✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ❑❍❖❆ ❚❖⑩◆ ✲ ❈❒ ✲ ❚■◆ ❍➴❈ ◆●❯❨➍◆ ✣Ù❈ ◆●⑨ ❳❹❨ ❉Ü◆● ❙■◆●❊❘ ❱⑨ ❇⑨■ ❚❖⑩◆ ❍■❚ tốt số ỵ t❤✉②➳t sè ▼➣ sè✿ ✻✵✹✻✵✶✵✹ ❍➔ ◆ë✐ ✲ ✷✵✶✽ ✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ❑❍❖❆ ❚❖⑩◆ ✲ ❈❒ ✲ ❚■◆ ❍➴❈ ◆●❯❨➍◆ ✣Ù❈ ◆●⑨ ❳❹❨ ❉Ü◆● ❙■◆●❊❘ ❱⑨ ❇⑨■ ❚❖⑩◆ ❍■❚ ▲✉➟♥ tốt số ỵ tt số số ữợ ❍Ú❯ ❱■➏❚ ❍×◆● ❍➔ ◆ë✐ ✲ ✷✵✶✽ ▲❮■ ❈❷▼ ❒◆ rữợ t tổ tọ ỏ t ỡ ỳ t ữ sỹ ữợ t➟♥ t➻♥❤✱ ♥❣❤✐➯♠ ❦❤➢❝ tr♦♥❣ ❦❤♦❛ ❤å❝ ✈➔ ♥❤ú♥❣ ♠è✐ q✉❛♥ t➙♠ ✤➦❝ ❜✐➺t tr♦♥❣ ❝✉ë❝ sè♥❣✳ ❚✐➳♣ t❤❡♦✱ tæ✐ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ ✤➳♥ ❝→❝ ❝→♥ ❜ë tr♦♥❣ ❑❤♦❛ ❚♦→♥✲❈ì✲❚✐♥ ❤å❝✱ ✤➦❝ ❜✐➺t ❧➔ ❝→❝ t❤➛② ❝ỉ t❤✉ë❝ ♠ỉ♥ ✣↕✐ sè✲❍➻♥❤ ❤å❝✲❚ỉ♣ỉ✱ ✈➲ sü ❣✐ó♣ ✤ï ❝❤➙♥ t❤➔♥❤ tr♦♥❣ q✉→ tr➻♥❤ tỉ✐ ❤å❝ t➟♣ t↕✐ tr÷í♥❣✳ ❈✉è✐ ❝ò♥❣✱ tỉ✐ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤ ✈➔ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ✤ë♥❣ ✈✐➯♥ tæ✐ tr♦♥❣ ❤å❝ t➟♣ ✈➔ ❝✉ë❝ sè♥❣✳ ✶ ỵ A F2 Ps F (k ) Rs M GLs Ds số tr rữớ ỗ ✷ ♣❤➛♥ tû ✣↕✐ sè ✤❛ t❤ù❝ tr➯♥ tr÷í♥❣ F2 s✐♥❤ ❜ð✐ s ♣❤➛♥ tû ▼æ✤✉♥ ❦❤æ♥❣ ê♥ ✤à♥❤ s✐♥❤ ❜ð✐ ♠ët ♣❤➛♥ tû ❜➟❝ k ❳➙② ❞ü♥❣ ❙✐♥❣❡r ❝õ❛ ♠æ✤✉♥ ❦❤æ♥❣ ê♥ ✤à♥❤ M ◆❤â♠ t✉②➳♥ t➼♥❤ tê♥❣ q✉→t ợ số tr trữớ F2 số s ử ỵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▲❮■ ▼Ð ✣❺❯ ■ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✶ ✷ ✸ ✣↕✐ sè ❙t❡❡♥r♦❞ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ▼æ✤✉♥ ❦❤æ♥❣ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❳➙② ❞ü♥❣ ❙✐♥❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✷ ✹ ✽ ✽ ✶✹ ✷✵ ■■ ❚→❝ ✤ë♥❣ ❝õ❛ ✤↕✐ sè ❙t❡❡♥r♦❞ ❧➯♥ R3F (k) ự ỵ t❤❛♠ ❦❤↔♦ ✹✷ ✶ ✷ ✶ ✷ ◗✉② ✈➲ tr÷í♥❣ ❤đ♣ s = ✈➔ ♠ỉ✤✉♥ F (k ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❚→❝ ✤ë♥❣ ❝õ❛ ✤↕✐ sè ❙t❡❡♥r♦❞ ❧➯♥ R3 F (k ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ự ỵ ❝❤➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❈❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✶✳✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ✷✺ ✷✼ ✸✷ ✸✼ ▲❮■ ▼Ð ✣❺❯ ◆❤✐➺♠ ✈ö ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ tr➻♥❤ ❜➔② ❧↕✐ ❦➳t q✉↔ ❝❤➼♥❤ ❝õ❛ ❜➔✐ ❜→♦ ❍÷♥❣✲ P♦✇❡❧❧ ❬✶✶❪✱ ✈➔ ✤÷❛ r❛ ởt ự ỵ ❜→♦ ✤â✳ ❉♦ ✤â✱ ❧í✐ ♥â✐ ✤➛✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❜→♠ s→t ♥ë✐ ❞✉♥❣ ❧í✐ ♥â✐ ✤➛✉ ❝õ❛ ❜➔✐ t tr ỗ ❝➜✉ ❍✉r❡✇✐❝③ π∗ (Ω∞ Σ∞ X ) → H∗ (Ω∞ X ; F2 ) tứ õ ỗ ê♥ ✤à♥❤ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❝â ✤✐➸♠ ❣è❝ X ✤➳♥ ỗ ổ ỏ ổ QX := Ω∞ Σ∞ X ✤÷đ❝ ❝→❝ ♥❤➔ tỉ♣ỉ ✤↕✐ sè ✤➦❝ ❜✐➺t q✉❛♥ t➙♠✳ ❈❤➥♥❣ ❤↕♥✱ ✈ỵ✐ X = S (QS ) t ỗ ❧✉➙♥ ê♥ ✤à♥❤ ❝õ❛ ♣❤ê ♠➦t ❝➛✉✳ ●✐↔ t❤✉②➳t ❝ê ✤✐➸♥ ✈➲ ❧ỵ♣ ❝➛✉ t✐➯♥ ✤♦→♥ r➡♥❣ ❝❤➾ ❝â ❝→❝ ♣❤➛♥ tû ❝õ❛ π∗ (QS ) ✈ỵ✐ ❜➜t ❜✐➳♥ ❍♦♣❢ ❜➡♥❣ ✶ ❤♦➦❝ ❜➜t ❜✐➳♥ ❑❡r✈❛✐r❡ ❜➡♥❣ ✶ ♥➡♠ tr ỗ r ỵ Q0 X t t❤ỉ♥❣ ❝õ❛ ✤✐➸♠ ❣è❝ tr♦♥❣ QX ✳ ◆❣✉②➵♥ ❍✳❱✳ ❍÷♥❣ ✤➣ ✤÷❛ r❛ ❣✐↔ t❤✉②➳t s❛✉ ✤➙②✳ ●✐↔ t❤✉②➳t ✵✳✶ tt tờ qt ợ ỵ X ởt ự õ ố õ ỗ ❝➜✉ ❍✉r❡✇✐❝③ π∗(Q0X ) → H∗(Q0X ) tr✐➺t t✐➯✉ tr➯♥ ❝→❝ ❧ỵ♣ ❝õ❛ π∗(Q0X ) ✈ỵ✐ ❧å❝ ❆❞❛♠s ❧ỵ♥ ❤ì♥ 2✳ ◆❤➟♥ ①➨t r➡♥❣✱ ✈ỵ✐ X = S ✱ ổ tỗ t ợ ữỡ ợ s ❜➡♥❣ ✵✱ ❝→❝ ♣❤➛♥ tû ❜➜t ❜✐➳♥ ❍♦♣❢ ❜➡♥❣ ✶ ❝â ❧å❝ ❆❞❛♠s ❜➡♥❣ ✶ ✈➔ ❝→❝ ♣❤➛♥ tû ❜➜t ❜✐➳♥ ❑❡r✈❛✐r❡ ❜➡♥❣ ✶ ❝â ❧å❝ ❆❞❛♠s ❜➡♥❣ ✷✳ ◆❣✉②➵♥ ❍✳❱✳ ❍÷♥❣ ✤➣ ✤÷❛ r❛ ❣✐↔ t❤✉②➳t ✈➲ ❞↕♥❣ ✤↕✐ sè ❝õ❛ ❧ỵ♣ ❝➛✉ ✈➔ ♠ët ❞↕♥❣ ②➳✉ ❝õ❛ ❣✐↔ t❤✉②➳t ✤â✳ ✣➸ ♣❤→t ❜✐➸✉ ❝→❝ ❣✐↔ t❤✉②➳t ♥➔②✱ t❛ ỗ r ỗ ❝➜✉ ▲❛♥♥❡s✲❩❛r❛t✐ ✭①❡♠ ❝❤✐ t✐➳t ❝❤➥♥❣ ❤↕♥ ❬✶✶❪✱ P❤ö ❧ö❝ ợ s ởt số ữỡ ỵ Ps ❧➔ ✤↕✐ sè ✤❛ t❤ù❝ F2 [x1 , xs ]✱ tr♦♥❣ ✤â ❜➟❝ ♠é✐ ❜✐➳♥ xi ❜➡♥❣ ✶ (1 ≤ i ≤ s)✳ ✣↕✐ sè ✤❛ t❤ù❝ Ps ✤÷đ❝ tr❛♥❣ ❜à ♠ët ❝➜✉ tró❝ ✤↕✐ sè ❦❤æ♥❣ ê♥ ✤à♥❤ ❝❤➼♥❤ t➢❝ tr➯♥ ✤↕✐ sè ❙t❡❡♥r♦❞ A ữ Ps ợ H (BZs ) ①❡♠ ♥❤÷ ♠ët ✤↕✐ sè ❦❤ỉ♥❣ ê♥ ✤à♥❤✳ ệ ệ ợ ộ số ữỡ s tỗ t↕✐ ♠ët ❤➔♠ tû ❙✐♥❣❡r Rs ✱ ❧➔ ♠ët ❤➔♠ tû ❦❤ỵ♣ ①→❝ ✤à♥❤ tr➯♥ ♣❤↕♠ trò ❝→❝ ♠ỉ✤✉♥ ❦❤ỉ♥❣ ê♥ ✤à♥❤✱ s❛♦ ❝❤♦ ✈ỵ✐ ♠é✐ ♠ỉ✤✉♥ ❦❤ỉ♥❣ ê♥ ✤à♥❤ M ✱ Rs M ❧➔ ♠ët ♠æ✤✉♥ ❝♦♥ ❝õ❛ Ps ⊗ M ✭❬✶✻❪✮✳ ▲❛♥♥❡s ✈➔ ❩❛r❛t✐ ❬✶✺❪ ✤➣ ①➙② ❞ü♥❣ ởt ỗ s F2 A Rs M TorA s (F2 , Σ M ) ✣è✐ ♥❣➝✉ t✉②➳♥ t➼♥❤ ❝õ❛ →♥❤ ①↕ ♥➔②✿ ExtsA (Σ−s M, F2 ) → (F2 A Rs M ) ữủ ỗ srt õ tữỡ ự ợ ởt ❦➳t ❝õ❛ →♥❤ ①↕ ❍✉r❡✇✐❝③ ✭✶✮ ❦❤✐ M ❧➔ ✤è✐ ỗ t ởt ổ õ ❣è❝ X ✳ ✭❈❤ù♥❣ ♠✐♥❤ ❝õ❛ ❦➳t q✉↔ ♥➔② ✤÷đ❝ tr➻♥❤ ❜➔② tr♦♥❣ ❬✶✹❪ ✈➔ ❬✸❪❀ ❝→❝ ❤➔♠ tû ❙✐♥❣❡r t ỗ t H (QX ) t H ∗ (X )✳ ✮ ◆❣✉②➵♥ ❍✳❱✳ ❍÷♥❣ ✤➣ ✤÷❛ r❛ ❞↕♥❣ ✤↕✐ sè s❛✉ ✤➙② ❝õ❛ ❣✐↔ t❤✉②➳t tê♥❣ q✉→t ✈➲ ❧ỵ♣ ❝➛✉ ✭①❡♠ ●✐↔ t❤✉②➳t ✶✳✷ ✈ỵ✐ M = F2 tr♦♥❣ ❬✺❪✱ ✈➔ ●✐↔ t❤✉②➳t ✶✳✷ ✈ỵ✐ M ❧➔ ♠ët ♠æ✤✉♥ ❦❤æ♥❣ ê♥ ✤à♥❤ ❜➜t ❦ý tr♦♥❣ ❬✶✸❪ ✮✳ ●✐↔ t❤✉②➳t ✵✳✷ ✭❉↕♥❣ ✤↕✐ sè ❝õ❛ ❣✐↔ t❤✉②➳t tờ qt ợ ỗ srt trt t t ố ữỡ ợ s > ợ ♠å✐ A ✲♠æ✤✉♥ ❦❤æ♥❣ ê♥ ✤à♥❤ M ✳ ●✐↔ t❤✉②➳t rt ự t tợ ữợ t ỹ ố ỗ sè ❙t❡❡♥r♦❞✳ ◆❣✉②➵♥ ❍✳❱✳ ❍÷♥❣ ❝ò♥❣ ❝→❝ ❝ë♥❣ sü ✤➣ ❦✐➸♠ ❝❤ù♥❣ ❣✐↔ t❤✉②➳t ♥➔② ❝❤♦ tr÷í♥❣ ❤đ♣ M = F2 ợ s = 3, ữủt tr ổ tr ỗ ❝➜✉ ▲❛♥♥❡s✲❩❛r❛t✐ tr✐➺t t✐➯✉ ✈ỵ✐ ♠å✐ s > ❦❤✐ M = F2 tr➯♥ ❝→❝ ♣❤➛♥ tû ♣❤➙♥ t➼❝❤ ✤÷đ❝ ❝õ❛ ExtsA (F2 , F2 ) ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ t tr ỗ ❝❤✉②➸♥ ❙✐♥❣❡r✳ ●✐↔ sû N ❧➔ ♠ët A ✲♠æ✤✉♥ ✳ ỗ r ố tü ♥❤✐➯♥ −s ψsN : TorA s (F2 , Σ N ) → F2 ⊗A (Ps ⊗ N ) ▲➜② N ❧➔ ♠ët ♠æ✤✉♥ ❦❤æ♥❣ ê♥ ✤à♥❤ M ✳ ❑❤✐ õ ố ủ t ỗ r ợ ỗ srt ủ t Rs M → Ps ⊗ M F2 ⊗A (Ps ⊗ M ) ❝õ❛ ♣❤➨♣ ♥❤ó♥❣ ❝❤➼♥❤ t➢❝ ✈➔ ♣❤➨♣ ❝❤✐➳✉ ①✉è♥❣ ❝→❝ ♣❤➛♥ tû A ✲❦❤ỉ♥❣ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ✭①❡♠ ❝❤➥♥❣ ❤↕♥ ▼➺♥❤ ✤➲ ❆✳✶ ❬✶✶❪✮✳ ❱ỵ✐ M = F2 = H ∗ (S )✱ ❦➳t q✉↔ ♥➔② ✤➣ ✤÷đ❝ ♥❤➟♥ r❛ ✈➔ ❝❤ù♥❣ ♠✐♥❤ ✤➛✉ t✐➯♥ tr♦♥❣ ❝æ♥❣ tr➻♥❤ ❬✺❪ ỵ ệ ệ tr tt ữợ tữỡ ữỡ ợ sỹ ỗ srt trt t tr ỗ ❙✐♥❣❡r ✈ỵ✐ ♠å✐ s > 2✳ ●✐↔ t❤✉②➳t ♥➔② ❝❤➾ ❜✐➸✉ ✤↕t ♠ët ♣❤➛♥ ❝õ❛ ●✐↔ t❤✉②➳t ✵✳✷✱ s♦♥❣ ÷✉ õ ổ ũ ố ỗ ❝õ❛ ✤↕✐ sè ❙t❡❡♥r♦❞ ✭①❡♠ t❤➯♠ ❬✶✸✱ ●✐↔ t❤✉②➳t ✶✳✻❪✮✳ ●✐↔ t❤✉②➳t ✵✳✸✳ ❈❤♦ M ❧➔ ♠ët A ✲♠æ✤✉♥ ❦❤æ♥❣ ê♥ ✤à♥❤ ✈➔ s ❧➔ ♠ët sè ♥❣✉②➯♥ ❧ỵ♥ ❤ì♥ 2✳ ❑❤✐ ✤â✱ ♠å✐ ♣❤➛♥ tû ❜➟❝ ❞÷ì♥❣ tr♦♥❣ ①➙② ❞ü♥❣ ❙✐♥❣❡r RsM ❧➔ A ✲♣❤➙♥ t➼❝❤ ✤÷đ❝ tr♦♥❣ Ps ⊗ M ✳ ●✐↔ t❤✉②➳t ✵✳✸ ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜❛♥ ✤➛✉ tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ M = F2 ✈➔ M = F2 [x1 , , xk ] ✭k ❧➔ ♠ët sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠✮ ❜ð✐ ◆❣✉②➵♥ ❍✳❱✳ ❍÷♥❣ ✈➔ ❚✳◆✳ ◆❛♠ ❧➛♥ ❧÷đt tr♦♥❣ ❬✽❪ ✈➔ ❬✾❪✳ ●✐↔ t❤✉②➳t ♥➔② ❦❤ỉ♥❣ ❝á♥ ✤ó♥❣ ✈ỵ✐ s ∈ {1, 2} ỵ s sỹ t t tứ sỹ tỗ t ợ ♠❛♥❣ t➯♥ ❜➜t ❜✐➳♥ ❍♦♣❢ ❜➡♥❣ ✶ ✈➔ ❝→❝ ❜➜t ❜✐➳♥ ❑❡r✈❛✐r❡ ❜➡♥❣ ✶✳ ●➛♥ ✤➙②✱ ●✐↔ t❤✉②➳t ✵✳✸ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥ ❜ð✐ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣ ✈➔ P ỵ ợ s > 2✱ →♥❤ ①↕ Rs M −→ F2 ⊗A (Ps ⊗ M ) ❧➔ t➛♠ t❤÷í♥❣ tr➯♥ ❝→❝ ♣❤➛♥ tû ❜➟❝ ữỡ ợ ổ ổ M ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ♥➔② ❧➔ ✤÷❛ r❛ ♠ët ❝❤ù♥❣ ỵ tr ỹ r ự õ ổ t ợ ữủ ỗ ự ✤➣ ❝â tr♦♥❣ ❬✾❪✳ ❈ư t❤➸✱ ❝❤ó♥❣ tỉ✐ ❝❤➾ r❛ r➡♥❣ ✈ỵ✐ ♠ët sè t❤❛② ✤ê✐ ✈➲ ❦ÿ t❤✉➟t ❤✐➸♥ ổ ữủ ỗ tr ❤♦➔♥ t♦➔♥ ❝â t❤➸ sû ❞ö♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✣à♥❤ ỵ ữ P t õ t q ự ỵ ✈➲ tr÷í♥❣ ❤đ♣ s = ✈➔ M ❧➔ ♠ỉ✤✉♥ ổ tỹ F (k ) ữủ ỗ ự ỵ ữ s❛✉✳ ▼é✐ ♣❤➛♥ tû R ∈ R3 F (k ) ữủ t tữỡ ự ợ ởt số ổ ❞✉② ♥❤➜t σ (R) ✭❈❤÷ì♥❣ ■■✱ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✮✱ s❛♦ ❝❤♦ i Sq2 (Hi ) + R= T, ✭ ✮ i≥0 tr♦♥❣ ✤â ❝→❝ Hi ✈➔ T ❧➔ ❝→❝ ♣❤➛♥ tû tr♦♥❣ P3 ⊗ F (k ) t❤ä❛ ♠➣♥ σ (T ) < σ (R) ✈➔ i ≤ σ (R) + 1✳ ✻ ▼Ư❈ ▲Ư❈ ❇➡♥❣ q✉② ♥↕♣ ❧ò✐ t❤❡♦ σ ✱ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ R ∈ A (P3 ⊗ F (k )), ✈ỵ✐ σ (R) = ✭❈❤÷ì♥❣ ■■■✱ ❇ê ✤➲ ✷✳✶✮✳ ◆❤÷ t❤➳✱ ♠é✐ ✤❛ t❤ù❝ R tr♦♥❣ R3 F (k ) i ✤➲✉ ✏❦❤û ✤÷đ❝✑ t tỷ tr Sq2 ợ i ổ ữủt q (R) + ú ỵ ự ♥➔② t❤✉➛♥ tó② sû ❞ư♥❣ ❝→❝ t♦→♥ tû ❙t❡❡♥r♦❞✱ ♠➔ ❦❤ỉ♥❣ ❞ò♥❣ ❝→❝ t♦→♥ tû ▼✐❧♥♦r Q0 ✈➔ Q1 ✱ ♥❤÷ tr♦♥❣ ❬✶✶❪✳ ❚r♦♥❣ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ❍÷♥❣ ✈➔ P♦✇❡❧❧ ❬✶✶❪✱ ✏❜✐➳♥ ✤✐➲✉ ❦❤✐➸♥✑ ✤÷đ❝ ✤ê✐ tø σ (R) t❤➔♥❤ f (R) t❤❡♦ ✣à♥❤ ♥❣❤➽❛ ✺✳✾ ❬✶✶❪✳ ✣✐➸♠ ❦❤→❝ ♥❤❛✉ ❣✐ú❛ ❤❛✐ ❝❤ù♥❣ ♠✐♥❤ ❧➔✱ ❦❤✐ f (R) = 0✱ ❍÷♥❣ ✈➔ P♦✇❡❧❧ ❦❤û R ❜➡♥❣ ❝→❝❤ ❞ò♥❣ ❝→❝ t♦→♥ tû Q0 ✈➔ Q1 ✱ ❝á♥ tr♦♥❣ tr÷í♥❣ ❤đ♣ σ (R) = 0✱ ❝❤ó♥❣ tỉ✐ ❞ò♥❣ ❝→❝ t♦→♥ tû Sq1 Sq2 ỗ ữỡ ữỡ ■ ❝â ♠ö❝ ✤➼❝❤ ♥➯✉ ♠ët sè ❦✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à✳ Ð ❝❤÷ì♥❣ ■■✱ s❛✉ ❦❤✐ tr➻♥❤ ❜➔② ✈✐➺❝ q✉② ỵ trữớ ủ s = M ❧➔ ♠ỉ✤✉♥ ❦❤ỉ♥❣ ê♥ ✤à♥❤ F (k )✱ ❝❤ó♥❣ tỉ✐ ✤÷❛ r❛ t→❝ ✤ë♥❣ ❝õ❛ ✤↕✐ sè ❙t❡❡♥r♦❞ A ❧➯♥ R3 F (k )✳ ❈✉è✐ ❝ò♥❣✱ ❝❤÷ì♥❣ ■■■ ✤÷đ❝ ự ỵ ũ ợ ❝→❝ ❜ê ✤➲ ❧✐➯♥ q✉❛♥✳ ✼ ❈❤÷ì♥❣ ■ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ❚r♦♥❣ t♦➔♥ ❜ë ❧✉➟♥ ✈➠♥ ♥➔②✱ t❛ ①➨t số trữớ F2 ỗ tỷ ❧➔ ✵ ✈➔ ✶✳ ◆➳✉ ❦❤æ♥❣ ❣✐↔✐ t❤➼❝❤ ❣➻ t❤➯♠ t ỵ ũ t tỡ tr tr÷í♥❣ F2 ✳ ✶ ✣↕✐ sè ❙t❡❡♥r♦❞ ▼ư❝ ✤➼❝❤ ❝õ❛ ♠ö❝ ♥➔② ❧➔ ♥❤➢❝ ❧↕✐ ✈➲ ✤↕✐ sè ❙t❡❡♥r♦❞✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❞ò♥❣ ❝❤♦ ♠ư❝ ♥➔② ❧➔ ❬✶✾❪✳ ❱ỵ✐ ♠å✐ ❝➦♣ sè ♥❣✉②➯♥ i, q ≥ ✈➔ ✈ỵ✐ ổ tổổ X tỗ t t ởt F2 ỗ Sqi : H q (X ) −→ H q+i (X ) t❤ä❛ ♠➣♥ ✶✳ Sqi ❧➔ ♠ët ♣❤➨♣ ❜✐➳♥ ✤ê✐ tü ♥❤✐➯♥ ❣✐ú❛ ❝→❝ ❤➔♠ tû H q (.) ✈➔ H q+i (.)❀ ✷✳ Sq0 = id❀ ✸✳ ◆➳✉ deg(x) = q t❤➻ Sqq (x) = x2 ❀ ✹✳ ◆➳✉ i > deg(x) t❤➻ Sqi (x) = 0❀ ✺✳ ❈æ♥❣ t❤ù❝ ❈❛rt❛♥ k k Sqi (x) Sqk−i (y ); Sq (xy ) = i=0 ✻✳ Sq1 ỗ st tữỡ ự ợ ợ ♥❤â♠ ❤➺ sè µ λ −→ Z2 −→ Z4 −→ Z2 −→ 0, tr♦♥❣ ✤â Zn = Z/n✱ µ(x) = 2x ✈➔ λ(y ) = y (mod 2)❀ ✽ ❈❤÷ì♥❣ ■■✳ ❚→❝ ✤ë♥❣ ❝õ❛ ✤↕✐ sè ❙t❡❡♥r♦❞ ❧➯♥ R3 F (k ) tr♦♥❣ ✤â h = h(K ) ✈➔ Kp ❧➔ ♠ët tr♦♥❣ ❝→❝ ❜➜t ❜✐➳♥ Q0 , Q1 , Q2 , W1 , , Wk ✈ỵ✐ ≤ p ≤ h✳ ❙û ❞ư♥❣ ❝ỉ♥❣ t❤ù❝ ❈❛rt❛♥ ✈➔ ▼➺♥❤ ✤➲ ✷✳✶ t❛ ❝â Sqi (K ) = Sqt1 (K1 ) · · · Sqth (Kh ) t1 +···+th =i S+ = h(S )=h(K ) T h(T )>h(K ) ❱➻ deg Q2 = ❧➔ ♥❤ä ♥❤➜t tr♦♥❣ sè ❜➟❝ ❝õ❛ ❝→❝ ✤❛ t❤ù❝ Q0 , Q1 , Q2 , W1 , , Wk ♥➯♥ t❛ ❝â i h(K ) < h(T ) ≤ h(K ) + , ✈ỵ✐ ♠å✐ T tr♦♥❣ tê♥❣✳ t ởt tỷ tũ ỵ S = Sqt1 (K1 ) · · · Sqth (Kh ) tr♦♥❣ tê♥❣✳ ❱➻ h(S ) = h(K ) ♥➯♥ t❛ ❝â = ✈ỵ✐ ♠å✐ p ♠➔ Kp ❧➔ ♠ët tr♦♥❣ ❝→❝ ❜➜t ❜✐➳♥ Q0 , W1 , , Wk ✳ ●✐↔ sû r➡♥❣ i2 (S ) ≥ i2 (K )✳ ❑❤✐ ✤â i2 (S ) = i2 (K ) ✈➻ h(S ) = h(K )✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✱ jp = ✈ỵ✐ ♠å✐ p ♠➔ Kp = Q2 ✳ ◆❤÷ t❤➳ jp ❝❤➾ ❝â t❤➸ ❦❤→❝ ❦❤ỉ♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ Kp = Q1 ✳ ❍ì♥ ♥ú❛✱ ✈➻ h(S ) = h(K ) ✈➔ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✱ ♥➳✉ jp = t❤➻ jp = 1✳ ❉♦ ✤â✱ i = t1 + · · · + th ≤ i1 (K ) ✣✐➲✉ ♥➔② ♠➙✉ t❤✉➝♥ ✈ỵ✐ ❣✐↔ t❤✐➳t i < i1 (R) (✈➻ i1 (R) = i1 (K )✮✳ ❈✉è✐ ❝ò♥❣✱ tø ❤➺ t❤ù❝ Sqi R = j1 s②♠ Sqi Qi00 Qi11 Qi22 W12 · · · Wk2 jk t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✹ ✭❬✾✱ ♥❣✉②➯♥ ❞÷ì♥❣ ✳ ❇ê ✤➲ ✸✳✹❪✮ ●✐↔ sû R ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ (R) t❤ä❛ ♠➣♥ i2(R) ≡ 2n − (mod 2n) ✈➔ 2hn− = R = Sq2 n+1 i (R)−2n−1 (RQ22 )+ ✈➔ n ❧➔ ♠ët sè ❑❤✐ ✤â S, tr♦♥❣ ✤â R := R/Qi2 (R) ✈➔ ♠é✐ S ❧➔ ♠ët R3F (k)✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ σ(S ) < n✳ ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â i (R ) h(R) = h(RQ22 ) = h(R) + i2 (R) ≡ h(R) + 2n − (mod 2n ) ✷✾ ❈❤÷ì♥❣ ■■✳ ❚→❝ ✤ë♥❣ ❝õ❛ ✤↕✐ sè ❙t❡❡♥r♦❞ ❧➯♥ R3 F (k ) ❉♦ ✤â h(RQ22 n−1 −1 ) = 2n−1 h(R) + 2n−1 − n+1 Sq2 = 2n−1 ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✸ ✭❛✮ ❝❤♦ RQ22 (RQ22 n−1 n−1 −1 −1 h(R) − 2n−1 = 2n−1 ✈➔ i = 2n−1 t❛ t❤✉ ✤÷đ❝ ) = RQ22 n −1 S, + tr♦♥❣ ✤â ♠é✐ S ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ n−1 i2 (S ) < i2 (RQ22 −1 ) + 2n−1 = 2n − ❇➜t ✤➥♥❣ t❤ù❝ ♥➔② ❦➨♦ t❤❡♦ σ (S ) < n✳ ✣➦t a := i2 (R) − (2n − 1) ≡ (mod 2n )✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ❈❛rt❛♥ ✈➔ ▼➺♥❤ ✤➲ ✷✳✶ t❛ ❝â Sq2 n+1 i (R)−2n−1 RQ22 = Sq2 n+1 (RQ22 n−1 −1 Qa2 ) = Sq2 n+1 (RQ22 n−1 −1 )Qa2 + RQ22 = RQ22 n −1 −1 n−1 S Qa2 + RQ22 + n−1 S Qa2 + RQ22 =R+ n−1 −1 −1 n+1 