(LUẬN văn THẠC sĩ) hệ nhân tử trong nhóm phạm trù phân bậc

➜➵✐ ❤ä❝ ❤✉Õ ❚r➢ê♥❣ ➤➵✐ ❤ä❝ s➢ ♣❤➵♠ ♣❤➵♠ t❤Þ ❝ó❝ ❍Ư ♥❤➞♥ tư tr♦♥❣ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❧✉❐♥ ➳♥ t✐Õ♥ sÜ t♦➳♥ ❤ä❝ ❍✉Õ ✲ ✷✵✶✹ download by : skknchat@gmail.com ➜➵✐ ❤ä❝ ❤✉Õ ❚r➢ê♥❣ ➤➵✐ ❤ä❝ s➢ ♣❤➵♠ ♣❤➵♠ t❤Þ ❝ó❝ ❍Ư ♥❤➞♥ tư tr♦♥❣ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❈❤✉②➟♥ ♥❣➭♥❤✿ ➜➵✐ sè ✈➭ ❧ý t❤✉②Õt sè ▼➲ sè✿ ✻✷✳ ✹✻✳ ✵✺✳ ✵✶ ❧✉❐♥ ➳♥ t✐Õ♥ sÜ t♦➳♥ ❤ä❝ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥ ❦❤♦❛ ❤ä❝✿ ✶✳ P●❙✳ ❚❙✳ ◆❣✉②Ô♥ ❚✐Õ♥ ◗✉❛♥❣ ✷✳ ●❙✳ ❚❙✳ ▲➟ ❱➝♥ ❚❤✉②Õt ❍✉Õ✱ ✷✵✶✹ download by : skknchat@gmail.com ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ ➤➞② ❧➭ ❝➠♥❣ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ t ợ ết t ữ ❦Õt q✉➯ ✈✐Õt ❝❤✉♥❣ ✈í✐ ❝➳❝ t➳❝ ❣✐➯ ❦❤➳❝ ➤➲ ➤➢ỵ❝ sù ♥❤✃t trÝ ❝đ❛ ❝➳❝ ➤å♥❣ t➳❝ ❣✐➯ ❦❤✐ ➤➢❛ ✈➭♦ ❧✉❐♥ ➳♥✳ ❈➳❝ sè ❧✐Ư✉✱ ❦Õt q✉➯ ➤➢ỵ❝ tr×♥❤ ❜➭② tr♦♥❣ ❧✉❐♥ ➳♥ ❧➭ tr✉♥❣ t❤ù❝ ✈➭ ❝❤➢❛ từ ợ ố tr t ỳ trì ♥➭♦ ❦❤➳❝✳ ❚➳❝ ❣✐➯ P❤➵♠ ❚❤Þ ❈ó❝ ✶ download by : skknchat@gmail.com ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ➳♥ ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ P●❙✳ ❚❙✳ ◆❣✉②Ơ♥ ❚✐Õ♥ ◗✉❛♥❣ ✈➭ ●❙✳ ❚❙✳ ▲➟ ❱➝♥ ❚❤✉②Õt✳ ▲ê✐ ➤➬✉ t✐➟♥✱ ❡♠ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ ❝❤➞♥ t❤➭♥❤ ✈➭ s➞✉ s➽❝ ♥❤✃t ➤Õ♥ ❝➳❝ ❚❤➬②✳ ❈➳❝ ❚❤➬② ❦❤➠♥❣ ❝❤Ø tr✉②Ò♥ ❝❤♦ ❡♠ ♥✐Ò♠ ➤❛♠ ♠➟ ♥❣❤✐➟♥ ❝ø✉ ❦❤♦❛ ❤ä❝✱ t❐♥ t×♥❤ ❤➢í♥❣ ❞➱♥ ✈➭ ❣✐ó♣ ➤ì ❡♠ ✈Ị ♠ä✐ ♠➷t✱ ♠➭ ❝ß♥ ❞➭♥❤ ❝❤♦ ❡♠ sù ❝ỉ ✈ị ✈➭ ➤é♥❣ ✈✐➟♥ tr♦♥❣ s✉èt q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ♠×♥❤✳ ❚➠✐ ①✐♥ tr➞♥ trä♥❣ ❝➯♠ ➡♥ ❝➳❝ t❤➬② ❝➠ tr♦♥❣ ❑❤♦❛ ❚♦➳♥✱ P❤ß♥❣ ❙❛✉ ➤➵✐ ❤ä❝ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠ ✲ ➜➵✐ ❤ä❝ ❍✉Õ✱ ❇❛♥ ➤➭♦ t➵♦ s❛✉ ➤➵✐ ❤ä❝ ✲ ➜➵✐ ❤ä❝ ❍✉Õ ✈➭ ❝➳❝ t❤➬② ❝➠ tr♦♥❣ ❇é ♠➠♥ ➜➵✐ sè✱ ❑❤♦❛ ❑❤♦❛ ❤ä❝ tù ♥❤✐➟♥ ✲ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❍å♥❣ ➜ø❝ ✲ ❚❤❛♥❤ ❍ã❛ ➤➲ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ t❤✉❐♥ ❧ỵ✐ ❝❤♦ t➠✐ ❤ä❝ t❐♣✱ ♥❣❤✐➟♥ ❝ø✉ ✈➭ ❤♦➭♥ t❤➭♥❤ ❝❤➢➡♥❣ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ ♠×♥❤✳ ❚➠✐ ❝ị♥❣ ①✐♥ ❣ư✐ ❧ê✐ ❝➯♠ ➡♥ ế sỹ ễ ủ ì ữ ú ➤ì ❝❤➞♥ t❤➭♥❤✳ ❈✉è✐ ❝ï♥❣✱ t➠✐ ♠✉è♥ ❜➭② tá ❧ß♥❣ ết s s ế ì t ì ữ sù ➤å♥❣ ❝➯♠✱ ➤é♥❣ ✈✐➟♥ ✈➭ ❝❤✐❛ s❰ ♥❤÷♥❣ ❦❤ã ❦❤➝♥ tr♦♥❣ s✉èt t❤ê✐ ❣✐❛♥ t➠✐ ❧➭♠ ♥❣❤✐➟♥ ❝ø✉ s✐♥❤ ✈➭ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ➳♥ ♥➭②✳ P❤➵♠ ❚❤Þ ❈ó❝ ✷ download by : skknchat@gmail.com ▼ô❝ ❧ô❝ ✶ ▼ét sè ❦✐Õ♥ t❤ø❝ ❝❤✉➮♥ ❜Þ ✶✻ ✶✳✶ ◆❤ã♠ ♣❤➵♠ trï ✭❜Ư♥✮ ♣❤➞♥ ❜❐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✶✳✶ ◆❤ã♠ ♣❤➵♠ trï ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✶✳✷ ◆❤ã♠ ♣❤➵♠ trï t❤✉ ❣ä♥ ✈➭ ❝➳❝ t➢➡♥❣ ➤➢➡♥❣ ❝❤Ý♥❤ t➽❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✶✳✶✳✸ ◆❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✶✳✹ ◆❤ã♠ ♣❤➵♠ trï ❜Ö♥ ♣❤➞♥ ❜❐❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✶✳✺ ❍➭♠ tö ♠♦♥♦✐❞❛❧✱ t➢➡♥❣ ➤➢➡♥❣ tù ♥❤✐➟♥ ♠♦♥♦✐❞❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✶✳✷ ❆♥♥✲♣❤➵♠ trï ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷✳✶ ❆♥♥✲♣❤➵♠ trï ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✷✳✷ ❆♥♥✲❤➭♠ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✷✳✸ ❆♥♥✲♣❤➵♠ trï t❤✉ ❣ä♥ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ P❤➞♥ ❧í♣ ❝➳❝ ❤➭♠ tư ♠♦♥♦✐❞❛❧ ❦✐Ĩ✉ (ϕ, f ) ✈➭ ø♥❣ ❞ơ♥❣ ✷✳✶ P❤➞♥ ❧í♣ ➤è✐ ➤å♥❣ ➤✐Ị✉ ❝➳❝ ❤➭♠ tư ♠♦♥♦✐❞❛❧ ❦✐Ĩ✉ ✷✺ (ϕ, f ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✷ P❤➞♥ ❧í♣ ❝➳❝ ♥❤ã♠ ♣❤➵♠ trï ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✸ P❤➞♥ ❧í♣ ❝➳❝ ♥❤ã♠ ♣❤➵♠ trï ❜Ö♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹ ✷✳✹ P❤➞♥ ❧í♣ ❝➳❝ ♥❤ã♠ ♣❤➵♠ trï ❜Ư♥ ♣❤➞♥ ❜❐❝ ❜ë✐ ❤Ư ♥❤➞♥ tö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✺ ➳♣ ❞ô♥❣ ✈➭♦ ❜➭✐ t♦➳♥ ♠ë ré♥❣ ♥❤ã♠ ❝ỉ ➤✐Ĩ♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✺✳✶ ◆❤ã♠ ♣❤➵♠ trï ❝ñ❛ ♠ét ❤➵t ♥❤➞♥ trõ✉ t➢ỵ♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺ ✷✳✺✳✷ ❍➭♠ tö ♠♦♥♦✐❞❛❧ ✈➭ ❜➭✐ t♦➳♥ ♠ë ré♥❣ ♥❤ã♠ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ◆❤ã♠ ♣❤➵♠ trï ❝❤➷t ❝❤Ï ✈➭ ♠ë ré♥❣ ♥❤ã♠ ❦✐Ó✉ ♠➠➤✉♥ ❝❤Ð♦ ✺✸ ✸✳✶ ◆❤ã♠ ♣❤➵♠ trï ❧✐➟♥ ❦Õt ✈í✐ ♠ét ♠➠➤✉♥ ❝❤Ð♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✸ ✸✳✷ P❤➞♥ ❧í♣ ❝➳❝ ♠➠➤✉♥ ❝❤Ð♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✸ ❇➭✐ t♦➳♥ ♠ë ré♥❣ ♥❤ã♠ ❦✐Ó✉ ♠➠➤✉♥ ❝❤Ð♦✿ ❧ý t❤✉②Õt ❝➯♥ trë ✈➭ ➤Þ♥❤ ❧ý ♣❤➞♥ ❧í♣ ✺✽ ✹ ◆❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❝❤➷t ❝❤Ï ✈➭ ♠ë ré♥❣ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ ❦✐Ó✉ ❝❤Ð♦ Γ✲♠➠➤✉♥ ✻✺ ✹✳✶ ▲ý t❤✉②Õt ➤è✐ ➤å♥❣ ➤✐Ị✉ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ ❝đ❛ ❈❡❣❛rr❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✸ download by : skknchat@gmail.com ✹✳✷ ◆❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ t❤✉ ❣ä♥ ✈➭ ❤➭♠ tö ♠♦♥♦✐❞❛❧ ♣❤➞♥ ❜❐❝ ❦✐Ó✉ (ϕ, f ) ✳ ✻✻ ✹✳✷✳✶ ❳➞② ❞ù♥❣ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ t❤✉ ❣ä♥ t❤➠♥❣ q✉❛ ♣❤➵♠ trï ❦❤✉♥❣ ✻✼ ✹✳✷✳✷ ❳➞② ❞ù♥❣ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ t❤✉ ❣ä♥ ❜➺♥❣ ♣❤➢➡♥❣ ♣❤➳♣ ❤Ư ♥❤➞♥ tư ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✾ ✹✳✷✳✸ P❤➞♥ ❧í♣ ❝➳❝ ❤➭♠ tư ♠♦♥♦✐❞❛❧ ♣❤➞♥ ❜❐❝ ❦✐Ĩ✉ ✹✳✸ Γ✲♠➠➤✉♥ ❝❤Ð♦ ✈➭ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❧✐➟♥ ❦Õt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✹✳✹ P❤➞♥ ❧í♣ ❝➳❝ Γ✲♠➠➤✉♥ ❝❤Ð♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✺ ❇➭✐ t♦➳♥ ♠ë ré♥❣ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ ❦✐Ĩ✉ ➤Þ♥❤ ❧ý ♣❤➞♥ ❧í♣ ✺ (ϕ, f ) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ Γ✲♠➠➤✉♥ ❝❤Ð♦✿ ❧ý t❤✉②Õt ❝➯♥ trë ✈➭ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ❆♥♥✲♣❤➵♠ trï ❝❤➷t ❝❤Ï ✈➭ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ✺✳✶ ▲ý t❤✉②Õt ➤è✐ ➤å♥❣ ➤✐Ị✉ ✈➭♥❤ ❝đ❛ ▼❛❝ ▲❛♥❡ ✈➭ ❙❤✉❦❧❛ ✽✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✽ ✺✳✷ ❙♦♥❣ ♠➠➤✉♥ ❝❤Ð♦ ✈➭ ❊✲❤Ö ❝❤Ý♥❤ q✉✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✶ ✺✳✸ P❤➞♥ ❧í♣ ❝➳❝ ❊✲❤Ö ❝❤Ý♥❤ q✉✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✹ ✺✳✹ ▼ë ré♥❣ ✈➭♥❤ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✹ download by : skknchat@gmail.com ❇➯♥❣ ❦ý ❤✐Ư✉ ❑ý ❤✐Ư✉ ◆❣❤Ü❛ ObG t❐♣ ❝➳❝ ✈❐t ❝đ❛ ♣❤➵♠ trï MorG t❐♣ ❝➳❝ ♠ị✐ t➟♥ ❝đ❛ ♣❤➵♠ trï (0, g, d) r➭♥❣ ❜✉é❝ ➤➡♥ ✈Þ ❝đ❛ ♣❤Ð♣ ❝é♥❣ (1, l, r) r➭♥❣ ❜✉é❝ ➤➡♥ ✈Þ ✭❝đ❛ ♣❤Ð♣ ♥❤➞♥✮ Π = π0 G t❐♣ ❝➳❝ ❧í♣ ✈❐t ➤➻♥❣ ❝✃✉ ❝ñ❛ A = π1 G t❐♣ ❝➳❝ tù ➤➻♥❣ ❝✃✉ ❝đ❛ ✈❐t ➤➡♥ ✈Þ SG ♣❤➵♠ trï t❤✉ ❣ä♥ ❝đ❛ ♣❤➵♠ trï Hom(ϕ,f ) [S, S ] t❐♣ ❝➳❝ ❧í♣ ➤å♥❣ ❧✉➞♥ ❝➳❝ ❤➭♠ tư ❦✐Ĩ✉ tõ Γ ♥❤ã♠ ♣❤➵♠ trï (Π, A, h) G G I G (ϕ, f ) S ➤Õ♥ S ♥❤ã♠ ♣❤➵♠ trï ❦✐Ó✉ (Π, A), (Π, A, k) G (Π, A) Γ✲♣❤➞♥ ❜❐❝ ❦✐Ó✉ (Π, A) (F, F ) (F, F˘ , F ) ❤➭♠ tö ♠♦♥♦✐❞❛❧ ✭Γ✲♣❤➞♥ ❜❐❝✮ (H, H), (G, G) ❝➳❝ t➢➡♥❣ ➤➢➡♥❣ ♠♦♥♦✐❞❛❧ ❝❤Ý♥❤ t➽❝ (HΓ , HΓ ), (GΓ , GΓ ) ❝➳❝ t➢➡♥❣ ➤➢➡♥❣ ♠♦♥♦✐❞❛❧ ❆♥♥✲❤➭♠ tö Γ✲♣❤➞♥ ❜❐❝ ❝❤Ý♥❤ t➽❝ (R, M ), (R, M, h) ❆♥♥✲♣❤➵♠ trï ❦✐Ĩ✉ H i (Π, A) ❝➳❝ ♥❤ã♠ ➤è✐ ➤å♥❣ ➤✐Ị✉ ♥❤ã♠ HΓi (Π, A) i HM acL (R, M ) i HShu (R, M ) ❝➳❝ ♥❤ã♠ ➤è✐ ➤å♥❣ ➤✐Ò✉ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ MA ✈➭♥❤ ❝➳❝ s♦♥❣ tÝ❝❤ ❝ñ❛ ✈➭♥❤ (R, M ) ❝➳❝ ♥❤ã♠ ➤è✐ ➤å♥❣ ➤✐Ị✉ ✈➭♥❤ ❝đ❛ ▼❛❝ ▲❛♥❡ ❝➳❝ ♥❤ã♠ ➤è✐ ➤å♥❣ ➤✐Ị✉ ✈➭♥❤ ❝đ❛ ❙❤✉❦❧❛ A t❐♣ ❝➳❝ ❧í♣ t➢➡♥❣ ➤➢➡♥❣ ❝➳❝ ♠ë ré♥❣ ♥❤ã♠ Ext(Π, A, ψ) d M, (B, D, d, θ), B → D ✭Γ✲✮♠➠➤✉♥ ❝❤Ð♦✱ ❊✲❤Ö ExtB→D (Q, B, ψ) t❐♣ ❝➳❝ ❧í♣ t➢➡♥❣ ➤➢➡♥❣ ❝➳❝ ♠ë ré♥❣ ♥❤ã♠ ❦✐Ĩ✉ ♠➠➤✉♥ ❝❤Ð♦ ExtΓB→D (Q, B, ψ) t❐♣ ❝➳❝ ❧í♣ t➢➡♥❣ ➤➢➡♥❣ ❝➳❝ ♠ë ré♥❣ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ ❦✐Ó✉ Γ✲♠➠➤✉♥ ❝❤Ð♦ ✺ download by : skknchat@gmail.com ❇➯♥❣ t❤✉❐t ♥❣÷ ❚❤✉❐t ♥❣÷ ❚✐Õ♥❣ ❆♥❤ ❆♥♥✲♣❤➵♠ trï ❆♥♥✲❝❛t❡❣♦r② ❆♥♥✲♣❤➵♠ trï ❝❤➷t ❝❤Ï str✐❝t ❆♥♥✲❝❛t❡❣♦r② ❆♥♥✲♣❤➵♠ trï ❝❤Ý♥❤ q✉✐ r❡❣✉❧❛r ❆♥♥✲❝❛t❡❣♦r② ❆♥♥✲♣❤➵♠ trï t❤✉ ❣ä♥ r❡❞✉❝❡❞ ❆♥♥✲❝❛t❡❣♦r② ❆♥♥✲❤➭♠ tö ❆♥♥✲❢✉♥❝t♦r ❆♥♥✲❤➭♠ tö ➤➡♥ s✐♥❣❧❡ ❆♥♥✲❢✉♥❝t♦r ❆♥♥✲♠ò✐ t➟♥ ❆♥♥✲♠♦r♣❤✐s♠ ❆♥♥✲t➢➡♥❣ ➤➢➡♥❣ ❆♥♥✲❡q✉✐✈❛❧❡♥❝❡ ❆♥♥✲t➢➡♥❣ ➤➢➡♥❣ ❝❤Ý♥❤ t➽❝ ❝❛♥♦♥✐❝❛❧ ❆♥♥✲❡q✉✐✈❛❧❡♥❝❡ ❝➯♥ trë ♦❜str✉❝t✐♦♥ ➤Ý♥❤ st✐❝❦ ➤✐Ị✉ ❦✐Ư♥ ❦❤í♣ ❝♦❤❡r❡♥❝❡ ❝♦♥❞✐t✐♦♥ ❊✲❤Ö ❊✲s②st❡♠ ❊✲❤Ö ❝❤Ý♥❤ q✉✐ r❡❣✉❧❛r ❊✲s②st❡♠ ❤➭♠ tö ♠♦♥♦✐❞❛❧ ♠♦♥♦✐❞❛❧ ❢✉♥❝t♦r ❤➭♠ tö ♠♦♥♦✐❞❛❧ ❝❤Ý♥❤ q✉✐ r❡❣✉❧❛r ♠♦♥♦✐❞❛❧ ❢✉♥❝t♦r ❤➭♠ tö ♠♦♥♦✐❞❛❧ ➤è✐ ①ø♥❣ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❢✉♥❝t♦r ❤➭♠ tö ♠♦♥♦✐❞❛❧ ♣❤➞♥ ❜❐❝ ❣r❛❞❡❞ ♠♦♥♦✐❞❛❧ ❢✉♥❝t♦r ❤➭♠ tö ♠♦♥♦✐❞❛❧ ♣❤➞♥ ❜❐❝ ❝❤Ý♥❤ q✉✐ r❡❣✉❧❛r ❣r❛❞❡❞ ♠♦♥♦✐❞❛❧ ❢✉♥❝t♦r ❤➵t ♥❤➞♥ trõ✉ t➢ỵ♥❣ ❛❜str❛❝t ❦❡r♥❡❧ ❤Ư ♥❤➞♥ tư ❢❛❝t♦r s❡t ❤Ư ♥❤➞♥ tư ❝❤Ý♥❤ q✉✐ r❡❣✉❧❛r ❢❛❝t♦r s❡t ❣✐➯ ❤➭♠ tö ♣s❡✉❞♦✲❢✉♥❝t♦r ♠➠➤✉♥ ❝❤Ð♦ ❝r♦ss❡❞ ♠♦❞✉❧❡ ♠➠➤✉♥ ❝❤Ð♦ ➤➻♥❣ ❜✐Õ♥ ❡q✉✐✈❛r✐❛♥t ❝r♦ss❡❞ ♠♦❞✉❧❡ ♠ë ré♥❣ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ ❡q✉✐✈❛r✐❛♥t ❣r♦✉♣ ❡①t❡♥s✐♦♥ ♠ë ré♥❣ tÝ❝❤ ❝❤Ð♦ ❝r♦ss❡❞ ♣r♦❞✉❝t ❡①t❡♥s✐♦♥ ♥❤ã♠ ♣❤➵♠ trï ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ ♥❤ã♠ ♣❤➵♠ trï ❝❤➷t ❝❤Ï str✐❝t ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ ♥❤ã♠ ♣❤➵♠ trï ❜Ö♥ ❜r❛✐❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ ♥❤ã♠ ♣❤➵♠ trï ❜Ö♥ ♣❤➞♥ ❜❐❝ ❣r❛❞❡❞ ❜r❛✐❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ ✻ download by : skknchat@gmail.com ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❣r❛❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❝❤➷t ❝❤Ï str✐❝t ❣r❛❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ ♥❤ã♠ ♣❤❛♠ trï rê✐ r➵❝ ❞✐s❝r❡t❡ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ ♥❤ã♠ ♣❤➵♠ trï t❤✉ ❣ä♥ r❡❞✉❝❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ ♣❤➵♠ trï ❦❤✉♥❣ s❦❡❧❡t❛❧ ❝❛t❡❣♦r② ♣❤➵♠ trï ♠♦♥♦✐❞❛❧ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r② ♣❤➵♠ trï ♠♦♥♦✐❞❛❧ ➤è✐ ①ø♥❣ s②♠♠❡tr✐❝ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r② ♣❤➵♠ trï P✐❝❛r❞ P✐❝❛r❞ ❝❛t❡❣♦r② ♣❤➞♥ ❜❐❝ ❣r❛❞❡❞ r➭♥❣ ❜✉é❝ ❝♦♥str❛✐♥t r➭♥❣ ❜✉é❝ ❜Ö♥ ❜r❛✐❞❡❞ ❝♦♥str❛✐♥t r➭♥❣ ❜✉é❝ ➤➡♥ ✈Þ ✉♥✐t ❝♦♥str❛✐♥t r➭♥❣ ❜✉é❝ ❣✐❛♦ ❤♦➳♥ ❝♦♠♠✉t❛t✐✈✐t② ❝♦♥str❛✐♥t r➭♥❣ ❜✉é❝ ❦Õt ❤ỵ♣ ❛ss♦❝✐❛t✐✈✐t② ❝♦♥str❛✐♥t s♦♥❣ ♠➠➤✉♥ ❝❤Ð♦ ❝r♦ss❡❞ ❜✐♠♦❞✉❧❡ s♦♥❣ tÝ❝❤ ❜✐♠✉❧t✐♣❧✐❝❛t✐♦♥ s♦♥❣ tÝ❝❤ ❣✐❛♦ ❤♦➳♥ ♣❡r♠✉t❛❜❧❡ ❜✐♠✉❧t✐♣❧✐❝❛t✐♦♥ sù t➢➡♥❣ t❤Ý❝❤ ❝♦♠♣❛t✐❜✐❧✐t② t✐Ò♥ ➤Ý♥❤ ♣r❡✲st✐❝❦ t➢➡♥❣ ➤➢➡♥❣ ♣❤➵♠ trï ❝❛t❡❣♦r✐❝❛❧ ❡q✉✐✈❛❧❡♥❝❡ t➢➡♥❣ ➤➢➡♥❣ tù ♥❤✐➟♥ ♠♦♥♦✐❞❛❧ ♠♦♥♦✐❞❛❧ ♥❛t✉r❛❧ ❡q✉✐✈❛❧❡♥❝❡ ✈❐t ♦❜❥❡❝t ✼ download by : skknchat@gmail.com s➡ ➤å ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝➳❝ ❦❤➳✐ ♥✐Ư♠✱ t❤✉❐t ♥❣÷ ◆❤ã♠ ♣❤➵♠ trï ❆♥♥✲♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❅ ❅ ❅         ❅ ❅   ◆❤ã♠ ♣❤➵♠ trï ✶✳ ◆❤ã♠ ♣❤➵♠ trï ✛ ⊃ ◆❤ã♠ ♣❤➵♠ trï ❝❤➷t ❝❤Ï ✛ ✲ ▼➠➤✉♥ ❝❤Ð♦     ❅ ❅   ❅ ❅ ❅     ▼ë ré♥❣ ♥❤ã♠ ✛ ✷✳ ◆❤ã♠ ♣❤➵♠ trï ✛ ♣❤➞♥ ❜❐❝ ⊃ ▼ë ré♥❣ ♥❤ã♠ ❦✐Ó✉ ♠➠➤✉♥ ❝❤Ð♦ ⊃ ✛ ◆❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❝❤➷t ❝❤Ï ✲ Γ✲♠➠➤✉♥ ❝❤Ð♦     ❅ ❅ ❅ ❅ ✸✳ ❆♥♥✲♣❤➵♠ trï       ▼ë ré♥❣ ♥❤ã♠ ✛ ➤➻♥❣ ❜✐Õ♥ ✛ ⊃ ⊃ ▼ë ré♥❣ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ ❦✐Ó✉ Γ✲♠➠➤✉♥ ❝❤Ð♦ ❆♥♥✲♣❤➵♠ trï ❝❤➷t ❝❤Ï ✛ ✲ ❊✲❤Ö ❝❤Ý♥❤ q✉✐     ❅ ❅   ❅     ❅ ❅ ▼ë ré♥❣ ✈➭♥❤ ✛ ▼ë ré♥❣ ✈➭♥❤ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ⊃ ✽ download by : skknchat@gmail.com (B, E, j, θ0 ) ❧➭ ♠ét ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ✈í✐ θ0 ❧➭ ♣❤Ð♣ ❧✃② s♦♥❣ tr♦♥❣ ➤ã ❞ß♥❣ tr➟♥ ❧➭ ❦❤í♣✱ ❤Ư tÝ❝❤✱ ❝➷♣ (id, ε) ❧➭ ♠ét ➤å♥❣ ❝✃✉ ❝đ❛ ❝➳❝ ❊✲❤Ư ❝❤Ý♥❤ q✉✐✳ ❍❛✐ ♠ë ré♥❣ ❝đ❛ B ❜ë✐ Q ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ (B, D, d, θ) ➤➢ỵ❝ ❣ä✐ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ♥Õ✉ ❜✐Ó✉ ➤å s❛✉ ❣✐❛♦ ❤♦➳♥ / j B / E p / Q / 0, E Q / 0, E ε / ✭✺✳✷✵✮ D α / ✈➭ j B /  E p / ε / D ε α = ε✳ ❍✐Ó♥ ♥❤✐➟♥ α ❧➭ ♠ét ➤➻♥❣ ❝✃✉✳ ❚r♦♥❣ ❜✐Ĩ✉ ➤å E: / B B ✈í✐ j d / / E  ε D p / Q / 0, ✭✺✳✷✶✮ ψ  q / ❈♦❦❡rd q ❧➭ ♣❤Ð♣ ❝❤✐Õ✉ ❝❤Ý♥❤ t➽❝✱ ❞♦ ❞ß♥❣ tr➟♥ ❧➭ ❦❤í♣ ✈➭ q ◦ ε ◦ j = q ◦ d = ♥➟♥ ❝ã ♠ét ➤å♥❣ ❝✃✉ ✈➭♥❤ ψ : Q → ❈♦❦❡rd s❛♦ ❝❤♦ ❤×♥❤ ✈✉➠♥❣ t❤ø ❤❛✐ ❣✐❛♦ ❤♦➳♥✳ ❍➡♥ ♥÷❛✱ ψ ❝❤Ø ♣❤ơ t❤✉é❝ ✈➭♦ ❧í♣ t➢➡♥❣ ➤➢➡♥❣ ❝đ❛ ♠ë ré♥❣ E ✳ ▼ơ❝ t✐➟✉ ❝đ❛ ❝❤ó♥❣ t➠✐ ❧➭ sư ❞ơ♥❣ ❧ý t❤✉②Õt ❝➯♥ trë ❝đ❛ ❝➳❝ ❆♥♥✲❤➭♠ tư ➤Ĩ ♥❣❤✐➟♥ ❝ø✉ t❐♣ ❝➳❝ ❧í♣ t➢➡♥❣ ➤➢➡♥❣ ❝➳❝ ♠ë ré♥❣ ✈➭♥❤ ❝đ❛ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ B ❜ë✐ Q B → D ❝➯♠ s✐♥❤ ψ ✱ ExtB→D (Q, B, ψ) ❚r♦♥❣ ♣❤➬♥ ♥➭②✱ ❝❤ó♥❣ t➠✐ t✐Õ♣ tơ❝ ♠ë ré♥❣ ❦ü t❤✉❐t ❤Ư ♥❤➞♥ tư ➤è✐ ✈í✐ ❜➭✐ t♦➳♥ ♠ë ré♥❣ ♥❤ã♠ ❦✐Ó✉ ♠➠➤✉♥ ❝❤Ð♦ ë ❈❤➢➡♥❣ ✸ s❛♥❣ ❝❤♦ ❜➭✐ t♦➳♥ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐✳ ❚r♦♥❣ ❜ỉ ➤Ị ❞➢í✐ ➤➞②✱ ❝➳❝ ❆♥♥✲❤➭♠ tư Dis Q ABD ệ ữ ệ ù ợ ể ❞ù♥❣ ❝➳❝ ♠ë ré♥❣ ♥❤➢ ✈❐②✳ ❇ỉ ➤Ị ✺✳✽✳ ❈❤♦ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ❆♥♥✲❤➭♠ tư ❜ë✐ (B, D, d, θ) ✈➭ ➤å♥❣ ❝✃✉ ✈➭♥❤ ψ : Q → Coker d✳ ỗ (F, F , F ) : Dis Q → A ❝➯♠ s✐♥❤ ❝➷♣ (ψ, 0) ➤Ò✉ tå♥ t➵✐ ♠ét ♠ë ré♥❣ EF Q ❦✐Ĩ✉ ❊✲❤Ư B → D ▼ë ré♥❣ ❈❤ø♥❣ ♠✐♥❤✳ ❝➯♠ s✐♥❤ ➤å♥❣ ❝✃✉ ❝ñ❛ B ψ : Q → Coker d✳ EF ➤➢ỵ❝ ❣ä✐ ❧➭ ♠ë ré♥❣ tÝ❝❤ ❝❤Ð♦ ❧✐➟♥ ❦Õt ✈í✐ ❆♥♥✲❤➭♠ tư F ✳ ❚❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶ (F, F˘ , F ) ❝➯♠ s✐♥❤ ❆♥♥✲❤➭♠ tư K : ❉✐sQ → SA ❦✐Ĩ✉ ˘ H) ❧➭ ❆♥♥✲❤➭♠ tö ❝❤Ý♥❤ t➽❝ SA → A ①➳❝ ➤Þ♥❤ ❜ë✐ ➤Ý♥❤ (xs , ix )✳ ❑❤✐ ➤ã (ψ, 0)✳ ●ä✐ (H, H, t❤❡♦ ✭✶✳✶✶✮ t❛ ❝ã ˘ s,r = −ixs ⊕xr , Hs,r = −ixs ·xr H(s) = xs , H(s, b) = b, H ✶✵✵ download by : skknchat@gmail.com ❈ị♥❣ t❤❡♦ ▼Ư♥❤ ➤Ị ✶✳✶✱ (F, F˘ , F ) ❧➭ ➤å♥❣ ❧✉➞♥ ✈í✐ ❝➳✐ ❤ỵ♣ t❤➭♥❤ ❉✐sQ K H − → SA − → A, ˘ , F ) ❝❤Ý♥❤ ❧➭ ❤ỵ♣ t❤➭♥❤ ♥➭②✳ ❑❤✐ ➤ã t❤❡♦ ❝➳❝❤ ①➳❝ ➤Þ♥❤ ❝đ❛ HK, ˘ HK ❉♦ ➤ã ❝ã t❤Ó ❧✃② (F, F t❛ ❝ã F˘u,v = f (u, v) = f (u, v) − ixs +xr , ✭✺✳✷✷✮ Fu,v = g(u, v) = g (u, v) − ixs ·xr ∈ B, ✭✺✳✷✸✮ ˘ u,v , g (u, v) = Ku,v ❉♦ tÝ♥❤ t➢➡♥❣ t❤Ý❝❤ u, v ∈ Q, s = ψ(u), r = ψ(v), f (u, v) = K ˘ , F ) ✈í✐ ❝➳❝ r➭♥❣ ❜✉é❝ ❝❤➷t ❝❤Ï ❝ñ❛ ❉✐sQ ✈➭ A ♥➟♥ f, g ❧➭ ♥❤÷♥❣ ❤➭♠ ❝❤✉➮♥ t➽❝ ❝đ❛ (F, F ✈í✐ t❤á❛ ♠➲♥ ❝➳❝ ➤➻♥❣ t❤ø❝✿ ❍➭♠ f (u, v + t) + f (v, t) − f (u, v) − f (u + v, t) = 0, ✭✺✳✷✹✮ f (u, v) = f (v, u), ✭✺✳✷✺✮ θF (u) g(v, t) − g(uv, t) + g(u, vt) − g(u, v)θF (t) = 0, ✭✺✳✷✻✮ g(u, v + t) − g(u, v) − g(u, t) + θF (u) f (v, t) − f (uv, ut) = 0, ✭✺✳✷✼✮ g(u + v, t) − g(u, t) − g(v, t) + f (u, v)θF (t) − f (ut, vt) = ✭✺✳✷✽✮ ϕ : Q → MB ①➳❝ ➤Þ♥❤ ❜ë✐ ϕ(u) = θF (u) = θxs (s = ψ(u)) t❤á❛ ♠➲♥ ❝➳❝ ❤Ö t❤ø❝ ϕ(u) + ϕ(v) = µf (u,v) + ϕ(u + v), ✭✺✳✷✾✮ ϕ(u)ϕ(v) = µg(u,v) + ϕ(uv) ✭✺✳✸✵✮ ❚❤❐t ✈❐②✱ t❛ sÏ ❝❤ø♥❣ ♠✐♥❤ ❤Ư t❤ø❝ ✭✺✳✷✾✮✱ ❝ß♥ ❤Ư t❤ø❝ ✭✺✳✸✵✮ ➤➢ỵ❝ s✉② r❛ t➢➡♥❣ tù tõ ❤Ư t❤ø❝ ✭✺✳✷✸✮✳ ❉♦ t❤❡♦ ✭✺✳✷✷✮ t❤× ˘ u,v ∈ Ker d ♥➟♥ t❤❡♦ ▼Ư♥❤ ➤Ị ✺✳✶ t❛ ❝ã f (u, v) ∈ CB ❑❤✐ ➤ã f (u, v) = K µf (u,v) = µ(−ixs +xr )✳ ❇ë✐ ✈❐②✱ ϕ(u) + ϕ(v) = θxs + θxr = θxs +xr = θ[d(−ixs +xr ) + xs+r ] = θ[d(−ixs +xr )] + θxs+r ✭✺✳✷✷✮ = µ(−ixs +xr ) + ϕ(u + v) = µf (u,v) + ϕ(u + v) ❉♦ ❤ä ❝➳❝ ❤➭♠ ①➞② ❞ù♥❣ tÝ❝❤ ❝❤Ð♦ (ϕ, f, g) t❤á❛ ♠➲♥ ❝➳❝ ❤Ö t❤ø❝ ✭✺✳✷✹✮ ✲ ✭✺✳✸✵✮ ♥➟♥ t❤❡♦ ❬✷✽❪ t❛ ❝ã t❤Ó E0 = [B, ϕ, f, g, Q], ♥❣❤Ü❛ ❧➭ E0 = B × Q ❝ï♥❣ ✈í✐ ❤❛✐ ♣❤Ð♣ t♦➳♥ (b, u) + (b , u ) = (b + b + f (u, u ), u + u ), ✶✵✶ download by : skknchat@gmail.com (b, u).