥❣s✮✱ ✶ ❚♦❦②♦ ❏✳ ▼❛t❤✳✱ ✱ ♥♦✳ ✷✱ ♣♣✳ ✷✸✼✲✷✻✶✳ ❬✶✻❪ ❍❡r③♦❣ ❏✳ ❛♥❞ ❙❜❛rr❛ ❊✳ ✭✷✵✵✶✮✱ ✧❙❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ❛♥❞ ❧♦❝❛❧ ❝♦❤♦♠♦❧♦❣②✧✱ ✐♥ ❆r✐t❤♠❡t✐❝ ❛♥❞ ●❡♦♠❡tr②✱ Pr♦❝❡❡❞✳ ♦❢ ■♥t❡r♥✳ ❈♦❧❧✳ ♦♥ ❆❧❣✳✱ ♣♣✳ ✸✷✼✲✸✹✵✳ ❬✶✼❪ ❍❡rr♠❛♥♥ ▼✳✱ ■❦❡❞❛ ❙✳✱ ❖r❜❛♥③ ❯✳ ✭✶✾✽✽✮ ❊q✉✐♠✉❧t✐♣❧✐❝✐t② ❛♥❞ ❇❧♦✇✐♥❣ ✉♣✱ ❙♣r✐♥❣❡r ♣r❡ss✳ ❬✶✽❪ ■❦❡❞❛✳ ❙ ✭✶✾✽✸✮✱✧❚❤❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②♥❡ss ♦❢ t❤❡ ❘❡❡s ❛❧❣❡❜r❛s ♦❢ ❧♦❝❛❧ r✐♥❣s✧✱ ◆❛❣♦②❛ ▼❛t❤✳❏✳✱ ✽✾ ✱ ♣♣✳ ✹✼✲✻✸✳ ❬✶✾❪ ▲❛t❧✐❢❢ ❏r ▲✳ ❏✳ ✭✶✾✼✾✮✱✧◆♦t❡ ♦♥ ❡ss❡♥t✐❛❧❧② ♣♦✇❡r ❢✐❧tr❛t✐♦♥s ✧✱ ▼✐❝❤✐❣❛♥ ▼❛t❤✳ ❏♦✉r♥❛❧ ✷✻✱ ♣♣✳ ✸✶✸✲✸✷✹✳ ❬✷✵❪ ▼❛ts✉♠✉r❛ ❍✳ ✭✶✾✽✵✮✱ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❙❡❝♦♥❞ ❊❞✐t✐♦♥✱ ❇❡♥❥❛♠✐♥✳ ❬✷✶❪ ▼❛ts✉♠✉r❛ ❍✳ ✭✶✾✽✻✮✱ ❈♦♠♠✉t❛t✐✈❡ r✐♥❣ t❤❡♦r②✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ❬✷✷❪ ❘❡❡s ❉✳ ✭✶✾✽✾✮✱ ▲❡❝t✉r❡ ♦♥ t❤❡ ❆s②♠♣t♦t✐❝ ❚❤❡♦r② ♦❢ ■❞❡❛❧s✱ ❈❛♠❜r✐❞❣❡ ❯♥✐✈❡rs✐t② Pr❡ss✳ ✹✸ ❬✷✸❪ ❘♦❜❡rts P✳ ✭✶✾✽✺✮✱ ✧❆ ♣r✐♠❡ ✐❞❡❛❧ ✐♥ ❛ ♣♦❧②♠♦♥✐❛❧ r✐♥❣ ✇❤♦s❡ s②♠❜♦❧✐❝ ❜❧♦✇✲✉♣ ✐s ♥♦t ◆♦❡t❤❡r✐❛♥✧✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ✾✹✱ ♥♦✳ ✹✱ ♣♣✳ ✺✽✾✲✺✾✷✳ ❬✷✹❪ ❙❝❤❡♥③❡❧ P✳ ✭✶✾✾✽✮✱ ✧❖♥ t❤❡ ❞✐♠❡♥s✐♦♥ ❢✐❧tr❛t✐♦♥ ❛♥❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❢✐❧t❡r❡❞ ♠♦❞✉❧❡s✧✱ ✐♥✿ Pr♦❝✳ ♦❢ t❤❡ ❋❡rr❛r❛ ▼❡❡t✐♥❣ ✐♥ ❤♦♥♦✉r ♦❢ ▼❛r✐♦ ❋✐♦r❡♥t✐♥✐✱ ❯♥✐✈❡rs✐t② ♦❢ ❆♥t✇❡r♣✱ ❲✐❧r✐❥❦✱ ❇❡❧❣✐✉♠✱ ♣♣✳ ✷✹✺✲✷✻✹✳ ❬✷✺❪ ❙t❛♥❧❡② ❘✳ P✳ ✭✶✾✾✻✮✱ ❈♦♠❜✐♥❛t♦r✐❝s ❛♥❤ ❈♦♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❙❡❝♦♥❞ ❡❞✐t✐♦♥✱ ❇✐r❦❤☎✉s❡r ❇♦st♦♥✳ ❬✷✻❪ ❚❛♥✐❣✉❝❤✐ ◆✳✱P❤✉♦♥❣ ❚✳ ❚✳✱ ❉✉♥❣ ◆✳❚✳ ❛♥❞ ❆♥ ❚✳ ◆✳ ✭✷✵✶✼✮✱ ✧❙❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❘❡❡s ❛❧❣❡❜r❛s✧✱ ❏♦✉r♥❛❧ ♦❢ t❤❡ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t② ♦❢ ❏❛♣❛♥✱ ✻✾✭✶✮✱ ♣♣✳ ✷✾✶ ✕ ✸✵✼✳ ❬✷✼❪ ❚❛♥✐❣✉❝❤✐ ◆✳✱ P❤✉♦♥❣ ❚✳ ❚✳ ✱ ❉✉♥❣ ◆✳ ❚✳ ❛♥❞ ❆♥ ❚✳ ◆✳ ✭✷✵✶✽✮✱ ✧❚♦♣✐❝ ♦♥ s❡q✉❡♥t✐❛❧❧② ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s✧✱ t♦ ❛♣♣❡❛r ✐♥ ❏♦✉r♥❛❧ ♦❢ ❈♦♠♠✉✲ t❛t✐✈❡ ❆❧❣❡❜r❛ ✭✾ ♣❛❣❡s✮✳ ❬✷✽❪ ❚r✉♥❣ ◆✳ ❱✳ ❛♥❞ ■❦❡❞❛ ❙✳✭✶✾✽✾✮✱ ✧❲❤❡♥ ✐s t❤❡ ❘❡❡s ❛❧❣❡❜r❛ ❈♦❤❡♥✲ ▼❛❝❛✉❧❛②❄✧✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✶✼✱ ♣♣✳ ✷✽✾✸✲✷✾✷✷✳ ❬✷✾❪ ❱✐❡t ❉✳ ◗✳ ✭✶✾✾✸✮✱ ✧❆ ♥♦t❡ ♦♥ t❤❡ ❈♦❤❡♥✲▼❛❝❛✉❧❛②♥❡ss ♦❢ ❘❡❡s ❆❧❣❡❜r❛ ♦❢ ❢✐❧tr❛t✐♦♥s✧✱ ❈♦♠♠✳ ❆❧❣❡❜r❛ ✷✶✱ ♣♣✳ ✷✷✶✲✷✷✾✳ ✹✹ ... ✣❸■ ❍➴❈ ❙× P❍❸▼ ✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕ ◆●❯❨➍◆ ❈❍➑ ❚❹▼ ❚➑◆❍ ❈❖❍❊◆✲▼❆❈❆❯▲❆❨ ❉❶❨ ❈Õ❆ ✣❸■ ❙➮ ❘❊❊❙ số ỵ tt số số ✹✻ ✵✶ ✵✹ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ❈→♥ ữợ P ❚❙✳ ❚r➛♥ ◆❣✉②➯♥ ❆♥ ❚❍⑩■ ◆●❯❨➊◆✱ ◆❿▼... ♠ët ❧å❝ ❧ơ② t❤ø❛ ❝èt ②➳✉❀ G tr tỗ t ởt số ♥❣✉②➯♥ ❞÷ì♥❣ k s❛♦ ❝❤♦ Fkn ⊆ (Rad(F1 ))n ❦❤✐ n 0❀ ✶✻ ✭✐✐✐✮ G ❧➔ ✈➔♥❤ ◆♦❡t❤❡r ✈➔ ✈ỵ✐ ộ số ữỡ n ổ tỗ t số ❞÷ì♥❣ ρ(n) s❛♦ ❝❤♦ Fρ(n) ⊆ (Rad(F1 ))n... R✲♠ỉ✤✉♥ ❤ú✉ ❤↕♥ s✐♥❤ ✈ỵ✐ M = {Mn }n≥0 ❧➔ ♠ët F ✲❧å❝✳ ◆➳✉ G(M, M) ❧➔ ♠ët G ✲♠æ✤✉♥ ỳ s ợ ộ số ữỡ n ổ tỗ t số ữỡ (n) s M(n) ⊆ (Rad(F1 ))n M1 t❤➻ M ❧➔ F ✲❧å❝ tèt✳ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ G(M, M) ❧➔ ♠ët G