Sq2 n+1 Sq2 Sq2 (Qa2 ) n+1 (Qa2 ) (Qa2 ), tr♦♥❣ ✤â ♠é✐ ❤↕♥❣ tû S Qa2 t❤ä❛ ♠➣♥ σ (S Qa2 ) < n ❜ð✐ ✈➻ σ (S ) < n ✈➔ a ≡ n+1 (mod 2n )✳ ▼➦t ❦❤→❝✱ →♣ ❞ö♥❣ ▼➺♥❤ ✤➲ ✷✳✶✱ ♥➳✉ Sq2 (Qa2 ) = t❤➻ ♥â ❦❤æ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ Q2 ✳ ❉♦ ✤â n−1 σ RQ22 −1 Sq2 n+1 (Qa2 ) = σ (Q22 n−1 −1 ) = n − < n ◆â✐ tâ♠ ❧↕✐✱ t❛ ❝â t❤➸ ✈✐➳t R = Sq2 n+1 i (R)−2n−1 (RQ22 S, )+ tr♦♥❣ ✤â ♠é✐ ❤↕♥❣ tû S t❤ä❛ ♠➣♥ σ (S ) < n✳ ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✷✳✺ ✭❬✾✱ ❇ê ✤➲ ✸✳✺❪✮✳ ●✐↔ sû R ❧➔ ♠ët R3F (k)✲✤ì♥ t❤ù❝ ❦❤ỉ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ Q2✱ ❝á♥ n ✈➔ i ❧➔ ❝→❝ sè ♥❣✉②➯♥ ❞÷ì♥❣ t❤ä❛ ♠➣♥ h(R) ≡ (mod 2n ), i1 (R) ≤ 2n − 1, 2n ≤ i ≤ 2n+1 ❑❤✐ ✤â Sqi (RQ22 n −1 )= S+ T, tr♦♥❣ ✤â ♠é✐ ❤↕♥❣ tû S ❧➔ ♠ët R3F (k)✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ σ(S ) < n✱ ❝á♥ ♠é✐ ❤↕♥❣ tû T ❧➔ ♠ët R3F (k)✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ i2(T ) ≡ 2n − (mod 2n) ✈➔ 2h(T ) = n−1 ✸✵ ❈❤÷ì♥❣ ■■✳ ❚→❝ ✤ë♥❣ ❝õ❛ ✤↕✐ sè ❙t❡❡♥r♦❞ ❧➯♥ R3 F (k ) ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ ❣✐↔ t❤✐➳t t❛ ❝â i ≥ 2n > 2n − ≥ i1 (R) = i1 (RQ22 ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✸ ✭❜✮ ❝❤♦ RQ22 Sqi (RQ22 n n −1 −1 n −1 ) ✈➔ i t❛ ❝â )= S+ T, tr♦♥❣ ✤â ♠é✐ S ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ i2 (S ) < i2 (RQ22 ❝á♥ ♠é✐ T ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ n h(RQ22 −1 ) < h(T ) ≤ h(RQ22 n −1 n −1 ) = 2n − 1, i )+ ❱ỵ✐ ♠é✐ S tr♦♥❣ tê♥❣✱ ✈➻ i2 (S ) < 2n − ♥➯♥ σ (S ) < n✳ ❈á♥ ✈ỵ✐ ♠é✐ T tr♦♥❣ tê♥❣ t❛ ❝â n h(RQ22 −1 ) = h(R) + 2n − < h(T ) ≤ h(R) + 2n − + n i n−1 ≤ h(R) + + (2 − 1) ❙✉② r❛ h(R) + 2n ≤ h(T ) ≤ h(R) + 2n + (2n−1 − 1)✳ ❑➳t ❤đ♣ ✤✐➲✉ ♥➔② ✈ỵ✐ ❣✐↔ t❤✐➳t h(R) ≡ (mod 2n )✱ t❛ t❤✉ ✤÷đ❝ h(T ) 2n−1 = ❈✉è✐ ❝ò♥❣✱ ❣✐↔ sû i2 (T ) = 2n − + b✱ tr♦♥❣ ✤â b ❧➔ ♠ët sè ♥❣✉②➯♥✳ ◆➳✉ b ≡ (mod 2n ) t❤➻ i2 (T ) ≡ 2n − (mod 2n )✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ b ≡ (mod 2n ) t❤➻ σ (T ) < n✳ ◆❤÷ t❤➳ T ✤÷đ❝ ①❡♠ ♥❤÷ ♠ët ♣❤➛♥ tû tr♦♥❣ tê♥❣ S✳ ✸✶ ❈❤÷ì♥❣ ■■■ ❈❤ù♥❣ ♠✐♥❤ ỵ ự ỵ ữợ tữỡ ữỡ ợ tr ❬✽❪✳ Ð ✤➙②✱ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët ❝❤ù♥❣ ♠✐♥❤ ❦❤→❝ ✈ỵ✐ ❝❤ù♥❣ ♠✐♥❤ ✤➣ ❝â tr♦♥❣ ❬✽❪✳ ❇ê ✤➲ ợ số ữỡ s H (Ps ; Sq1 ) = ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝â ♣❤➙♥ t➼❝❤ ❝→❝ Sq1 ✲♠♦❞✉❧❡ Ps = F2 ⊕ P s , ð ✤➙② F2 ❧➔ Sq1 ✲♠♦❞✉❧❡ t➛♠ t❤÷í♥❣✳ ❉♦ ✤â H∗ (Ps ; Sq1 ) = H∗ (F2 ; Sq1 ) ⊕ H∗ (P s ; Sq1 ) ❱➻ t❤➳✱ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲✱ t❛ ❝❤➾ ❝➛♥ ❝❤➾ r❛ r➡♥❣ H∗ (Ps ; Sq1 ) = H∗ (F2 ; Sq1 ) = F2 ✳ ◆❤➟♥ ①➨t r➡♥❣✱ t❤❡♦ ❝æ♥❣ t❤ù❝ ❈❛rt❛♥✱ t❛ ❝â ✤➥♥❣ ❝➜✉ ❝→❝ Sq1 ✲♠♦❞✉❧❡✿ Ps = P1 ⊗ · · · ⊗ P1 (s ❧➛♥) ✭◆❤➟♥ ①➨t r➡♥❣ ✤✐➲✉ ♥➔② ❦❤ỉ♥❣ ✤ó♥❣ ♥➳✉ t❤❛② Ps ❜ð✐ P s ✈➔ P1 ❜ð✐ P ✳ ✣â ỵ t tứ P s s Ps ✳✮ ❚ø ✤â✱ t❤❡♦ ❝ỉ♥❣ t❤ù❝ ❑✉♥♥❡t❤ →♣ ❞ư♥❣ ❝❤♦ ❤➺ sè tr♦♥❣ tr÷í♥❣ F2 ✱ t❛ ❝â H∗ (Ps ; Sq ) = H∗ (P1 ; Sq1 ) ⊗ · · · ⊗ H∗ (P1 ; Sq1 ), (s ❧➛♥) ❉♦ ✤â✱ sü ❦✐➺♥ H∗ (Ps ; Sq1 ) = F2 ✤÷đ❝ q✉② ✈➲ ✤➥♥❣ t❤ù❝ ✤â ợ s = ứ ổ tự rt ữớ ✤➲✉ ❜✐➳t r➡♥❣ k Sq xk = ✸✷ xk+1 ữỡ ự ỵ ❞➔♥❣ s✉② r❛ tr♦♥❣ P1 ✿ Ker(Sq ) = Span{xk |k ❝❤➤♥, k ≥ 0}, Im(Sq ) = Span{xk |k ❝❤➤♥ > 0} ❑➳t q✉↔ ❧➔✱ H∗ (P1 ; Sq ) = Span{xk |k ❝❤➤♥, k ≥ 0} Span{xk |k ❝❤➤♥ > 0} = F2 ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ tr➯♥ ❝â t❤➸ ①❡♠ ởt trữớ ủ r ữợ ✈ỵ✐ M = F2 ✳ ❇ê ✤➲ ✶✳✷✳ ❱ỵ✐ ♠å✐ số ữỡ s ợ ổ ổ ✤à♥❤ M t❛ ❝â H∗ (P s ⊗ M ; Sq1 ) = ❈❤ù♥❣ ♠✐♥❤✳ ❚❤❡♦ q✉❛♥ ❤➺ ❆❞❡♠✱ Sq1 ❧➔ ♠ët ✈✐ ♣❤➙♥ tr➯♥ A✲♠♦❞✉❧❡ ❦❤æ♥❣ ê♥ ✤à♥❤ ❜➜t ❦ý✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ❈❛rt❛♥✱ P s ⊗ M ✈ỵ✐ ✈✐ ♣❤➙♥ Sq1 ❧➔ t➼❝❤ t❡♥s♦r ❝õ❛ ✷ ♠♦❞✉❧❡ P ✈➔ M ❝ò♥❣ ✈ỵ✐ ✈✐ ♣❤➙♥ Sq1 ✳ ỵ Sq0 = id tr P s M ✳✮ ❉♦ ✤â✱ H∗ (P s ⊗ M ; Sq1 ) = H∗ (P s ; Sq1 ) ⊗ H∗ (M, Sq1 ) ❚❤❡♦ ❇ê ✤➲ ✶✳✶ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✶✳✸✳ ●✐↔ sû R ❧➔ t➼❝❤ ❝õ❛ ❝→❝ ♣❤➙♥ tû ♣❤➙♥ ❜✐➺t tr♦♥❣ t➟♣ ❤ñ♣ {Q0, Q1, Q2}✳ ❑❤✐ ✤â ✭❛✮ R ∈ Sq1 P3 + Sq2 P3❀ ✭❜✮ R W12 · · · Wk2 j1 jk sym ❈❤ù♥❣ ♠✐♥❤✳ ∈ Sq1 (P3 ⊗ F (k )) + Sq2 (P3 ⊗ F (k ))✳ ✭❛✮ ❚❛ ❝â t❤❡♦ ❬▼➺♥❤ ✤➲ ✷✳✶✭❛✮✱❈❤÷ì♥❣ ■■❪ Q0 = Sq1 Q1 , Q1 = Sq2 Q2 , Q0 Q1 = Sq2 (Q0 Q2 ), Q0 Q2 = Sq1 (Sq4 (Q1 )) ❚❤❡♦ ❬✽✱ ❇ê ✤➲ ❇❪ Q1 Q2 = Sq1 (A) + Sq2 (B ) ✈ỵ✐ A, B ∈ P3 ❱➻ Sq1 Q2 = ♥➯♥ t❤❡♦ ❬✽✱ ❇ê ✤➲ ✷✳✺❪ t❛ ❝â Q2 = Sq1 C ✈ỵ✐ C ∈ P3 ❈✉è✐ ❝ò♥❣✱ Q0 Q1 Q2 = Q0 Sq1 (A) + Sq2 (B ) = Sq1 (Q0 A) + Sq2 (Q0 B ) ✸✸ ❈❤÷ì♥❣ ■■■✳ ❈❤ù♥❣ ỵ ữ ỵ r Sq1 j1 jk j1 jk W12 · · · Wk2 =0 sym ✈➔ Sq2 W12 · · · Wk2 = sym ❉♦ ✤â✱ →♣ ❞ö♥❣ ♣❤➛♥ ✭❛✮ t❛ ❝â ✤✐➲✉ ♣❤↔✐ ự ữợ õ ởt trỏ tt tr ự ỵ ự ❝õ❛ ❇ê ✤➲ ♥➔② ✤÷đ❝ ❞➔♥❤ r✐➯♥❣ ð ♠ư❝ s❛✉✳ ❇ê ✤➲ ✶✳✹ ✭❬✾✱ ✳ ●✐↔ sû R ❧➔ ♠ët R3F (k)✲✤ì♥ t❤ù❝ ✱ u = ❧➔ ♠ët ❇ê tr P3 n tỷ tũ ỵ ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣✳ ✭❛✮ ◆➳✉ σ(R) < n t❤➻ Ru2 ∈ A (P3 ⊗ F (k))✳ ✭❜✮ ◆➳✉ i2(R) ≡ 2n − (mod 2n) ✈➔ 2h(R) = t❤➻ Ru2 ∈ A (P3 ⊗ F (k))✳ ✭❝✮ ◆➳✉ i2(R) = 2n − ≥ i1(R), h(R) ≡ 2n − (mod 2n) ✈➔ u ∈ Sq1 P3 + Sq2 P3 t❤➻ Ru2 ∈ A (P3 ⊗ F (k)) ự ỵ sỷ R = Qi0 Qi1 Qi2 W12 · · · Wk2 ❧➔ ♠ët n n n−1 n j1 jk R3 F (k )✲✤ì♥ t❤ù❝✳ ❚❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ R ∈ A (P3 ⊗ F (k )) ❚❛ ❝❤✐❛ t❤➔♥❤ ❝→❝ tr÷í♥❣ ❤đ♣ s❛✉✳ ✶ Qi00 Qi11 Qi22 = 1✳ ❑❤✐ ✤â j1 jk sym tr♦♥❣ ✤â f = j1 W12 · · · Wk2 = f + ⊗ ( R= jk u21 · · · u2k )8 , sym ⊗ xi ∈ P3 ⊗ F (k ) ✈➔ ❝→❝ ✤➲✉ ❝â ❜➟❝ ❞÷ì♥❣✳ i ❚❛ ❜✐➳t r➡♥❣ Sq1 R = ✈➔ Sq1 ⊗ ( sym j1 Sq1 f = ❚❤❡♦ ❇ê ✤➲ ✶✳✷ t❛ ❝â f ∈ Sq1 (P3 ⊗ F (k )) ✸✹ j u21 · · · uk2 k )8 = 0✳ õ ữỡ ự ỵ t ❦❤→❝✱ ♥➳✉ ✤➦t d = 2j1 + · · · + 2jk t❤➻ j1 jk j1 u21 · · · u2k )8 = Sq4d ⊗ ( 1⊗( sym jk u21 · · · uk2 )4 ∈ A (P3 ⊗ F (k )) sym ❍➺ q✉↔ ❧➔ j1 jk W12 · · · Wk2 ∈ A (P3 ⊗ F (k )) R= sym ✷ Qi00 Qi11 Qi22 = 1✳ ✣➦t (R) = n tự tỗ t t ♠ët sè ♥❣✉②➯♥ ❦❤æ♥❣ ➙♠ k s❛♦ ❝❤♦ i2 (R) = 2n − + k 2n+1 ✭✷✳✶✮ ❳➨t trữớ ủ k > t R ữợ n+1 R = R(Qk2 )2 , tr♦♥❣ ✤â R ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ σ (R) = n < n + 1✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❛✮ ❝❤♦ ❜ë ❜❛ (R, Qk2 , n + 1) t❛ ❝â R ∈ A (P3 ⊗ F (k )) ✭✷✳✷✮ ❳➨t tr÷í♥❣ ❤đ♣ k = 0✳ ❚❛ ①➨t trữớ ủ ữợ i0 (R) 2n+1 ❤♦➦❝ i1 (R) ≥ 2n+1 ❚❛ ✈✐➳t R ữợ n+1 R = RA2 , tr õ A = Q0 ❤♦➦❝ A = Q1 tò② t❤❡♦ i0 (R) ≥ 2n+1 ❤❛② i1 (R) ≥ 2n+1 ✱ ❝á♥ R ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ σ (R) = n < n + ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❛✮ ❝❤♦ ❜ë ❜❛ (R, A, n + 1) t❛ ❝â R ∈ A (P3 ⊗ F (k )) ✭✷✳✷✳✷✮ i0 (R) ≤ 2n+1 − ✈➔ i1 (R) ≤ 2n+1 − 1❀ i0 (R) ≥ 2n ✳ ❚❛ ❝❤✐❛ t❤➔♥❤ ❜❛ tr÷í♥❣ ❤đ♣ ❝♦♥✳ ❤♦➦❝ i1(R) ≥ 2n • n = ✳ ⑩♣ ❞ư♥❣ ❇ê ✤➲ ✶✳✸ ✭❜✮ t❛ ❝â R ∈ A (P3 ⊗ F (k )) n > t tỗ t↕✐ m ✈ỵ✐ < m ≤ n ✈➔ m R = RA2 , ✸✺ h(R) 2m−1 = 0✳ ❈❤÷ì♥❣ ự ỵ tr õ A = Q0 ❤♦➦❝ A = Q1 tò② t❤❡♦ i0 (R) ≥ 2n ❤❛② i1 (R) ≥ 2n ✱ ❝á♥ R ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ i2 (R) ≡ 2m − (R) )−2m (R ) (mod 2m ) ✈➔ 2hm− = h(2Rm− = 2hm− = 0✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ 1 ✶✳✹ ✭❜✮✱ t❛ ❝â R ∈ A (P3 ⊗ F (k )) • n>0 ❚❛ ❝â ✈➔ h(R) 2m−1 =1 ✈ỵ✐ ♠å✐ m t❤ä❛ ♠➣♥ < m ≤ n✳ h(R) ≡ 2n − (mod 2n ) t R ữợ n R = Rv , tr♦♥❣ ✤â v ❧➔ t➼❝❤ ♣❤➙♥ ❜✐➺t ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ t➟♣ ❤ñ♣ {Q0 , Q1 }✱ ✈➔ R ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ i0 (R, i1 (R), i2 (R) 2n ú ỵ r i2 (R) = i2 (R) − 2n i2 (v ) = 2n − ≥ i1 (R), h(R) = h(R) − 2n h(v ) ≡ 2n − (mod 2n ) ❚❤❡♦ ❇ê ✤➲ ✶✳✸ ✭❛✮ t❛ ❝â v ∈ Sq1 P3 + Sq2 P3 ✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❝✮ ❝❤♦ ❜ë ❜❛ (R, v, n) t❛ ❝â R ∈ A (P3 ⊗ F (k )) ✭✷✳✷✳✸✮ i0 (R) ≤ 2n − ✈➔ i1 (R) ≤ 2n − 1✳ ◆➳✉ n = t❤➻ t❛ q✉❛② ❧↕✐ ✶ ✳ ❉♦ ✤â s❛✉ ✤➙② t❛ ❝❤➾ ①➨t n ≥ 1✳ • ♥❂✶✳ ❑❤✐ ✤â✱ →♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❜✮ t❛ ❝â R ∈ A (P3 ⊗ F (k )) (R) n tỗ t m ợ < m < n ✈➔ 2hm− = 0✳ m m ❍✐➸♥ ♥❤✐➯♥ ❧➔ i2 (R) ≡ − (mod )✳ ❚❛ ✈✐➳t m R = RQ22 , tr♦♥❣ ✤â R ❧➔ ♠ët RF3 (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ h(R) 2m−1 h(R) − 2m = 2m−1 = h(R) 2m−1 = 0, ✈➔ i2 (R) = 2n − − 2m ≡ 2m − (mod 2m ) ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❜✮ ❝❤♦ ❜ë ❜❛ (R, Q2 , m) t❛ ❝â R ∈ A (P3 ⊗ F (k )) ữỡ ự ỵ ❝❤➼♥❤ • n ≥ ✈➔ ❑❤✐ ➜②✱ h(R) 2m−1 =1 ✈ỵ✐ ♠å✐ m t❤ä❛ ♠➣♥ < m < n✳ h(R) ≡ 2n−1 − (mod 2n−1 ) ❚❛ t R ữợ R = Ru2 n1 , tr ✤â u ❧➔ t➼❝❤ ♣❤➙♥ ❜✐➺t ❝õ❛ ❝→❝ ♣❤➛♥ tû tr♦♥❣ t➟♣ ❤ñ♣ {Q0 , Q1 , Q2 }✱ ❝á♥ R ❧➔ ♠ët R3 F (k )−✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ i2 (R) = i2 (R) − 2n−1 i2 (u) = 2n − − 2n−1 = 2n−1 − ≥ i1 (R), h(R) = h(R) − 2n−1 h(u) ≡ 2n−1 − (mod 2n−1 ) ❚❤❡♦ ❇ê ✤➲ ✶✳✸ ✭❛✮✱ t❛ ❝â u ∈ Sq1 P3 + Sq2 P3 ✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❝✮ ❝❤♦ ❜ë ❜❛ (R, u, n − 1) t❛ ❝â R ∈ A (P3 F (k )) ỵ ữủ ự ❤♦➔♥ t♦➔♥✳ ✷ ❈❤ù♥❣ ♠✐♥❤ ❇ê ✤➲ ✶✳✹ ❚r♦♥❣ ♣❤➛♥ ♥➔② t❛ tr➻♥❤ ❜➔② ♠ët ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❇ê ✤➲ ✶✳✹✳ ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣✳ ❚r÷í♥❣ ủ t q ữợ ởt rë♥❣ ❝õ❛ ❇ê ✤➲ ✺✳✶ tr♦♥❣ ❬✾❪✳ ❇ê ✤➲ ✷✳✶✳ ●✐↔ sû R ❧➔ ♠ët R3F (k)✲✤ì♥ t❤ù❝ ✈ỵ✐ σ(R) = 0✱ ✈➔ u = ❧➔ ♠ët ♣❤➛♥ tû tũ ỵ tr P3 õ Ru2 Sq1 (P3 ⊗ F (k )) + Sq2 (P3 ⊗ F (k )) ❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤✐❛ t❤➔♥❤ ❤❛✐ tr÷í♥❣ ❤đ♣✳ ✶ i1 (R) (mod 2) t R ữợ ❞↕♥❣ j1 R = Qi00 Qi11 Qi22 W12 · · · Wk2 jk sym ❙✉② r❛ j1 Ru2 = Qi00 Qi11 Qi22 u2 W12 · · · Wk2 sym jk ữỡ ự ỵ ❱➻ Sq1 (Qi00 Qi11 Qi22 u2 ) = ♥➯♥ t❤❡♦ ❬✽✱ ❇ê ✤➲ ✷✳✺❪ t❛ ❝â Qi00 Qi11 Qi22 u2 = Sq1 (A) ợ A P3 ữ ✈➟② j1 Ru2 = Sq1 (A) jk W12 · · · Wk2 j1 = Sq1 A jk W12 · · · Wk2 sym sym ✷ i1 (R) ≡ (mod 2)✳ ✣➦t S = R/Q1 Qi22 ✱ ✈ỵ✐ i2 = i2 (R)✳ ❱➻ σ (R) = ♥➯♥ i2 (R) ❧➔ ❝❤➤♥✳ ❉♦ ✤â t❛ ❝â Ru2 = SQi22 u2 Q1 = SQi22 u2 Sq2 (Q2 ) = Sq2 (SQi22 +1 u2 ) + Sq2 (SQi22 u2 )Q2 = Sq2 (SQi22 +1 u2 ) + Sq2 (Su2 )Qi22 +1 ❚❛ ❝â Sq2 (Su2 ) = Sq2 (S )u2 + S (Sq1 u)2 ú ỵ r i1 (S ) = i1 (Sq2 S ) ≡ (mod 2) ◆➯♥ t❤❡♦ ✶ t❛ ❝â Sq2 (Su2 ) = Sq1 v ợ v P3 F (k ) ữ ✈➟② Sq2 (Su2 )Qi22 +1 = Sq1 (v )Qi22 +1 = Sq1 (vQi22 +1 ) ❇ê ✤➲ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇➙② ❣✐í✱ t❛ tr➻♥❤ ❜➔② ❝❤ù♥❣ ♠✐♥❤ ❝❤♦ ❇ê ✤➲ ự ữủ ữợ ❇ê ✤➲ 1.4 ✭❛✮ ✈➔ ❇ê ✤➲ 1.4 ✭❜✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ n ≤ N t❤➻ ❇ê ✤➲ 1.4 ✭❝✮ ❝ô♥❣ ✈➟②✳ ●✐↔ sû u = Sq1 v1 + Sq2 v2 ✈ỵ✐ v1 , v2 ∈ P3 ✳ ❚❛ ❝â n Ru2 = R(Sq1 v1 + Sq2 v2 )2 n n = R(Sq1 v1 )2 + R(Sq2 v2 )2 n n n n n n+1 n = Sq2 (Rv12 ) + Sq2 (R)v12 = Sq2 (Rv12 ) + Sq2 n + Sq2 n n+1 n n (Rv22 ) + Sq2 (R)(Sq1 v2 )2 ✸✽ n n n+1 (Rv22 ) + Sq2 (R)(Sq1 v2 )2 + Sq2 n n n + Sq2 (R)v12 + Sq2 n+1 (R)v22 n n (R)v22 ữỡ ự ỵ ú ỵ r Sq2 (R)(Sq1 v2 )2 = Sq2 n n n R(Sq1 v2 )2 n = Sq2 n R(Sq1 v2 )2 + R(Sq1 Sq1 v2 )2 n n ❉♦ ✤â✱ n n n Ru2 + Sq2 (R)v12 + Sq2 n+1 n (R)v22 ∈ A (P3 ⊗ F (k )) n ✣➦t R := R/Q22 −1 ✳ ❍✐➸♥ ♥❤✐➯♥ R ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ ❦❤ỉ♥❣ ❝❤✐❛ ❤➳t ❝❤♦ Q2 ✈ỵ✐ h(R) = h(R) − (2n − 1) ≡ (mod 2n ) ✈➔ i1 (R) = i1 (R) ≤ 2n − 1✳ ⑩♣ ❞ư♥❣ ❬❇ê ✤➲ ✷✳✸✱ ❈❤÷ì♥❣ ■■❪ t❛ t❤✉ ✤÷đ❝ n n Sq2 (R) = Sq2 (RQ22 n+1 Sq2 (R) = Sq2 n+1 n −1 (RQ22 n S1 + )= −1 T1 , S2 + )= T2 , tr♦♥❣ ✤â ♠é✐ ❤↕♥❣ tû S1 ❤❛② S2 ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ σ (S1 ) < n ✈➔ σ (S2 ) < n✱ ❝á♥ ♠é✐ ❤↕♥❣ tû T1 ❤❛② T2 ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ ✈ỵ✐ i2 (T1 ) ≡ T1 ) T2 ) i2 (T2 ) ≡ 2n − (mod 2n ) ✈➔ h2(n− = h2(n− = 0✳ 1 ❉♦ ✤â n Ru2 + n Sv12 + n n S2 v22 + n T1 v12 + T2 v22 ∈ A (P3 ⊗ F (k )) ❚❤❡♦ ❣✐û t❤✐➳t✱ ❇ê ✤➲ ✶✳✹ ✭❛✮ ✤ó♥❣ ✈ỵ✐ ❝→❝ ❜ë ❜❛ (S1 , v1 , n) ✈➔ (S2 , v2 , n)❀ ♥❤÷ n n t❤➳ S1 v12 ✈➔ S2 v22 ✤➲✉ t❤✉ë❝ A (P3 ⊗ F (k ))✳ ❚÷ì♥❣ tü✱ ❇ê ✤➲ ✶✳✹ ✭❜✮ ✤ó♥❣ ❝❤♦ ❝→❝ ❜ë ❜❛ (T1 , v1 , n) ✈➔ (T2 , v2 , n) ❝❤♦ ♥➯♥ n n T1 v12 ✈➔ T2 v22 ✤➲✉ t❤✉ë❝ A (P3 ⊗ F (k ))✳ ◆â✐ tâ♠ ❧↕✐✱ n Ru2 A (P3 F (k )) ữợ ữủ ❝❤ù♥❣ ♠✐♥❤✳ ✷ ◆➳✉ ❇ê ✤➲ 1.4 ✭❛✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ n ≤ N t❤➻ ❇ê ✤➲ 1.4 ✭❜✮ ❝ô♥❣ ✈➟②✳ ⑩♣ ❞ư♥❣ ❬❇ê ✤➲ ✷✳✷✱ ❈❤÷ì♥❣ ■■❪ t❛ ❝â R = Sq2 i (R ) tr♦♥❣ ✤â R := R/Q22 σ (S ) < n✳ ❉♦ ✤â n n+1 i (R)−2n−1 RQ22 + S, ✈➔ ♠é✐ S tr♦♥❣ tê♥❣ ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ t❤ä❛ ♠➣♥ n+1 Ru2 = Sq2 i (R)−2n−1 RQ22 n u2 + n Su2 n ❱➻ σ (S ) < n ♥➯♥ →♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❛✮ ❝❤♦ ❜ë ❜❛ (S, u, n) t❛ ❝â Su2 ∈ A (P3 ⊗ F (k )) ợ S tr tờ ữỡ ự ỵ i (R)2n1 t R1 := RQ22 Sq2 n+1 n ✳ ❚❤❡♦ ❝æ♥❣ t❤ù❝ ❈❛rt❛♥ t❛ ❝â (R1 )u2 = Sq2 n+1 (R1 u2 ) + Sq2 (R1 )(Sq1 u)2 + R1 (Sq2 u)2 = Sq2 n+1 (R1 u2 ) + Sq2 = Sq2 n+1 (R1 u2 ) + Sq2 ❱➻ ✈➟② Sq2 n+1 n n n n n R1 (Sq1 u)2 n + R1 (Sq1 Sq1 u)2 + R1 (Sq2 u)2 n n R1 (Sq1 u)2 n + R1 (Sq2 u)2 n n n n n n (R1 u2 ) + R1 (Sq2 u)2 ∈ A (P3 ⊗ F (k )) i (R)−2n−1 ❚❛ ❝â σ (R1 ) = σ (Q22 ) = n − < n✳ ❉♦ ✤â →♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❛✮ t❛ ❝â n R1 (Sq2 u)2 ∈ A (P3 ⊗ F (k )) ❚â♠ ❧↕✐ n Ru2 = Sq2 n+1 n n (R1 )u2 + Su2 ∈ A (P3 F (k )) ữợ ữủ ự ①♦♥❣✳ ✸ ❇ê ✤➲ 1.4✭❛✮ ✤ó♥❣ ✈ỵ✐ ♠å✐ n✳ ❑❤➥♥❣ ✤à♥❤ ♥➔② ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ q✉② ♥↕♣ t❤❡♦ n✳ ❱ỵ✐ n = 1✱ tø ❣✐↔ t❤✐➳t σ (R) < t❛ ❝â σ (R) = 0✳ ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✶ t❛ ❝â Ru2 ∈ Sq1 (P3 ⊗ F (k )) + Sq2 (P3 ⊗ F (k )) ❉♦ ✤â✱ ❇ê ✤➲ 1.4 ✭❛✮ ✤ó♥❣ ✈ỵ✐ n = 1✳ ❇➙② ❣✐í ①➨t n > 1✱ ❣✐↔ sû ❇ê ✤➲ 1.4 ✭❛✮ ✤ó♥❣ ✈ỵ✐ t➜t ❝↔ ❝→❝ ❣✐→ trà ♥❤ä ỡ n ữợ ữợ 1.4 ✭❜✮ ✈➔ ❇ê ✤➲ 1.4 ✭❝✮ ❝ơ♥❣ ✤ó♥❣ ✈ỵ✐ t➜t ❝↔ ❝→❝ ❣✐→ trà ❜➨ ❤ì♥ n✳ ❚❛ ①➨t trữớ ủ s rữớ ủ (R) = 0✳ ❚❤❡♦ ❇ê ✤➲ ✷✳✶✱ t❛ ❝â n Ru2 = R(u2 • n−1 )2 ∈ A (P3 ⊗ F (k )) rữớ ủ ỗ t ởt số m ✈ỵ✐ ≤ m < σ(R) ✈➔ h(R ) = 2m σ (R)+1 ❑➳t ❤đ♣ ✈ỵ✐ sü ❦✐➺♥ m < σ (R) < n ✈➔ i2 (R) ≡ 2σ(R) − (mod ❝â m + < n ✈➔ i2 (R) ≡ 2m+1 − (mod 2m+1 )✳ 0✳ )✱ t❛ ❱➻ m + < n ✈➔ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣✱ →♣ ❞ö♥❣ ❇ê ✤➲ 1.4 ✭❜✮ ❝❤♦ ❜ë ❜❛ n−m−1 (R, u2 , m + 1) t❛ ❝â n Ru2 = R(u2 n−m−1 m+1 )2 ✹✵ ∈ A (P3 ⊗ F (k )) ❈❤÷ì♥❣ ■■■✳ ự ỵ rữớ ủ (R) > ✈➔ h(R) 2m =1 ✈ỵ✐ ♠å✐ m t❤ä❛ ♠➣♥ ≤ m < σ(R)✳ ❑❤✐ ✤â h(R) ≡ 2(R) (mod 2(R) ) ỵ p := (R) t R t ữợ p R = RS , tr♦♥❣ ✤â R ❧➔ ♠ët R3 F (k )✲✤ì♥ t❤ù❝ ✈ỵ✐ i0 (R), i1 (R) ≤ 2p − 1, i2 (R) = 2p − 1✱ ❝á♥ S ❧➔ ♠ët ✤ì♥ t❤ù❝ t❤❡♦ ❝→❝ ❜✐➳♥ Q0 , Q1 , Q2 t❤ä❛ ♠➣♥ σ (S ) = ⑩♣ ❞ö♥❣ ❇ê ✤➲ ✷✳✶ ❝❤♦ S ✈➔ v := u2 n−p−1 = 1✱ t❛ t❤✉ ✤÷đ❝ Sv ∈ A (P3 ⊗ F (k )) ▼➦t ❦❤→❝✱ h(R) = h(R) − 2p h(S ) ≡ 2p − (mod 2p ), i2 (R) = 2p − ≥ i1 (R) ❙û ❞ư♥❣ ❣✐↔ t❤✐➳t q✉② ♥↕♣ ❝ò♥❣ ✈ỵ✐ p = σ (R) < n✱ t❛ →♣ ❞ö♥❣ ❇ê ✤➲ ✶✳✹ ✭❝✮ ❝❤♦ ❜ë ❜❛ (R, Sv , p) t❛ t❤✉ ✤÷đ❝ p n Ru2 = R(Sv )2 A (P3 F (k )) ữợ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤✳ ❇ê ✤➲ ✶✳✹ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❤♦➔♥ t♦➔♥✳ ✹✶ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❊❞✇❛r❞ ❇✳ ❈✉rt✐s✱ ❚❤❡ ❉②❡r✲▲❛s❤♦❢ ▼❛t❤✳ ✭✶✾✼✺✮✱ ✷✸✶✕✷✹✻✳ ✶✾ ❛❧❣❡❜r❛ ❛♥❞ t❤❡ Λ✲❛❧❣❡❜r❛✱ ■❧❧✐♥♦✐s ❏✳ ❆ ❢✉♥❞❛♠❡♥t❛❧ s②st❡♠ ♦❢ ✐♥✈❛r✐❛♥ts ♦❢ t❤❡ ❣❡♥❡r❛❧ ♠♦❞✉❧❛r ❧✐♥❡❛r ❣r♦✉♣ ✇✐t❤ ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❢♦r♠ ♣r♦❜❧❡♠✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✱ ❙♦❝✳ ✶✷ ❬✷❪ ▲✳ ❊✳ ❉✐❝❦s♦♥✱ ✭✶✾✶✶✮✱ ✼✺✲✾✽✳ ❬✸❪ P❛✉❧ ●✳ ●♦❡rss✱ ❯♥st❛❜❧❡ ♣r♦❥❡❝t✐✈❡s ❛♥❞ st❛❜❧❡ Ext✿ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✭✸✮ ✭✶✾✽✻✮✱ ♥♦✳ ✸✱ ✺✸✾✕✺✻✶✳ ✺✸ ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s✱ Pr♦❝✳ ❚❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ ❙t❡❡♥r♦❞ sq✉❛r❡s ♦♥ t❤❡ ♠♦❞✉❧❛r ✐♥✈❛r✐❛♥ts ♦❢ ❧✐♥❡❛r ❣r♦✉♣s✱ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✶✶✸ ✭✶✾✾✶✮✱ ♥♦✳ ✹✱ ✶✵✾✼✕ ❬✹❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣✱ ✶✶✵✹✳ ❬✺❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣✱ ❙♣❤❡r✐❝❛❧ ❝❧❛ss❡s ❛♥❞ t❤❡ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✭✶✾✾✼✮✱ ♥♦✳ ✶✵✱ ✸✽✾✸✕✸✾✶✵✳ ✸✹✾ ❬✻❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣✱ ❚❤❡ ✭✶✾✾✾✮✱ ♥♦✳ ✹✱ ✼✷✼✕✼✹✸✳ ❛❧❣❡❜r❛✐❝ tr❛♥s❢❡r✱ ❚r❛♥s✳ ✇❡❛❦ ❝♦♥❥❡❝t✉r❡ ♦♥ s♣❤❡r✐❝❛❧ ❝❧❛ss❡s✱ ▼❛t❤✳ ❩✳ ✷✸✶ ❬✼❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣✱ ❖♥ tr✐✈✐❛❧✐t② ♦❢ ❉✐❝❦s♦♥ ✐♥✈❛r✐❛♥ts ✐♥ t❤❡ ❙t❡❡♥r♦❞ ❛❧❣❡❜r❛✱ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s✳ ❙♦❝✳ ✶✵✸✕✶✶✸✳ t❤❡ ❤♦♠♦❧♦❣② ♦❢ ✶✸✹ ✭✷✵✵✸✮✱ ♥♦✳ ✶✱ ❬✽❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣ ❛♥❞ ❚r➛♥ ◆✳ ◆❛♠✱ ❚❤❡ ❤✐t ♣r♦❜❧❡♠ ❢♦r t❤❡ ❛❧❣❡❜r❛✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✭✷✵✵✶✮✱ ♥♦✳ ✶✷✱ ✺✵✷✾✕✺✵✹✵✳ ❉✐❝❦s♦♥ ❬✾❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣ ❛♥❞ ❚r➛♥ ◆✳ ◆❛♠✱ ❚❤❡ ❤✐t ♣r♦❜❧❡♠ ❢♦r t❤❡ ✐♥✈❛r✐❛♥ts ♦❢ ❧✐♥❡❛r ❣r♦✉♣s✱ ❏✳ ❆❧❣❡❜r❛ ✭✷✵✵✶✮✱ ♥♦✳ ✶✱ ✸✻✼✕✸✽✹✳ ♠♦❞✉❧❛r ✸✺✸ ✷✹✻ ❬✶✵❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣ ❛♥❞ ❋r❛♥❦❧✐♥ P✳ P❡t❡rs♦♥✱ ❙♣❤❡r✐❝❛❧ ❝❧❛ss❡s ❛♥❞ t❤❡ ❉✐❝❦s♦♥ ❛❧❣❡❜r❛✱ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜r✐❞❣❡ P❤✐❧♦s✳ ❙♦❝✳ ✭✶✾✾✽✮✱ ♥♦✳ ✷✱ ✷✺✸✕ ✷✻✹✳ ✶✷✹ ❬✶✶❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣ ❛♥❞ ●❡♦❢❢r❡② P♦✇❡❧❧✱ ❚❤❡ A −❞❡❝♦♠♣♦s❛❜✐❧✐t② ❙✐♥❣❡r ❝♦♥str✉❝t✐♦♥✱ ❏✳ ❆❧❣❡❜r❛ ✭✷✵✶✾✮✱ ✶✽✻✲✷✵✻✳ ✺✶✼ ✹✷ ♦❢ t❤❡ ❚⑨■ ▲■➏❯ ❚❍❆▼ ❑❍❷❖ ❬✶✷❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣✱ ❱ã ❚✳ ◆✳ ◗✉ý♥❤✱ ❛♥❞ ◆❣ỉ ❆✳ ❚✉➜♥✱ ❖♥ t❤❡ ♦❢ t❤❡ ▲❛♥♥❡s✲❩❛r❛t✐ ❤♦♠♦♠♦r♣❤✐s♠✱ ❈✳ ❘✳ ▼❛t❤✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s ♥♦✳ ✸✱ ✷✺✶✕✷✺✹✳ ✈❛♥✐s❤✐♥❣ ✸✺✷ ✭✷✵✶✹✮✱ ❚✉➜♥✱ ❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ❛❧✲ ❣❡❜r❛✐❝ ❝♦♥❥❡❝t✉r❡ ♦♥ s♣❤❡r✐❝❛❧ ❝❧❛ss❡s✱ ✺✵ ♣❛❣❡s✱ ♣r❡♣r✐♥t ✶✺✻✹ ❢t♣✿✴✴❢✐❧❡✳✈✐❛s♠✳♦r❣✴❲❡❜✴❚✐❡♥❆♥P❤❛♠✲✶✺✴❀ s✉❜♠✐tt❡❞✱ ✷✵✶✺✳ ❬✶✸❪ ◆❣✉②➵♥ ❍✳ ❱✳ ❍÷♥❣ ❛♥❞ ◆❣æ ❆✳ ❬✶✹❪ ❏❡❛♥ ▲❛♥♥❡s✱ ❙✉r ❧❡ n✲❞✉❛❧ ❞✉ n✲➧♠❡ s♣❡❝tr❡ ❞❡ ❇r♦✇♥✲●✐t❧❡r✱ ▼❛t❤✳ ❩✳ ✭✶✾✽✽✮✱ ♥♦✳ ✶✱ ✷✾✕✹✷✳ ❬✶✺❪ ❏❡❛♥ ▲❛♥♥❡s ❛♥❞ ❙❛☎✙❞ ❩❛r❛t✐✱ ❙✉r ▼❛t❤✳ ❩✳ ✭✶✾✽✼✮✱ ♥♦✳ ✶✱ ✷✺✕✺✾✳ ✶✾✹ ✶✾✾ ❧❡s ❢♦♥❝t❡✉rs ❞➨r✐✈➨s ❞❡ ❧❛ ❞➨st❛❜✐❧✐s❛t✐♦♥✱ ❬✶✻❪ ●❡♦❢❢r❡② ▼✳ ▲✳ P♦✇❡❧❧✱ ❖♥ ✉♥st❛❜❧❡ ♠♦❞✉❧❡s ♦✈❡r t❤❡ ❉✐❝❦s♦♥ ❛❧❣❡❜r❛s✱ t❤❡ ❙✐♥❣❡r ❢✉♥❝t♦rs Rs ❛♥❞ t❤❡ ❢✉♥❝t♦rs Fixs✱ ❆❧❣❡❜r✳ ●❡♦♠✳ ❚♦♣♦❧✳ ✭✷✵✶✷✮✱ ♥♦✳ ✹✱ ✷✹✺✶✕✷✹✾✶✳ ✶✷ ❯♥st❛❜❧❡ ♠♦❞✉❧❡s ♦✈❡r t❤❡ ❙t❡❡♥r♦❞ ❛❧❣❡❜r❛ ❛♥❞ ❙✉❧❧✐✈❛♥✬s ❢✐①❡❞ ♣♦✐♥t s❡t ❝♦♥❥❡❝t✉r❡✱ ❈❤✐❝❛❣♦ ▲❡❝t✉r❡s ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❯♥✐✈❡rs✐t② ♦❢ ❬✶✼❪ ▲✐♦♥❡❧ ❙❝❤✇❛rt③✱ ❈❤✐❝❛❣♦ Pr❡ss✱ ❈❤✐❝❛❣♦✱ ■▲✱ ✶✾✾✹✳ ❬✶✽❪ ❲✐❧❧✐❛♠ ▼✳ ❙✐♥❣❡r✱ ✭✶✾✽✾✮✱ ♥♦✳ ✹✱ ✹✾✸✕✺✷✸✳ ❚❤❡ tr❛♥s❢❡r ✐♥ ❤♦♠♦❧♦❣✐❝❛❧ ❛❧❣❡❜r❛✱ ▼❛t❤✳ ❩✳ ✷✵✷ ❬✶✾❪ ◆♦r♠❛♥ ❊✳ ❙t❡❡♥r♦❞✱ ❈♦❤♦♠♦❧♦❣② ♦♣❡r❛t✐♦♥s✱ ▲❡❝t✉r❡s ❜② ◆✳ ❊✳ ❙t❡❡♥r♦❞ ✇r✐tt❡♥ ❛♥❞ r❡✈✐s❡❞ ❜② ❉❛✈✐❞ ❇✳ ❆✳ ❊♣st❡✐♥✳ ❆♥♥❛❧s ♦❢ ▼❛t❤❡♠❛t✐❝s ❙t✉❞✐❡s✱ ◆♦✳ ✺✵✱ Pr✐♥❝❡t♦♥ ❯♥✐✈❡rs✐t② Pr❡ss✱ Pr✐♥❝❡t♦♥✱ ◆✳❏✳✱ ✶✾✻✷✳ ❬✷✵❪ ❘♦❜❡rt ❏✳ ❲❡❧❧✐♥❣t♦♥✱ ❚❤❡ ✉♥st❛❜❧❡ ❆❞❛♠s s♣❡❝tr❛❧ s❡q✉❡♥❝❡ ❢♦r ❢r❡❡ ✐t❡r❛t❡❞ ❧♦♦♣ s♣❛❝❡s✱ ▼❡♠✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳ ✭✶✾✽✷✮✱ ♥♦✳ ✷✺✽✱ ✈✐✐✐✰✷✷✺✳ ✸✻ ❬✷✶❪ ❈❧❛r❡♥❝❡ ❲✐❧❦❡rs♦♥✱ ❆ ♣r✐♠❡r ♦♥ t❤❡ ❉✐❝❦s♦♥ ✐♥✈❛r✐❛♥ts✱ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ◆♦rt❤✇❡st❡r♥ ❍♦♠♦t♦♣② ❚❤❡♦r② ❈♦♥❢❡r❡♥❝❡ ✭❊✈❛♥st♦♥✱ ■❧❧✳✱ ✶✾✽✷✮✱ ❈♦♥t❡♠♣✳ ▼❛t❤✳✱ ✈♦❧✳ ✶✾✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ Pr♦✈✐❞❡♥❝❡✱ ❘■✱ ✶✾✽✸✱ ♣♣✳ ✹✷✶✕✹✸✹✳ ✹✸