(b , u ) = (b.b + bϕ(u ) + ϕ(u)b + g(u, u ), uu ) E0 t❤á❛ ♠➲♥ ❝➳❝ t✐➟♥ ➤Ị ❝đ❛ ✈➭♥❤ ♥❤ê ❝➳❝ ❤Ư t❤ø❝ tr➟♥✳ ❚r♦♥❣ ➤ã✱ ➤➳♥❣ ❝❤ó ý ❧➭ tÝ♥❤ ❦Õt ❤ỵ♣ ❝đ❛ ♣❤Ð♣ ♥❤➞♥ tr♦♥❣ E0 ❝ã ➤➢ỵ❝ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❊✲❤Ư (B → D) ❧➭ ❝❤Ý♥❤ q✉✐✳ ❚❤❐t ✈❐②✱ t❛ ❝ã ❝➳❝ tÝ❝❤ [(b, u)(b , u )](b , u ) = ((bb )b + bϕ(u )ϕ(u ) + [ϕ(u)b ]ϕ(u ) + g(u, u )ϕ(u ) + ϕ(uu )b + g(uu , u ), (uu )u ), (b, u)[(b , u )(b , u )] = (b(b b ) + bϕ(u u ) + ϕ(u)[b ϕ(u )] + ϕ(u)ϕ(u )b + ϕ(u)g(u, u ) + g(u, u u ), u(u u )) ❉♦ ✭✺✳✷✻✮✱ ✭✺✳✸✵✮✱ ❧✉❐t ❦Õt ❤ỵ♣ ➤è✐ ✈í✐ tÝ❝❤ tr♦♥❣ tr♦♥❣ B, Q, ✈➭ ❧✉❐t ❣✐❛♦ ❤♦➳♥ ➤è✐ ✈í✐ tỉ♥❣ B, ➤➷❝ ❜✐Öt ❞♦ ❤Ö t❤ø❝ ✭✺✳✹✮✿ [ϕ(u)b ]ϕ(u ) = ϕ(u)[b ϕ(u )], t❛ t❤✉ ➤➢ỵ❝ ❧✉❐t ❦Õt ❤ỵ♣ ➤è✐ ✈í✐ tÝ❝❤ tr♦♥❣ E0 ❚❛ ➤➢ỵ❝ ❞➲② ❦❤í♣ ❝➳❝ ➤å♥❣ ❝✃✉ ✈➭♥❤ j0 p0 EF : → B → E0 → Q → ✈í✐ j0 (b) = (b, 0); p0 (b, u) = u, b ∈ B, u ∈ Q ❉♦ j0 (B) ❧➭ ✐➤➟❛♥ ❤❛✐ ♣❤Ý❛ tr♦♥❣ E0 ♥➟♥ j0 : B → E0 ❧➭ ♠ét ❊✲❤Ö ❝❤Ý♥❤ q✉✐ ✈í✐ θ0 : E0 → MB ❧➭ ♣❤Ð♣ ❧✃② s♦♥❣ tÝ❝❤✳ ➜å♥❣ ❝✃✉ ✈➭♥❤ ε : E0 → D ợ ị (b, u) = db + xψ(u) , (b, u) ∈ E0 , tr♦♥❣ ➤ã xψ(u) ❧➭ ➤➵✐ ❞✐Ư♥ ❝đ❛ ψ(u) tr♦♥❣ D✳ ❚❛ ❝❤ø♥❣ tá ❝➷♣ (idB , ε) t❤á❛ ♠➲♥ ❝➳❝ ❤Ö t❤ø❝ ✭✺✳✼✮✱ ✭✺✳✽✮✳ ❉Ô t❤✃② r➺♥❣ ε ◦ j0 = d✳ ❍➡♥ ♥÷❛✱ ✈í✐ ♠ä✐ (b, u) ∈ E0 , c ∈ B t❛ ❝ã θ(b,u) c = j0−1 [(b, u)(c, 0)] = bc + ϕ(u)c, θε(b,u) c = θdb+xψ(u) c = bc + ϕ(u)c ❉♦ ➤ã θ(b,u) c = θε(b,u) c✳ ❚➢➡♥❣ tù✱ cθ(b,u) = cθε(b,u) ✳ ❱× ✈❐② (idB , ε) ❧➭ ♠ét ➤å♥❣ ❝✃✉ ❝ñ❛ ❝➳❝ ❊✲❤Ư ❝❤Ý♥❤ q✉✐✱ ♥❣❤Ü❛ ❧➭ t❛ ❝ã ♠ë ré♥❣ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ✭✺✳✷✶✮✱ tr♦♥❣ ➤ã ❜ë✐ E ➤➢ỵ❝ t❤❛② E0 ✳ ❉♦ ✈í✐ ♠ä✐ u ∈ Q t❛ ❝ã qε(0, u) = q(xψ(u) ) = ψ(u) ♥➟♥ ♠ë ré♥❣ EF ❝➯♠ s✐♥❤ ψ : Q → ❈♦❦❡r d✳ ❚r♦♥❣ ❜ỉ ➤Ị tr➟♥✱ ❜é ❜❛ (ϕ, f, g) ♠➠ t➯ ♠ét ❆♥♥✲❤➭♠ tư tõ Dis Q tí✐ A ❧➭ ♠ét ❤Ư ♥❤➞♥ tư ➤è✐ ✈í✐ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ĩ✉ ❊✲❤Ư (B → D)✳ ◆ã ❧➭ ♠ë ré♥❣ ❝đ❛ ❦❤➳✐ ♥✐Ư♠ ❤Ư ♥❤➞♥ tư ➤è✐ ✈í✐ ♠ë ré♥❣ ♥❤ã♠ ❦✐Ĩ✉ ♠➠➤✉♥ ❝❤Ð♦ ➤➢ỵ❝ ①Ðt ë ❈❤➢➡♥❣ ✸ s❛♥❣ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ❤❛✐ ♣❤Ð♣ t♦➳♥✳ ➜Þ♥❤ ❧ý ❞➢í✐ ➤➞② ❧➭ ♣❤✐➟♥ ❜➯♥ ❝đ❛ ❧ý t❤✉②Õt ❙❝❤r❡✐❡r ❝❤♦ ❝➳❝ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐✳ ✶✵✷ download by : skknchat@gmail.com ➜Þ♥❤ ❧ý ✺✳✾ ✭▲ý t❤✉②Õt ❙❝❤r❡✐❡r ❝❤♦ ❝➳❝ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ó✉ ❊✲❤Ö ❝❤Ý♥❤ q✉✐✮✳ ❈ã ♠ét s♦♥❣ ➳♥❤ Ω : HomAnn (ψ,0) [DisQ, A] → ExtB→D (Q, B, ψ) ❈❤ø♥❣ ♠✐♥❤✳ ❇➢í❝ ✶✿ ❈➳❝ ❆♥♥✲❤➭♠ tư EF , EF ♠ë ré♥❣ ❧✐➟♥ ❦Õt t➢➡♥❣ ø♥❣ ●✐➯ sö (F, F˘ , F )✱ (F , F˘ , F ) ➤å♥❣ ❧✉➞♥ ❦❤✐ ✈➭ ❝❤Ø ❦❤✐ ❝➳❝ ❧➭ t➢➡♥❣ ➤➢➡♥❣ F, F : ❉✐sQ → A ❧➭ ❤❛✐ ❆♥♥✲❤➭♠ tư ➤å♥❣ ❧✉➞♥✱ ✈í✐ ➤å♥❣ ❧✉➞♥ α : F → F ✳ ❑❤✐ ➤ã✱ t❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ❝đ❛ ❆♥♥✲♠ị✐ t➟♥✱ ❝➳❝ ❜✐Ĩ✉ ➤å s❛✉ ❣✐❛♦ ❤♦➳♥ F˘u,v F (u + v) ✲ αu+v F (u) + F (v) F (uv) ✲ F˘ αu ⊗αv F (u) + F (v), ❄ F (uv) u,v ❚❤❡♦ ➤Þ♥❤ ♥❣❤Ü❛ ♣❤Ð♣ t♦➳♥ F (u)F (v) ❄ ❄ F (u + v) ✲ αuv αu +αv ❄ Fu,v ✲ F F (u)F (v) u,v ⊗ tr➟♥ A t❛ ❝ã αu ⊗ αv = αu αv + αu θF (v) + θF (u) αv ❚õ ➤ã✱ ✈í✐ f (u, v) = F˘u,v , f (u, v) = F˘ u,v , g(u, v) = Fu,v , g (u, v) = Fu,v t❛ ❝ã f (u, v) − f (u, v) = αu − αu+v + αv , ✭✺✳✸✶✮ g (u, v) − g(u, v) = αu αv + αu θF (v) + θF (u) αv − αuv ✭✺✳✸✷✮ ❇➞② ❣✐ê t❛ ➤➷t α∗ : EF → EF (b, u) → (b − αu , u) ❈❤ó ý r➺♥❣ θF (u) ợ = àu + F (u) sử ụ ❝➳❝ ❤Ö t❤ø❝ ✭✺✳✸✶✮✱ ✭✺✳✸✷✮✱ t❛ ❝❤ø♥❣ ♠✐♥❤ α∗ ❧➭ ♠ét ➤➻♥❣ ❝✃✉✳ ❍➡♥ ♥÷❛✱ ❜✐Ĩ✉ ➤å ✭✺✳✷✵✮ ❣✐❛♦ ❤♦➳♥✱ tr♦♥❣ ➤ã E ✈➭ E ❧➬♥ ❧➢ỵt ➤➢ỵ❝ t❤❛② ❜ë✐ EF ✈➭ EF ✳ ❚❛ ❝ß♥ ♣❤➯✐ ❝❤Ø r❛ ε α∗ = ε✳ ❉♦ α : F → F ❧➭ ♠ét ➤å♥❣ ❧✉➞♥ ♥➟♥ F (u) = xψ(u) = F (u)✳ ❇ë✐ ✈❐② xψ(u) = d(αu ) + xψ(u) ✱ ❤❛② d(αu ) = 0✳ ❑❤✐ ➤ã✱ ε α∗ (b, u) = ε (b − αu , u) = d(b − αu ) + xψ(u) = d(b) − d(αu ) + xψ(u) = d(b) + xψ(u) = ε(b, u) ❱❐② ❤❛✐ ♠ë ré♥❣ EF ✈➭ EF ❧➭ t➢➡♥❣ ➤➢➡♥❣✳ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ ❤❛✐ ♠ë ré♥❣ EF ✈➭ EF ❧➭ t➢➡♥❣ ➤➢➡♥❣ t❤× tå♥ t➵✐ ➤➻♥❣ ❝✃✉ ✈➭♥❤ (b, u) → (b − αu , u) tõ EF ➤Õ♥ EF ✳ ❉♦ ✈❐②✱ ❜➺♥❣ ❝➳❝❤ ❧❐♣ ❧✉❐♥ ♥❣➢ỵ❝ ❧➵✐ tõ♥❣ ❜➢í❝✱ t❛ t❤✉ ➤➢ỵ❝ ➤å♥❣ ❧✉➞♥ α:F →F ✳ ✶✵✸ download by : skknchat@gmail.com ❇➢í❝ ✷✿ ●✐➯ sư Ω ❧➭ t♦➭♥ ➳♥❤✳ E ❧➭ ♠ét ♠ë ré♥❣ E ❝ñ❛ B ❜ë✐ Q ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ (B, D, d, θ) ❝➯♠ s✐♥❤ ψ : Q → Coker d ♥❤➢ tr♦♥❣ ❜✐Ó✉ ➤å ❣✐❛♦ ❤♦➳♥ ✭✺✳✷✶✮✳ ❚❛ sÏ ❝❤ø♥❣ tá r➺♥❣ E ❝ã ♠ét ❤Ư ♥❤➞♥ tư ❧✐➟♥ ❦Õt✱ ♥❣❤Ü❛ ❧➭ t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ♠ét ♠ë ré♥❣ tÝ❝❤ ❝❤Ð♦ tư EF ❧✐➟♥ ❦Õt ✈í✐ ♠ét ❆♥♥✲❤➭♠ (F, F˘ , F ) : DisQ → A ♥➭♦ ➤ã✳ ●ä✐ A = AB→E ❧➭ ❆♥♥✲♣❤➵♠ trï ❝❤➷t ❝❤Ï ❧✐➟♥ ❦Õt ✈í✐ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ (B, E, j, θ )✳ ❑❤✐ ➤ã t❤❡♦ ▼Ư♥❤ ➤Ị ✺✳✸✱ ❝➷♣ ➤å♥❣ ❝✃✉ ➤➡♥ (idB , ε) tr♦♥❣ ❜✐Ó✉ ➤å ✭✺✳✷✶✮ ①➳❝ ➤Þ♥❤ ♠ét ❆♥♥✲❤➭♠ tư ˘ K) : A → A✳ (K, K, ❉♦ π0 A = Q, π1 A = ♥➟♥ ❆♥♥✲♣❤➵♠ trï t❤✉ ❣ä♥ SA ❝❤Ý♥❤ ❧➭ ❆♥♥✲♣❤➵♠ trï Dis Q ❚r♦♥❣ A t❛ ❝❤ä♥ ➤Ý♥❤ (eu , ie ), e ∈ E, u ∈ Q ✭♥❣❤Ü❛ ❧➭ {eu } ❧➭ ♠ét ❤Ö ➤➵✐ ❞✐Ö♥ ❝ñ❛ Q tr♦♥❣ E ✮✳ ❑❤✐ ➤ã t❤❡♦ ✭✶✳✶✶✮ ❆♥♥✲❤➭♠ tö ❝❤Ý♥❤ t➽❝ (H , H˘ , H ) : DisQ → A ➤➢ỵ❝ ❝❤♦ ❜ë✐ H (u) = eu , H˘ ❑❤✐ ➤ã ❤ỵ♣ t❤➭♥❤ u,v = −ieu +ev = g (u, v), Hu,v = −ieu ev = h (u, v) F = K ◦ H ①➳❝ ➤Þ♥❤ ♠ét ❆♥♥✲❤➭♠ tư DisQ → A✱ ✈í✐ F (u) = ε(eu ), F˘u,v = H˘ u,v = g (u, v), Fu,v = Hu,v = h (u, v) ❚❤❡♦ ♣❤Ð♣ ❝❤ø♥❣ ♠✐♥❤ ị ý t ị ợ rộ EF ❝ñ❛ tÝ❝❤ ❝❤Ð♦ E0 = [B, ϕ, g , h , Q]✱ ❧✐➟♥ ❦Õt ✈í✐ (F, F˘ , F )✳ ❇➞② ❣✐ê t❛ ❝❤ø♥❣ tá E ✈➭ EF t➢➡♥❣ ➤➢➡♥❣✱ tø❝ ❧➭ ❝ã ❜✐Ó✉ ➤å s❛✉ ❣✐❛♦ ❤♦➳♥ / EF : B j0 / E0 p0 / Q / 0, E0 p / Q / 0, E ε0 / D ε / D α / E: ✈➭ B j  /E = ỗ tư ❝đ❛ E ✈✐Õt ➤➢ỵ❝ ❞✉② ♥❤✃t ❞➢í✐ ❞➵♥❣ b + eu , b ∈ B, ♥➟♥ t❛ ❝ã t❤Ó ①➳❝ ➤Þ♥❤ ♠ét ➳♥❤ ①➵ α : E0 → E, (b, u) → b + eu ➜Ó ❝❤Ø r❛ α ❧➭ ♠ét ➤å♥❣ ❝✃✉ ✈➭♥❤✱ tr➢í❝ ❤Õt t❛ t❤✃② r➺♥❣ ❤Ö ➤➵✐ ❞✐Ö♥ {eu } ❝ã ♠ét sè tÝ♥❤ ❝❤✃t s❛✉✿ ϕ(u)c = θeu (c), cϕ(u) = cθeu , c ∈ B, ✭✺✳✸✸✮ eu + ev = −ieu +ev + eu+v = g (u, v) + eu+v , ✭✺✳✸✹✮ eu ev = −ieu ev + eu.v = h (u, v) + euv ✭✺✳✸✺✮ ✭➜➻♥❣ t❤ø❝ ✭✺✳✸✸✮ ❝ã ➤➢ỵ❝ ❞♦ ❝➷♣ (idB , ε) ❧➭ ➤å♥❣ ❝✃✉ ♠➠➤✉♥ ❝❤Ð♦✳ tứ ó ợ ị ĩ ủ ♠ị✐ t➟♥ tr♦♥❣ A ✮✳ ❱× ✈❐②✱ t❛ ❝ã α[(b, u) + (c, v)] = α(b + c + g (u, v), u + v) = b + c + g (u, v) + eu+v (5.34) = b + c + eu + ev = (b + eu ) + (c + ev ) = α(b, u) + α(c, v) ✶✵✹ download by : skknchat@gmail.com α[(b, u)(c, v)] = α(bc + bϕ(v) + ϕ(u)c + h (u, v), uv) = bc + bϕ(v) + ϕ(u)c + h (u, v) + euv (5.33),(5.35) bc + bθev + θeu c + eu ev = = bc + b.ev + eu c + eu ev = (b + eu ).(c + ev ) = α(b, u).α(c, v) ❈✉è✐ ❝ï♥❣✱ t❛ ❝❤ä♥ ➤➵✐ ❞✐Ö♥ eu s❛♦ ❝❤♦ ✭✺✳✷✶✮ t❛ ❝ã ε(eu ) = xψ(u) ✳ ➜✐Ị✉ ♥➭② ❝ã t❤Ĩ t❤ù❝ ❤✐Ư♥ ➤➢ỵ❝ ❞♦ tõ q(ε(eu )) = ψp(eu ) = ψ(u)✳ ❑❤✐ ➤ã✿ εα(b, u) = ε(b + eu ) = ε(b) + ε(eu ) = d(b) + xψ(u) = ε0 (b, u), ♥❣❤Ü❛ ❧➭ E ✈➭ EF ❧➭ ❤❛✐ ♠ë ré♥❣ t➢➡♥❣ ➤➢➡♥❣✳ ●✐➯ sö A = AB→D ❧➭ ❆♥♥✲♣❤➵♠ trï ❝❤➷t ❝❤Ï ❧✐➟♥ ❦Õt ✈í✐ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ B → D✳ ❉♦ π0 A = Coker d ✈➭ π1 A = Ker d ♥➟♥ ❆♥♥✲♣❤➵♠ trï t❤✉ ❣ä♥ SA ❝ã ❞➵♥❣ SA = (Cokerd, Kerd, k), tr♦♥❣ ➤ã ➤å♥❣ ❝✃✉ k ∈ HShu (Cokerd, Kerd) ❞♦ A ✈➭ SA ❧➭ ♥❤÷♥❣ ❆♥♥✲♣❤➵♠ trï ❝❤Ý♥❤ q✉✐✳ ❑❤✐ ➤ã ψ : Q → Cokerd ❝➯♠ s✐♥❤ ♠ét ❝➯♥ trë ψ ∗ k ∈ HShu (Q, Kerd) ❉➢í✐ ➤➞②✱ ❝❤ó♥❣ t➠✐ ♣❤➳t ❜✐Ĩ✉ ❦Õt q✉➯ ❝❤Ý♥❤ ❝đ❛ ♠ơ❝ ♥➭②✱ ❧➭ ♠ét ♠ë ré♥❣ ủ ị ý ữ ỗ ột í q✉✐ ❦❤✐ ∂ ✲♠ë ré♥❣ ➤➢ỵ❝ ①Ðt tr♦♥❣ ❬✻❪ t❤ù❝ r❛ ❧➭ tr➢ê♥❣ ❤ỵ♣ r✐➟♥❣ ❝đ❛ ❊✲❤Ư Q = Cokerd, ✈➭ ψ = idCokerd ♥➟♥ ❦Õt q✉➯ ❝đ❛ ❝❤ó♥❣ t➠✐ ❧➭ ❝❤ø❛ ➜Þ♥❤ ❧ý ✹✳✹✳✷ ❬✻❪✳ ➜Þ♥❤ ❧ý ✺✳✶✵✳ ❈❤♦ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ sù tr✐Ưt t✐➟✉ ❝đ❛ ❝đ❛ B ❜ë✐ Q✱ ψ∗k tr♦♥❣ ❦✐Ĩ✉ ❊✲❤Ư (B, D, d, θ) ✈➭ ➤å♥❣ ❝✃✉ ✈➭♥❤ ψ : Q → Cokerd✳ ❑❤✐ ➤ã HShu (Q, Kerd) B →D ❧➭ ➤✐Ị✉ ❦✐Ư♥ ❝➬♥ ✈➭ ➤đ ➤Ĩ tå♥ t➵✐ ♠ë ré♥❣ ✈➭♥❤ ❝➯♠ s✐♥❤ ữ k t tì tồ t ♠ét s♦♥❣ ➳♥❤ ExtB→D (Q, B, ψ) ↔ HShu (Q, Kerd) k = tì t ị ❧ý ✶✳✷ tå♥ t➵✐ ♠ét ❆♥♥✲❤➭♠ tö (Ψ, Ψ) : Dis Q → ˘ H) : SA → A t❛ ➤➢ỵ❝ ♠ét ❆♥♥✲❤➭♠ SA ✳ ▲✃② ❤ỵ♣ t❤➭♥❤ ✈í✐ ❆♥♥✲❤➭♠ tö ❝❤Ý♥❤ t➽❝ (H, H, ˘ , F ) : Dis Q → A✱ ✈➭ t❤❡♦ ❇ỉ ➤Ị ✺✳✽ t❤✉ ➤➢ỵ❝ ♠ë ré♥❣ ❧✐➟♥ ❦Õt EF ✳ tư (F, F ❈❤ø♥❣ ♠✐♥❤✳ ◆Õ✉ ◆❣➢ỵ❝ ❧➵✐✱ ❣✐➯ sư ❝ã ♠ë ré♥❣ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ t❤á❛ ♠➲♥ ❜✐Ĩ✉ ➤å ✭✺✳✷✶✮✳ ●ä✐ ❆♥♥✲♣❤➵♠ trï ❝❤➷t ❝❤Ï ❧✐➟♥ ❦Õt ✈í✐ ❊✲❤Ư A ❧➭ B → E ✳ ❚❤Õ t❤× t❤❡♦ ▼Ư♥❤ ➤Ị ✺✳✸✱ tå♥ t➵✐ ♠ét ✶✵✺ download by : skknchat@gmail.com ❆♥♥✲❤➭♠ tö ➤Ị ✶✳✶✱ F : A → A✳ ❇ë✐ ✈× ❆♥♥✲♣❤➵♠ trï t❤✉ ❣ä♥ ❝ñ❛ A ❧➭ Dis Q ♥➟♥ t❤❡♦ ▼Ư♥❤ F ❝➯♠ s✐♥❤ ♠ét ❆♥♥✲❤➭♠ tư ❦✐Ĩ✉ (ψ, 0) tõ Dis Q tí✐ (Coker d, Ker d, k)✳ ❇➞② ❣✐ê✱ t❤❡♦ ➜Þ♥❤ ❧ý ✶✳✷✱ ❝➳✐ ❝➯♥ trë ❝đ❛ ❝➷♣ (Q, Ker d), ♥❣❤Ü❛ ❧➭ (ψ, 0) ♣❤➯✐ tr✐Öt t✐➟✉ tr♦♥❣ HShu ψ ∗ k = 0✳ ❇➞② ❣✐ê✱ s ó tr ị ý ợ s r s❛✉✳ ❚r➢í❝ ❤Õt✱ ❝ã ♠ét s♦♥❣ ➳♥❤ tù ♥❤✐➟♥ Hom[DisQ, A] ↔ Hom[DisQ, SA ] ❚õ ➤ã ❞♦ π0 (DisQ) = Q, π1 (SA ) = Kerd ♥➟♥ t❤❡♦ ➜Þ♥❤ ❧ý ✺✳✾ ✈➭ ➜Þ♥❤ ❧ý ✶✳✷ t❛ ❝ã s♦♥❣ ➳♥❤ ExtB→D (Q, B, ψ) ↔ HShu (Q, Kerd) ❑Õt ❧✉❐♥ ❝đ❛ ❈❤➢➡♥❣ ✺ ❚r♦♥❣ ❝❤➢➡♥❣ ♥➭② ❝❤ó♥❣ t➠✐ ➤➲ t ợ ột số ết q í s ã ➜➢❛ r❛ ❦❤➳✐ ♥✐Ö♠ ❊✲❤Ö✱ ❊✲❤Ö ❝❤Ý♥❤ q✉✐✱ ❝❤Ø r❛ sù t➢➡♥❣ ➤➢➡♥❣ ❣✐÷❛ ❝➳❝ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ✈➭ ❝➳❝ s♦♥❣ ♠➠➤✉♥ ❝❤Ð♦ tr➟♥ ✈➭♥❤✳ • ❇✐Ĩ✉ ❞✐Ơ♥ ❊✲❤Ư ❝❤Ý♥❤ q q ữ ủ trù t ẽ ã P❤➳t ❜✐Ĩ✉ ♠è✐ ❧✐➟♥ ❤Ư ❣✐÷❛ ❝➳❝ ➤å♥❣ ❝✃✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ✈➭ ❝➳❝ ❆♥♥✲❤➭♠ tư ❣✐÷❛ ❝➳❝ ❆♥♥✲♣❤➵♠ trï ❧✐➟♥ ❦Õt✳ • P❤➞♥ ❧í♣ ❝➳❝ ❊✲❤Ư ❝❤Ý♥❤ q✉✐✳ • P❤➳t ❜✐Ó✉ ✈➭ ❣✐➯✐ ❜➭✐ t♦➳♥ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ó✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ♥❤ê ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❧ý t❤✉②Õt ❆♥♥✲♣❤➵♠ trï✳ ✶✵✻ download by : skknchat@gmail.com ❑Õt ❧✉❐♥ ▼➠➤✉♥ ❝❤Ð♦ ✈➭ ♥❤ã♠ ♣❤➵♠ trï✱ ♠ét ❝➳❝❤ ➤é❝ ❧❐♣✱ ➤➲ ➤➢ỵ❝ sư ❞ơ♥❣ ré♥❣ r➲✐ tr♦♥❣ ♥❤✐Ị✉ ❦❤✉♥❣ ❝➯♥❤ ❦❤➳❝ ♥❤❛✉✳ ❈➳❝ ❦Õt q✉➯ ✈Ị ♥❤ã♠ ♣❤➵♠ trï ❝đ❛ ❍✳ ❳✳ ❙Ý♥❤ ✭✶✾✼✺✮ ➤➲ ➤➢ỵ❝ ♥➞♥❣ ❧➟♥ ❝❤♦ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❜ë✐ ❆✳ ▼✳ ❈❡❣❛rr❛ ✈➭ ❝➳❝ ❝é♥❣ sù✱ ✈➭ ❝❤♦ tr➢ê♥❣ ❤ỵ♣ ✈➭♥❤ ♣❤➵♠ trï ✭❤❛② ❆♥♥✲♣❤➵♠ trï✮ ❜ë✐ ◆✳ ❚✳ ◗✉❛♥❣✳ ❇➟♥ ❝➵♥❤ ➤ã✱ ❘✳ ❇r♦✇♥ ✈➭ ❈✳ ❙♣❡♥❝❡r ✭✶✾✼✻✮ ➤➲ ❝❤Ø r❛ r➺♥❣ ♠➠➤✉♥ ❝❤Ð♦ ❝ã t❤Ĩ ➤➢ỵ❝ ♥❣❤✐➟♥ ❝ø✉ ❜ë✐ ❝➳❝ ♥❤ã♠ ♣❤➵♠ trï ❝❤➷t ❝❤Ï✳ ➜✐Ị✉ ♥➭② ❣ỵ✐ ý ❝❤♦ ❝❤ó♥❣ t➠✐ r➺♥❣ ❝ã t❤Ĩ ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❧í♣ ♣❤➵♠ trï ♣❤ø❝ t➵♣ ❤➡♥ ♥❤➢✿ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❝❤➷t ❝❤Ï✱ ❆♥♥✲♣❤➵♠ trï ❝❤➷t ❝❤Ï✱ ➤Ó tõ ➤ã ♥❣❤✐➟♥ ❝ø✉ ❝➳❝ ❝✃✉ tró❝ ❣➬♥ ✈í✐ ♠➠➤✉♥ ❝❤Ð♦ ♥❤➢✿ ♠➠➤✉♥ ❝❤Ð♦ ➤➻♥❣ ❜✐Õ♥✱ ❊✲❤Ư✳ ▲✉❐♥ ➳♥ ➤➲ ❣✐➯✐ q✉②Õt ✈✃♥ ➤Ị ♥➭② ✈í✐ ữ ết q í s ị ể ❝đ❛ ♠ét ❤➭♠ tư ♠♦♥♦✐❞❛❧ ❣✐÷❛ ❤❛✐ ♥❤ã♠ ♣❤➵♠ trï ✈➭ ❧ý t❤✉②Õt ❝➯♥ trë ❝đ❛ ♠ét ❤➭♠ tư✳ ❚õ ➤ã ➤➢❛ r❛ ➤Þ♥❤ ❧ý ♣❤➞♥ ❧í♣ ❝❤Ý♥❤ ①➳❝ ❝❤♦ ♣❤➵♠ trï ❝➳❝ ♥❤ã♠ ♣❤➵♠ trï ✈➭ ♣❤➵♠ trï ❝➳❝ ♥❤ã♠ ♣❤➵♠ trï ❜Ư♥✳ ✷✳ P❤➞♥ ❧í♣ ❝➳❝ ♠➠➤✉♥ ❝❤Ð♦ ✈➭ ①➞② ❞ù♥❣ ❧ý t❤✉②Õt ❙❝❤r❡✐❡r ❝❤♦ ❝➳❝ ♠ë ré♥❣ ♥❤ã♠ ❦✐Ĩ✉ ♠➠➤✉♥ ❝❤Ð♦ ❞ù❛ tr➟♥ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❧ý t❤✉②Õt ♥❤ã♠ ♣❤➵♠ trï ❝❤➷t ❝❤Ï✳ ❈➳❝ ❦Õt q✉➯ t❤✉ ➤➢ỵ❝ ❧➭ ♠ë ré♥❣ ❝➳❝ ❦Õt q✉➯ ❝đ❛ ❘✳ ❇r♦✇♥ ✈➭ ❝➳❝ ➤å♥❣ t➳❝ ❣✐➯✳ ✸✳ ◆❣❤✐➟♥ ❝ø✉ ♥❤ã♠ ♣❤➵♠ trï ♣❤➞♥ ❜❐❝ ❝❤➷t ❝❤Ï✱ ✈➭ tõ ➤ã ♣❤➞♥ ❧í♣ ❝➳❝ ✈➭ ①➞② ❞ù♥❣ ❧ý t❤✉②Õt ❙❝❤r❡✐❡r ❝❤♦ ❝➳❝ ♠ë ré♥❣ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ ❦✐Ó✉ Γ✲♠➠➤✉♥ ❝❤Ð♦ Γ✲♠➠➤✉♥ ❝❤Ð♦✳ ❈➳❝ ❦Õt q✉➯ t❤✉ ➤➢ỵ❝ ❧➭ ❜❛♦ ❤➭♠ ❧ý t❤✉②Õt ♠ë ré♥❣ ♥❤ã♠ ➤➻♥❣ ❜✐Õ♥ ❝ñ❛ ❆✳ ▼✳ ❈❡❣❛rr❛ ✲ ❏✳ ▼✳ ●❛r❝Ý❛✲❈❛❧❝✐♥❡s ✲ ❏✳ ❆✳ ❖rt❡❣❛ ✈➭ ❧ý t❤✉②Õt ♠ë ré♥❣ ♥❤ã♠ ❦✐Ĩ✉ ♠➠➤✉♥ ❝❤Ð♦ ❝đ❛ ❘✳ ❇r♦✇♥ ✲ ❖✳ ▼✉❝✉❦✳ ✹✳ ◆❣❤✐➟♥ ❝ø✉ ❆♥♥✲♣❤➵♠ trï ❝❤➷t ❝❤Ï✱ tõ ➤ã ♣❤➞♥ ❧í♣ ❝➳❝ ❊✲❤Ư ❝❤Ý♥❤ q✉✐ ✈➭ ❝➳❝ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐✳ ✺✳ P❤➞♥ ❧í♣ ♣❤➵♠ trï ❝➳❝ ♥❤ã♠ ♣❤➵♠ trï ❜Ư♥ ❤Ư tư tr♦♥❣ ♥❤ã♠ ♣❤➵♠ trï ❜Ư♥ ❦✐Ĩ✉ Γ✲♣❤➞♥ ❜❐❝ ♥❤ê ❝➳❝ ❤Ư ♥❤➞♥ tư tr➟♥ Γ ✈í✐ (M, N )✳ ✶✵✼ download by : skknchat@gmail.com ❉❛♥❤ ♠ơ❝ ❝➳❝ ❝➠♥❣ tr×♥❤ ❝đ❛ t➳❝ ❣✐➯ ❝ã ❧✐➟♥ q✉❛♥ ➤Õ♥ ❧✉❐♥ ➳♥ ✶✳ ◆✳ ❚✳ ◗✉❛♥❣✱ ◆✳ ❚✳ ❚❤✉②✱ P✳ ❚✳ ❈✉❝✱ ▼♦♥♦✐❞❛❧ ❢✉♥❝t♦rs ❜❡t✇❡❡♥ ✭❜r❛✐❞❡❞✮ ●r✲❝❛t❡❣♦r✐❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✱ ❊❛st✲❲❡st ❏✳ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✸✱ ◆♦ ✷ ✭✷✵✶✶✮✱ ✶✻✸✲✶✽✻✳ ✷✳ ◆✳ ❚✳ ◗✉❛♥❣✱ P✳ ❚✳ ❈✉❝✱ ❈r♦ss❡❞ ❜✐♠♦❞✉❧❡s ♦✈❡r r✐♥❣s ❛♥❞ ❙❤✉❦❧❛ ❝♦❤♦♠♦❧♦❣②✱ ▼❛t❤✳ ❈♦♠♠✉♥✳✱ ✶✼ ◆♦✳ ✷ ✭✷✵✶✷✮✱ ✺✼✺✲✺✾✽✳ ✸✳ ◆✳ ❚✳ ◗✉❛♥❣✱ P✳ ❚✳ ❈✉❝✱ ❈❧❛ss✐❢✐❝❛t✐♦♥ ♦❢ ❣r❛❞❡❞ ❜r❛✐❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣s ❜② ♣s❡✉❞♦✲ ❢✉♥❝t♦rs✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡✱ ❍✉❡ ❯♥✐✈❡rs✐t②✱ ❱♦❧✳ ✼✼✱ ◆♦✳ ✽ ✭✷✵✶✷✮✱ ✺✾✲✻✽✳ ✹✳ ◆✳ ❚✳ ◗✉❛♥❣✱ P✳ ❚✳ ❈✉❝✱ ◆✳ ❚✳ ❚❤✉②✱ ❈r♦ss❡❞ ♠♦❞✉❧❡s ❛♥❞ str✐❝t ●r✲❝❛t❡❣♦r✐❡s✱ ❈♦♠♠✉✲ ♥✐❝❛t✐♦♥s ♦❢ ❑♦r❡❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱♦❧✳ ✷✾ ✭✷✵✶✹✮✱ ◆♦✳✶✱ ✾✲✷✷✳ ✺✳ ◆✳ ❚✳ ◗✉❛♥❣✱ P✳ ❚✳ ❈✉❝✱ ❊q✉✐✈❛r✐❛♥t ❝r♦ss❡❞ ♠♦❞✉❧❡s ❛♥❞ ❝♦❤♦♠♦❧♦❣② ♦❢ ❣r♦✉♣s ✇✐t❤ ♦♣❡r❛t♦rs✱ ❛r❳✐✈✿ ✶✸✵✷✳✹✺✼✸✈✶ ❬♠❛t❤✳❈❚❪ ✶✾ ❋❡❜ ✷✵✶✸✳ ❈➳❝ ❦Õt q✉➯ tr♦♥❣ ❧✉❐♥ ➳♥ ➤➲ ➤➢ỵ❝ ❜➳♦ ❝➳♦ ✈➭ t❤➯♦ ❧✉❐♥ t➵✐✿ ✲ ❍é✐ t❤➯♦ ❦❤♦❛ ❤ä❝ ✈Ò ▼ét sè ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ♠í✐ tr♦♥❣ t♦➳♥ ❤ä❝ ❤✐Ư♥ ➤➵✐ ✈➭ ø♥❣ ❞ô♥❣✱ ❚❤❛♥❤ ❍ã❛ ✭❞♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❍å♥❣ ➜ø❝✱ ❱✐Ö♥ ❚♦➳♥ ❤ä❝ ❱✐Öt ◆❛♠ ✈➭ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠ ❍➭ ◆é✐ ♣❤è✐ ❤ỵ♣ tỉ ❝❤ø❝✮✱ ✺✴✷✵✶✶✳ ✲ ộ ị t qố ề số ì ọ ✲ ❚➠♣➠✱ ❚❤➳✐ ◆❣✉②➟♥✱ ✶✶✴✷✵✶✶✳ ✲ ➜➵✐ ❤é✐ ❚♦➳♥ ❤ä❝ ❱✐Öt ◆❛♠ ❧➬♥ t❤ø ✽✱ ◆❤❛ ❚r❛♥❣✱ ✽✴✷✵✶✸✳ ✲ ❳❡♠✐♥❛r ❇é ♠➠♥ ➜➵✐ sè ✲ ❍×♥❤ ❤ä❝✱ ❑❤♦❛ ❚♦➳♥✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ s➢ ♣❤➵♠✱ ➜➵✐ ❤ä❝ ❍✉Õ✳ ✲ ❳❡♠✐♥❛r ❇é ♠➠♥ ➜➵✐ sè✱ ❑❤♦❛ ❑❤♦❛ ❤ä❝ tù ♥❤✐➟♥✱ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❍å♥❣ ➜ø❝✱ ❚❤❛♥❤ ❍ã❛✳ ✶✵✽ download by : skknchat@gmail.com ❚➭✐ ❧✐Ö✉ t❤❛♠ ❦❤➯♦ ❬✶❪ ◆✳ ❚✳ ◗✉❛♥❣✱ ❆♥♥✲♣❤➵♠ trï✱ ▲✉❐♥ ➳♥ ❚✐Õ♥ sÜ✱ ✶✾✽✽✱ ❍➭ ◆é✐✳ ❚✐Õ♥❣ ❆♥❤ ❬✷❪ rs r ă s O ❚❤r❡❡✲❝r♦ss❡❞ ♠♦❞✉❧❡s✱ ❍♦♠♦❧♦❣② ❍♦♠♦t♦♣② ❛♥❞ ❆♣♣❧✳✱ ✷ ✭✷✵✵✾✮✱ ✶✻✶✲✶✽✼✳ ❬✸❪ ❏✳ ❈✳ ❇❛❡③✱ ❆✳ ❉✳ ▲❛✉❞❛✱ ❍✐❣❤❡r ❞✐♠❡♥s✐♦♥❛❧ ❛❧❣❡❜r❛ ❱✿ ✷✲❣r♦✉♣s✱ ❚❤❡♦r② ❆♣♣❧✳ ❈❛t❡❣✳✱ ✶✷ ✭✷✵✵✹✮✱ ✹✷✸✲✹✾✶✳ ❬✹❪ ❍✳✲❏✳ ❇❛✉❡s✱ ❙❡❝♦♥❞❛r② ❝♦❤♦♠♦❧♦❣② ❛♥❞ t❤❡ ❙t❡❡♥r♦❞ sq✉❛r❡✱ ❍♦♠♦❧♦❣② ❍♦♠♦t♦♣② ❆♣♣❧✳✱ ✹ ✭✷✵✵✷✮✱ ◆♦✳ ✷✱ ♣❛rt ✶✱ ✷✾✲✻✷✳ ❬✺❪ ❍✳✲❏✳ ❇❛✉❡s✱ ❊✳ ●✳ ▼✐♥✐❛♥✱ ❈r♦ss❡❞ ❡①t❡♥s✐♦♥s ♦❢ ❛❧❣❡❜r❛s ❛♥❞ ❍♦❝❤s❝❤✐❧❞ ❝♦❤♦♠♦❧♦❣②✱ ❍♦♠♦❧♦❣② ❍♦♠♦t♦♣② ❆♣♣❧✳✱ ✹ ✭✷✵✵✷✮✱ ◆♦✳ ✷✱ ✻✸✲✽✷✳ ❬✻❪ ❍✲❏✳ ❇❛✉❡s✱ ❚✳ P✐r❛s❤✈✐❧✐✱ ❙❤✉❦❧❛ ❝♦❤♦♠♦❧♦❣② ❛♥❞ ❛❞❞✐t✐✈❡ tr❛❝❦ t❤❡♦r✐❡s✱ ❛r❳✐✈✿✵✹✵✶✶✺✽✈✶ ❬♠❛t❤✳❈❚❪ ✶✹ ❏❛♥ ✷✵✵✹✳ ❬✼❪ ❘✳ ❇r♦✇♥✱ ●r♦✉♣♦✐❞s ❛♥❞ ❝r♦ss❡❞ ♦❜❥❡❝ts ✐♥ ❛❧❣❡❜r❛✐❝ t♦♣♦❧♦❣②✱ ❍♦♠♦❧♦❣② ❍♦♠♦t♦♣② ❛♥❞ ❆♣♣❧✐❡❞✱ ✶ ✭✶✮ ✭✶✾✾✾✮✱ ✶✲✼✽✳ ❬✽❪ ❘✳ ❇r♦✇♥✱ ❈✳ ❙♣❡♥❝❡r✱ ♦❢ ❛ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣✱ G ✲❣r♦✉♣♦✐❞s✱ ❝r♦ss❡❞ ♠♦❞✉❧❡s ❛♥❞ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❣r♦✉♣♦✐❞ Pr♦❝✳ ❑♦♥♥✳ ◆❡❞✳ ❆❦❛❞✳ ✈✳ ❲❡t✳✱ ✼✾ ✭✶✾✼✻✮✱ ✷✾✻ ✲ ✸✵✷✳ ❬✾❪ ❘✳ ❇r♦✇♥✱ ❖✳ ▼✉❝✉❦✱ ❈♦✈❡r✐♥❣ ❣r♦✉♣s ♦❢ ♥♦♥✲❝♦♥♥❡❝t❡❞ t♦♣♦❧♦❣✐❝❛❧ ❣r♦✉♣s r❡✈✐s✐t❡❞✱ ▼❛t❤✳ Pr♦❝✳ ❈❛♠❜✳ P❤✐❧✳ ❙♦❝✳✱ ✶✶✺ ✭✶✾✾✹✮✱ ✾✼✲✶✶✵✳ ❬✶✵❪ ❘✳ ❇r♦✇♥✱ ❚✳ P♦rt❡r✱ ❖♥ t❤❡ ❙❝❤r❡✐❡r t❤❡♦r② ♦❢ ♥♦♥❛❜❡❧✐❛♥ ❡①t❡♥s✐♦♥s✿ ❣❡♥❡r❛❧✐s❛t✐♦♥s ❛♥❞ ❝♦♠♣✉t❛t✐♦♥s✱ Pr♦❝❡❡❞✐♥❣ ❘♦②❛❧ ■r✐s❤ ❆❝❛❞❡♠②✱ ✾✻❆ ✭✶✾✾✻✮✱ ✷✶✸✲✷✷✼✳ ❬✶✶❪ ❘✳ ❇r♦✇♥✱ P✳ ❏✳ ❍✐❣❣✐♥s✱ ❘✳ ❙✐✈❡r❛✱ ◆♦♥❛❜❡❧✐❛♥ ❛❧❣❡❜r❛✐❝ t♦♣♦❧♦❣②✿ ❢✐❧t❡r❡❞ s♣❛❝❡s✱ ❝r♦ss❡❞ ❝♦♠♣❧❡①❡s✱ ❝✉❜✐❝❛❧ ❤♦♠♦t♦♣② ❣r♦✉♣♦✐❞s✱ ❊▼❙ ❚r❛❝ts ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ✶✺✱ ✼✵✸ ♣❛❣❡s ✭❆✉❣✉st ✷✵✶✶✮✳ ❬✶✷❪ ▼✳ ❈❛❧✈♦✱ ❆✳ ▼✳ ❈❡❣❛rr❛✱ ◆✳ ❚✳ ◗✉❛♥❣✱ ❍✐❣❤❡r ❝♦❤♦♠♦❧♦❣✐❡s ♦❢ ♠♦❞✉❧❡s✱ ❆❧❣❡❜r❛✐❝ ✫ ●❡♦♠❡tr✐❝ ❚♦♣♦❧♦❣② ✶✷ ✭✷✵✶✷✮ ✸✹✸ ✲ ✹✶✸✳ ✶✵✾ download by : skknchat@gmail.com ❬✶✸❪ ❆✳ ▼✳ ❈❡❣❛rr❛✱ ❆✳ ❘✳ ●❛r③ã♥✱ ❏✳ ❆✳ ❖rt❡❣❛✱ ●r❛❞❡❞ ❡①t❡♥s✐♦♥s ♦❢ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r✐❡s✱ ❏✳ ❆❧❣❡❜r❛✱ ✷✹✶ ✭✷✮ ✭✷✵✵✶✮✱ ✻✷✵✲✻✺✼✳ ❬✶✹❪ ❆✳ ▼✳ ❈❡❣❛rr❛✱ ❏✳ ▼✳ ●❛r❝Ý❛ ✲ ❈❛❧❝✐♥❡s✱ ❏✳ ❆✳ ❖rt❡❣❛✱ ❖♥ ❣r❛❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣s ❛♥❞ ❡q✉✐✈❛r✐❛♥t ❣r♦✉♣ ❡①t❡♥s✐♦♥s✱ ❈❛♥❛❞✳ ❏✳ ▼❛t❤✳✱ ✺✹ ✭✺✮ ✭✷✵✵✷✮✱ ✾✼✵✲✾✾✼✳ ❬✶✺❪ ❆✳ ▼✳ ❈❡❣❛rr❛✱ ❏✳ ▼✳ ●❛r❝Ý❛ ✲ ❈❛❧❝✐♥❡s✱ ❏✳ ❆✳ ❖rt❡❣❛✱ ❈♦❤♦♠♦❧♦❣② ♦❢ ❣r♦✉♣s ✇✐t❤ ♦♣❡r❛t♦rs✱ ❍♦♠♦❧♦❣② ❍♦♠♦t♦♣② ❆♣♣❧✳ ❱♦❧✳ ✶✱ ◆♦✳ ✹ ✭✷✵✵✷✮✱ ✶✲✷✸✳ ❬✶✻❪ ❆✳ ▼✳ ❈❡❣❛rr❛✱ ❆✳ ❘✳ ●❛r③ã♥✱ ❙♦♠❡ ❛❧❣❡❜r❛✐❝ ❛♣♣❧✐❝❛t✐♦♥s ♦❢ ❣r❛❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣ t❤❡♦r②✱ ❚❤❡♦r② ❛♥❞ ❆♣♣❧✐❝❛t✐♦♥s ♦❢ ❈❛t❡❣♦r✐❡s✱ ✶✶ ✭✶✵✮ ✭✷✵✵✸✮✱ ✷✶✺✲✷✺✶✳ ❬✶✼❪ ❆✳ ▼✳ ❈❡❣❛rr❛✱ ❊✳ ❑❤♠❛❧❛❞③❡✱ ❍♦♠♦t♦♣② ❝❧❛ss✐❢✐❝❛t✐♦♥ ♦❢ ❜r❛✐❞❡❞ ❣r❛❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣s✱ ❏✳ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ❆❧❣❡❜r❛✱ ✷✵✾ ✭✷✵✵✼✮✱ ✹✶✶✲✹✸✼✳ ❬✶✽❪ ❆✳ ▼✳ ❈❡❣❛rr❛✱ ❊✳ ❑❤♠❛❧❛❞③❡✱ ❍♦♠♦t♦♣② ❝❧❛ss✐❢✐❝❛t✐♦♥ ♦❢ ❣r❛❞❡❞ P✐❝❛r❞ ❝❛t❡❣♦r✐❡s✱ ❆❞✈✳ ▼❛t❤✳✱ ✷✶✸ ✭✷✵✵✼✮✱ ✻✹✹✲✻✽✻✳ ❬✶✾❪ ❙✳ ❊✐❧❡♥❜❡r❣✱ ❙✳ ▼❛❝▲❛♥❡✱ ❖♥ t❤❡ ❣r♦✉♣s H(Π, n) ■✱ ■■✱ ❆♥♥✳ ♦❢ ▼❛t❤✳ ✺✽ ✭✶✾✺✸✮ ✺✺✲ ✶✵✻✱ ro ă ❲❛❧❧✱ ●r❛❞❡❞ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r✐❡s✱ ❈♦♠♣♦s✐t✐♦ ▼❛t❤❡♠❛t✐❝❛✱ ✷✽✱ ◆♦✳ ✸ ✭✶✾✼✹✮✱ ✷✷✾✲✷✽✺✳ ❬✷✶❪ ❆✳ ❏♦②❛❧✱ ❘✳ ❙tr❡❡t✱ ❇r❛✐❞❡❞ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r✐❡s✱ ▼❛❝q✉❛r✐❡ ▼❛t❤❡♠❛t✐❝s ❘❡♣♦rt ◆♦✳ ✽✻✵✵✽✶✱ ◆♦✈❡♠❜❡r ✶✾✽✻✳ ❬✷✷❪ ❆✳ ❏♦②❛❧✱ ❘✳ ❙tr❡❡t✱ ❇r❛✐❞❡❞ t❡♥s♦r ❝❛t❡❣♦r✐❡s✱ ❆❞✈✳ ▼❛t❤✳✱ ❱♦❧✳ ✷✱ ◆♦✳ ✶ ✭✶✾✾✸✮✱ ✷✵✲✼✽✳ ❬✷✸❪ ●✳ ▼✳ ❑❡❧❧②✱ ❚❡♥s♦r ♣r♦❞✉❝ts ✐♥ ❝❛t❡❣♦r✐❡s✳ ❏✳ ❆❧❣❡❜r❛✱ ✷✱ ◆♦✳✶ ✭✶✾✻✺✮✱ ✶✺✲✸✼✳ ❬✷✹❪ ▼✳ ▲✳ ▲❛♣❧❛③❛✱ ❈♦❤❡r❡♥❝❡ ❢♦r ❝❛t❡❣♦r✐❡s ✇✐t❤ ❣r♦✉♣ str✉❝t✉r❡✿ ❛♥ ❛❧t❡r♥❛t✐✈❡ ❛♣✲ ♣r♦❛❝❤✱ ❏✳ ❆❧❣❡❜r❛✱ ✽✹ ✭✶✾✽✸✮✱ ✸✵✺✲✸✷✸✳ ❬✷✺❪ ❙✳ ▼❛❝▲❛♥❡✱ ❈♦❤♦♠♦❧♦❣② t❤❡♦r② ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✱ Pr♦❝✳ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❣r❡ss ♦❢ ▼❛t❤❡♠❛t✐❝✐❛♥s✱ ❱♦❧✳ ■■ ✭✶✾✺✵✮✱ ✽✲✶✹✳ ❬✷✻❪ ❙✳ ▼❛❝▲❛♥❡✱ ◆❛t✉r❛❧ ❛ss♦❝✐❛t✐✈✐t② ❛♥❞ ❝♦♠♠✉t❛t✐✈✐t②✱ ❘✐❝❡ ❯♥✐✈❡rs✐t② st✉❞✐❡s✱ ✹✾ ◆♦✳ ✹ ✭✶✾✻✸✮✱ ✷✽✲✹✻✳ ❬✷✼❪ ❙✳ ▼❛❝ ▲❛♥❡✱ ❍♦♠♦❧♦❣②✱ ❙♣✐♥❣❡r ✲ ❱❡r❧❛❣✱ ❇❡r❧✐♥✲●➡tt✐♥❣❡♥✲❍❡✐❞❡✐❜❡r❣✱ ✶✾✻✸✳ ❬✷✽❪ ❙✳ ▼❛❝ ▲❛♥❡✱ ❊①t❡♥s✐♦♥s ❛♥❞ ♦❜str✉❝t✐♦♥s ❢♦r r✐♥❣s✱ ■❧❧✐♥♦✐s ❏✳ ▼❛t❤❡♠❛t✐❝s✱ ✷ ✭✶✾✺✽✮✱ ✸✶✻✲✸✹✺✳ ❬✷✾❪ ❇✳ ◆♦♦❤✐✱ ◆♦t❡s ♦♥ ✷✲❣r♦✉♣♦✐❞s✱ ✷✲❣r♦✉♣s ❛♥❞ ❝r♦ss❡❞ ♠♦❞✉❧❡s✱ ❍♦♠♦❧♦❣②✱ ❍♦♠♦t♦♣② ❆♣♣❧✳ ✾ ✭✷✵✵✼✮✱ ✼✺✲✶✵✻✳ ❬✸✵❪ ❇✳ ◆♦♦❤✐✱ ●r♦✉♣ ❝♦❤♦♠♦❧♦❣② ✇✐t❤ ❝♦❡❢❢✐❝✐❡♥ts ✐♥ ❛ ❝r♦ss❡❞ ♠♦❞✉❧❡✱ ❏♦✉r♥❛❧ ♦❢ t❤❡ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ♦❢ ❏✉ss✐❡✉✱ ❱♦❧✉♠❡ ✶✵ ✲ ■ss✉❡ ✵✷✱ ❆♣r✐❧ ✷✵✶✶✱ ✸✺✾✲✹✵✹✳ ✶✶✵ download by : skknchat@gmail.com ❬✸✶❪ ❚✳ P✐r❛s❤✈✐❧✐✱ ❆❧❣❡❜r❛ ❝♦❤♦♠♦❧♦❣② ♦✈❡r ❛ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛✱ ❛r❳✐✈✿✵✸✵✾✶✽✹✈✶ ❬♠❛t❤✳❈❚❪ ✶✵ ❙❡♣ ✷✵✵✸✳ ❬✸✷❪ ❚✳ P♦rt❡r✱ ❊①t❡♥s✐♦♥s✱ ❝r♦ss❡❞ ♠♦❞✉❧❡s ❛♥❞ ✐♥t❡r♥❛❧ ❝❛t❡❣♦r✐❡s ✐♥ ❝❛t❡❣♦r✐❡s ♦❢ ❣r♦✉♣s ✇✐t❤ ♦♣❡r❛t✐♦♥s✱ Pr♦❝✳ ❊❞✐♥❜✳ ▼❛t❤✳ ❙♦❝✳✱ ✸✵ ✭✶✾✽✼✮ ✸✼✸✲✸✽✶✳ ❬✸✸❪ ◆✳ ❚✳ ◗✉❛♥❣✱ ❙tr✉❝t✉r❡ ♦❢ ❆♥♥✲❝❛t❡❣♦r✐❡s ❛♥❞ ▼❛❝ ▲❛♥❡✲❙❤✉❦❧❛ ❝♦❤♦♠♦❧♦❣②✱ ❊❛st✲ ❲❡st ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✺✱ ◆♦ ✶ ✭✷✵✵✸✮✱ ✺✶✲✻✻✳ ❬✸✹❪ ◆✳ ❚✳ ◗✉❛♥❣✱ ❚❤❡ ❢❛❝t♦r s❡ts ♦❢ ●r✲❝❛t❡❣♦r✐❡s ♦❢ t❤❡ t②♣❡ (Π, A)✱ ■♥t❡r♥❛t✐♦♥❛❧ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛✱ ✹ ✭✷✵✶✵✮✱ ◆♦✳ ✶✹✱ ✻✺✺✲✻✻✽✳ ❬✸✺❪ ◆✳ ❚✳ ◗✉❛♥❣✱ ❉✳ ❉✳ ❍❛♥❤✱ ❍♦♠♦❧♦❣✐❝❛❧ ❝❧❛ss✐❢✐❝❛t✐♦♥ ♦❢ ❆♥♥✲❢✉♥❝t♦rs✱ ❊❛st✲❲❡st ❏✳ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✶✱ ◆♦✳ ✷ ✭✷✵✵✾✮✱ ✶✾✺✲✷✶✵✳ ❬✸✻❪ ◆✳ ❚✳ ◗✉❛♥❣✱ ◆✳ ❚✳ ❚❤✉②✱ P✳ ❚✳ ❈✉❝✱ ▼♦♥♦✐❞❛❧ ❢✉♥❝t♦rs ❜❡t✇❡❡♥ ✭❜r❛✐❞❡❞✮ ●r✲❝❛t❡❣♦r✐❡s ❛♥❞ t❤❡✐r ❛♣♣❧✐❝❛t✐♦♥s✱ ❊❛st✲❲❡st ❏✳ ♦❢ ▼❛t❤❡♠❛t✐❝s✱ ✶✸✱ ◆♦ ✷ ✭✷✵✶✶✮✱ ✶✻✸✲✶✽✻✳ ❬✸✼❪ ◆✳ ❚✳ ◗✉❛♥❣✱ ❈♦❤♦♠♦❧♦❣✐❝❛❧ ❝❧❛ss✐❢✐❝❛t✐♦♥ ♦❢ ❆♥♥✲❝❛t❡❣♦r✐❡s✱ ▼❛t❤✳ ❈♦♠♠✉♥✳✱ ✶✽ ✭✷✵✶✸✮✱ ✶✲✶✽✳ ❬✸✽❪ ◆✳ ❚✳ ◗✉❛♥❣✱ P✳ ❚✳ ❈✉❝✱ ❈r♦ss❡❞ ❜✐♠♦❞✉❧❡s ♦✈❡r r✐♥❣s ❛♥❞ ❙❤✉❦❧❛ ❝♦❤♦♠♦❧♦❣②✱ ▼❛t❤✳ ❈♦♠♠✉♥✳✱ ✶✼ ◆♦✳ ✷ ✭✷✵✶✷✮✱ ✺✼✺✲✺✾✽✳ ❬✸✾❪ ◆✳ ❚✳ ◗✉❛♥❣✱ P✳ ❚✳ ❈✉❝✱ ❊q✉✐✈❛r✐❛♥t ❝r♦ss❡❞ ♠♦❞✉❧❡s ❛♥❞ ❝♦❤♦♠♦❧♦❣② ♦❢ ❣r♦✉♣s ✇✐t❤ ♦♣❡r❛t♦rs✱ ❛r❳✐✈✿ ✶✸✵✷✳✹✺✼✸✈✶ ❬♠❛t❤✳❈❚❪ ✶✾ ❋❡❜ ✷✵✶✸✳ ❬✹✵❪ ◆✳ ❚✳ ◗✉❛♥❣✱ P✳ ❚✳ ❈✉❝✱ ◆✳ ❚✳ ❚❤✉②✱ ❈r♦ss❡❞ ♠♦❞✉❧❡s ❛♥❞ str✐❝t ●r✲❝❛t❡❣♦r✐❡s✱ ❈♦♠✲ ♠✉♥✐❝❛t✐♦♥s ♦❢ ❑♦r❡❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❙♦❝✐❡t②✱ ❱♦❧✳ ✷✾✱ ◆♦✳✶ ✭✷✵✶✹✮✱ ✾✲✷✷✳ ❬✹✶❪ ◆✳ ❚✳ ◗✉❛♥❣✱ P✳ ❚✳ ❈✉❝✱ ❈❧❛ss✐❢✐❝❛t✐♦♥ ♦❢ ❣r❛❞❡❞ ❜r❛✐❞❡❞ ❝❛t❡❣♦r✐❝❛❧ ❣r♦✉♣s ❜② ♣s❡✉❞♦✲ ❢✉♥❝t♦rs✱ ❏♦✉r♥❛❧ ♦❢ ❙❝✐❡♥❝❡✱ ❍✉❡ ❯♥✐✈❡rs✐t②✱ ❱♦❧✳ ✼✼✱ ◆♦✳ ✽ ✭✷✵✶✷✮✱ ✺✾✲✻✽✳ ❬✹✷❪ ❘✳ ▲✳ ❚❛②❧♦r✱ ❈♦♠♣♦✉♥❞ ❣r♦✉♣ ❡①t❡♥s✐♦♥s ■✱ ❚r❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✼✺ ✭✶✾✺✸✮✱ ✶✵✻✲ ✶✸✺✳ ❬✹✸❪ ❏✳ ❍✳ ❈✳ ❲❤✐t❡❤❡❛❞✱ ❈♦♠❜✐♥❛t♦r✐❛❧ ❤♦♠♦t♦♣② ■■✱ ❇✉❧❧✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✺✺ ✭✶✾✹✾✮✱ ✹✺✸✲✹✾✻✳ ❚✐Õ♥❣ P❤➳♣ ❬✹✹❪ ❏✳ ❇Ð♥❛❜♦✉✱ ❈❛tÐ❣♦r✐❡s ❛✈❡❝ ♠✉❧t✐♣❧✐❝❛t✐♦♥✱ ❈✳❘✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s✱ ✷✺✻✱ ◆♦✳ ✾ ✭✶✾✻✸✮✱ ✶✽✾✼✲✶✽✾✵✳ ❬✹✺❪ ❉✳ ❈♦♥❞✉❝❤Ð✱ ▼♦❞✉❧❡s ❝r♦✐sÐs ❣Ð♥Ðr❛❧✐sÐs ❞❡ ❧♦♥❣❡✉r ✷✱ ❏ P✉r❡ ❆♣♣❧✳ ❆❧❣✳✱ ✸✹ ✭✶✾✽✹✮✱ ◆♦ ✷✲✸✱ ✶✺✺✲✶✼✽✳ ❬✹✻❪ P✳ ❉❡❞❡❝❦❡r✱ ▲❡s ❢♦♥❝t❡✉rs ExtΠ ✱ HΠ2 ❡t HΠ2 ♥♦♥ ❛❜Ð❧✐❡♥s✱ ❈✳ ❘✳ ❆❝❛❞✳ ❙❝✐✳ P❛r✐s✱ ✷✺✽ ✭✶✾✻✹✮✱ ✹✽✾✶✲✹✽✾✹✳ ✶✶✶ download by : skknchat@gmail.com ❬✹✼❪ ❆✳ ●r♦t❤❡♥❞✐❡❝❦✱ ❈❛tÐ❣♦r✐❡s ❢✐❜rÐ❡s ❡t ❞Ðs❝❡♥t❡✱ ✭❙●❆ ■✱ ❡①♣♦sÐ ❱■✮✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✷✷✹ ✭✶✾✼✶✮✱ ❙♣r✐♥❣❡r✱ ❇❡r❧✐♥✱ ✶✹✺✲✶✾✹✳ ❬✹✽❪ ❙✳ ▼❛❝ ▲❛♥❡✱ ❍♦♠♦❧♦❣✐❡ ❞❡s ❛♥♥❡❛✉① ❡t ❞❡s ♠♦❞✉❧❡s✱ ❈♦❧❧♦q✉❡ ❞❡ ❚♦♣♦❧♦❣✐❡ ❛❧✲ ❣Ð❜r✐q✉❡✱ ▲♦✉✈❛✐♥ ✭✶✾✺✻✮✱ ✺✺✲✽✵✳ ❬✹✾❪ ◆✳ ❙❛❛✈❡❞r❛ ❘✐✈❛♥♦✱ ❈❛tÐ❣♦r✐❡s ❚❛♥♥❛❦✐❡♥♥❡s✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳✱ ✷✻✺ ✭✶✾✼✷✮✱ ❙♣r✐❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥ ❛♥❞ ◆❡✇ ❨♦r❦✳ ❬✺✵❪ ❍✳ ❳✳ ❙✐♥❤✱ ●r✲❝❛tÐ❣♦r✐❡s✱ ❚❤❒s❡ ❞❡ ❞♦❝t♦r❛t✱ ❯♥✐✈❡rs✐tÐ P❛r✐s ❱■■✱ ✶✾✼✺✳ ❬✺✶❪ ❍✳ ❳✳ ❙✐♥❤✱ ●r✲❝❛tÐ❣♦r✐❡s str✐❝t❡s✱ ❆❝t❛ ♠❛t❤❡♠❛t✐❝❛ ❱✐❡t♥❛♠✐❝❛✱ ❚♦♠✳ ✸✱ ◆♦✳ ✷ ✭✶✾✼✽✮✱ ✹✼✲✺✾✳ ❬✺✷❪ ❯✳ ❙❤✉❦❧❛✱ ❈♦❤♦♠♦❧♦❣✐❡ ❞❡s ❛❧❣❡❜r❛s ❛ss♦❝✐❛t✐✈❡s✱ ❆♥♥✳ ❙❝✐✳ ❊❝♦❧❡ ◆♦r♠✳ ế ứ rr Uă ❜❡r ❞✐❡ ❊r✇❡✐t❡r✉♥❣ ✈♦♥ ●r✉♣♣❡♥ ■✱ ▼♦♥❛ts❤✳ ▼❛t❤✳ P❤②s✳ ✸✹ ✭✶✾✷✻✮✱ ✶✻✺✲✶✽✵✱ ✶✶✷ download by : skknchat@gmail.com ❈❤Ø sè HΓn (Π, A)✱ ✻✻ ❝đ❛ ❤➭♠ tư ♠♦♥♦✐❞❛❧✱ ✷✾ HoAnnstr✱ ✾✾ ❝đ❛ ❆♥♥✲❤➭♠ tư✱ ✷✹ Γ✲♠➠➤✉♥ ❝❤Ð♦✱ ✼✸ ❝➷♣ ❤➭♠ ❧✐➟♥ ❦Õt✱ ✷✹✱ ✷✾ ❝✃✉ tró❝ ❝đ❛ ❆♥♥✲♣❤➵♠ trï✱ ✷✸ ❧✐➟♥ ❦Õt✱ ✼✻ Γ✲♥❤ã♠✱ ✶✽ ❊✲❤Ö✱ ✾✷ Γ✲➤å♥❣ ❝✃✉✱ ✶✽ ❝❤Ý♥❤ q✉✐✱ ✾✷ Hom(ϕ,f ) [S, S ]✱ ✼✷ ❧✐➟♥ ❤ỵ♣✱ ✾✺ Π✲♠➠➤✉♥ Γ✲➤➻♥❣ ❜✐Õ♥✱ ✶✽ Γ (Π, A, h)✱ ✻✼ ❤➭♠ tö ♠♦♥♦✐❞❛❧✱ ✶✾ Annstr✱ ✾✾ ❤➭♠ tö ESyst✱ ✾✾ ❦✐Ó✉ Psd(Γ, Brg)✱ ✸✽ ♣❤➞♥ ❜❐❝ ✸✽ ❝❤Ý♥❤ q✉✐✱ ✼✾ ❤➭♠ tư ♠♦♥♦✐❞❛❧ ❆♥♥✲❤➭♠ tư ❞➵♥❣ ❜Ư♥✱ ✷✵ (f1 , f0 )✱ ✾✺ ♣❤➞♥ ❜❐❝✱ ✷✵ ➤➡♥✱ ✾✼ ➤å♥❣ ❧✉➞♥ ♠➵♥❤✱ ✾✽ ❆♥♥✲♠ò✐ t➟♥✱ ✷✷ ❆♥♥✲♣❤➵♠ trï✱ ✷✵ ❝❤Ý♥❤ q✉✐✱ ✺✼ ❤➵t ♥❤➞♥ trõ✉ t➢ỵ♥❣✱ ✹✺ ❤Ư ♥❤➞♥ tư✱ ✸✽✱ ✺✶ ❝❤Ý♥❤ q✉✐✱ ✼✹ ❝❤➷t ❝❤Ï✱ ✷✶ ❝❤Ý♥❤ q✉✐✱ ✷✶ ❦✐Ĩ✉ (ϕ, f )✱ ✼✷ ❍➭♠ tư ♠♦♥♦✐❞❛❧ Psd(Γ, C)✱ ✸✽ Γ BCG ✱ Γ✲♣❤➞♥ ❜❐❝ ♠➠➤✉♥ ❝❤Ð♦✱ ✺✸ (R, M )✱ ✷✸ ❝➳❝ tù ➤➻♥❣ ❝✃✉✱ ✺✹ ❧✐➟♥ ❦Õt✱ ✾✹ ❧✐➟♥ ❦Õt✱ ✺✺ t❤✉ ❣ä♥✱ ✷✷ ➤➻♥❣ ❜✐Õ♥✱ ✼✸ ❆♥♥✲t➢➡♥❣ ➤➢➡♥❣✱ ✷✷ ♠ë ré♥❣ ❝❤Ý♥❤ t➽❝✱ ✷✸ tÝ❝❤ ❝❤Ð♦✱ ✻✵ ♠ë ré♥❣ ♥❤ã♠ ❝➯♥ trë ❝đ❛ ❤➭♠ tư ❦✐Ĩ✉ ♠➠➤✉♥ ❝❤Ð♦✱ ✺✾ Γ✲♣❤➞♥ ❜❐❝✱ ✼✷ t➢➡♥❣ ➤➢➡♥❣✱ ✺✾ ✶✶✸ download by : skknchat@gmail.com ♠ë ré♥❣ tÝ❝❤ ❝❤Ð♦✱ ✼✵✱ ✶✵✵ s♦♥❣ tÝ❝❤✱ ✾✶ ❣✐❛♦ ❤♦➳♥✱ ✾✷ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ó✉ ❊✲❤Ư tr♦♥❣✱ ✾✷ t➢➡♥❣ ➤➢➡♥❣✱ ✶✵✵ ♠ë ré♥❣ ✈➭♥❤ ❦✐Ĩ✉ ❊✲❤Ư ❝❤Ý♥❤ q✉✐✱ ✾✾ ♠ë ré♥❣ ➤➻♥❣ ❜✐Õ♥ t✐➟♥ ➤Ị ♥❣ò ❣✐➳❝✱ ✶✻ tÝ❝❤ ❝❤Ð♦✱ ✽✷ ♠ë ré♥❣ ➤➻♥❣ ❜✐Õ♥ ❦✐Ó✉ t➢➡♥❣ ➤➢➡♥❣✱ ✽✶ ♥❤ã♠ ♣❤➵♠ trï✱ ✶✼ sù t➢➡♥❣ t❤Ý❝❤✱ ✷✷ Γ✲♠➠➤✉♥ ❝❤Ð♦✱ ✽✶ t✐➟♥ ➤Ò t❛♠ ❣✐➳❝✱ ✶✻ t✐Ị♥ ➤Ý♥❤ ❦✐Ĩ✉ (Π, A)✱ ✸✷ t➞♠✱ ✹✺ t➢➡♥❣ ➤➢➡♥❣ ❝❤Ý♥❤ t➽❝✱ ✶✽ ❜Ö♥✱ ✶✾ t➢➡♥❣ ➤➢➡♥❣ ♠♦♥♦✐❞❛❧✱ ✷✵ ❜Ö♥ t❤✉ ❣ä♥✱ ✸✹ t➢➡♥❣ ➤➢➡♥❣ ♣❤➞♥ ❜❐❝ ❝❤Ý♥❤ t➽❝✱ ✻✾ ❦✐Ó✉ t➢➡♥❣ ➤➢➡♥❣ tù ♥❤✐➟♥ ♠♦♥♦✐❞❛❧✱ ✶✾ (Π, A)✱ ✶✽ ♣❤➞♥ ❜❐❝✱ ✷✵ ❧✐➟♥ ❦Õt✱ ✺✹ ❜Ö♥✱ ✷✵ ♣❤➞♥ ❜❐❝✱ ✶✾ ❝❤➷t ❝❤Ï✱ ✼✹ ❦✐Ó✉ (Π, A)✱ ✼✷ ✈❐t ➤➡♥ ✈Þ✱ ✶✻ ❧✐➟♥ ❦Õt✱ ✼✹ ➤Ý♥❤✱ ✶✼✱ ✷✸ t❤✉ ❣ä♥✱ ✼✷ ➤å♥❣ ❝✃✉ rê✐ r➵❝✱ ✶✽ Γ✲♠➠➤✉♥ ❝❤Ð♦✱ ✼✻ t❤✉ ❣ä♥✱ ✶✽ ❊✲❤Ö✱ ✾✷ ➤è✐ ①ø♥❣✱ ✶✾ ♠➠➤✉♥ ❝❤Ð♦✱ ✺✺ s♦♥❣ ♠➠➤✉♥ ❝❤Ð♦✱ ✾✶ ♣❤➞♥ ❜❐❝✱ ✶✽ ỉ♥ ➤Þ♥❤✱ ✶✽ ♣❤➵♠ trï ♠♦♥♦✐❞❛❧✱ ✶✻ t♦➳♥ tö✱ ✾✷ ➤å♥❣ ❧✉➞♥✱ ✶✾✱ ✷✷ ❝❤➷t ❝❤Ï✱ ✶✼ ♣❤➵♠ trï ♠♦♥♦✐❞❛❧ ♣❤➞♥ ❜❐❝✱ ✶✽ ❜Ö♥✱ ✶✾ ♣❤➵♠ trï P✐❝❛r❞✱ ✶✾ r➭♥❣ ❜✉é❝ ❜Ö♥✱ ✶✾ r ộ ết ợ r ộ ị s♦♥❣ ♠➠➤✉♥ ❝❤Ð♦✱ ✾✶ s♦♥❣ t➞♠✱ ✾✷ ✶✶✹ download 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P❤➵♠ trï ♠♦♥♦✐❞❛❧ ợ ị ó ể trở t trù tró❝ ♥❤ã♠ ❦❤✐ ❜ỉ s✉♥❣ t❤➟♠ ❦❤➳✐ ♥✐Ư♠ ✈❐t ❦❤➯ ị r trờ ợ trù ề ột r ✭♥❣❤Ü❛ ❧➭ ♠ä✐ ♠ị✐ t➟♥ tr♦♥❣ ♣❤➵♠ trï ➤Ị✉ ❧➭ tì t t ợ ột trù q✉❛♥ trä♥❣✱ ➤ã ❧➭ ♥❤ã♠ ♣❤➵♠... ➤➻♥❣ ❝✃✉ ♥➭② ➤Ị✉ ❧➭ ➤å♥❣ ♥❤✃t t❤× t❛ ♥ã✐ ❝➳❝ r➭♥❣ ❜✉é❝ ❧➭ ❝❤➷t ❝❤Ï✱ ✈➭ ♣❤➵♠ trï ➤❛♥❣ ①Ðt ❧➭ trù t ẽ ỗ trù ề t➢➡♥❣ ➤➢➡♥❣ ✈í✐ ♠ét ♣❤➵♠ trï ♠♦♥♦✐❞❛❧ ❝❤➷t ❝❤Ï✳ P❤Ð♣ t♦➳♥ t❡♥s➡ t❤➠♥❣ t❤➢ê♥❣ ❧➭♠ ❝❤♦... ❝❤✐Ị✉ ✸ ❝đ❛ ❝➳❝ ♥❤ã♠✳ ❉♦ tr♦♥❣ ❧í♣ ♣❤➵♠ trï ♥➭② ♠ä✐ ♠ị✐ t➟♥ ➤Ị✉ ❧➭ ➤➻♥❣ t ế tr ủ ỗ trù tộ ề ợ ị ó ỗ ó trù t ủ G ➤Þ♥❤ ❤♦➭♥ t♦➭♥ ❜❛ ❜✃t ❜✐Õ♥✿ ♥❤ã♠ Π = π0 G ❝➳❝ ❧í♣ G✱ Π✲♠➠➤✉♥ A = π1 G ❝➳❝