❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛
▼ó❞✉❧♦s ❞❡ ❯❧r✐❝❤
♣♦r
▼❛r✐❛♥❛ ❞❡ ❇r✐t♦ ▼❛✐❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛
❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛
▼ó❞✉❧♦s ❞❡ ❯❧r✐❝❤
♣♦r
▼❛r✐❛♥❛ ❞❡ ❇r✐t♦ ▼❛✐❛
s♦❜ ❛ ♦r✐❡♥t❛çã♦ ❞❡
Pr♦❢✳ ❉r✳ ❈❧❡t♦ ❇r❛s✐❧❡✐r♦ ▼✐r❛♥❞❛ ◆❡t♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ ❈❈❊◆ ✲ ❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
M217m Maia, Mariana de Brito.
Módulos de Ulrich / Mariana de Brito Maia.- João Pessoa, 2013.
60f.
Orientador: Cleto Brasileiro Miranda Falcão Dissertação (Mestrado) - UFPB/CCEN
1. Matemática. 2. Módulo Ulrich. 3. Módulo Cohen-Macaulay maximal. 4. Número mínimo de geradores. 5. Multiplicidade.
❆❣r❛❞❡❝✐♠❡♥t♦s
●♦st❛r✐❛ ❞❡ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✳
❆ ♠❡✉s ♣❛✐s ❡ ♠✐♥❤❛ ✐r♠ã ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ❝♦♠♣r❡❡♥sã♦ ♣♦r t♦❞♦s ♦s ♠♦♠❡♥t♦s ❡♠ q✉❡ ❡✉ ♥ã♦ ♣✉❞❡ ❡st❛r ❧á✳
❆ ♠✐♥❤❛ ❢❛♠í❧✐❛✳ ▼❡✉s ❛✈ós✿ ❆♥tô♥✐♦✱ ❍✉♠❜❡rt♦✱ ■♥ês ❡ ▲✐♥❞❛❧✈❛✳ ▼❡✉s t✐♦s✿ ❏♦s✐♥❛❧❞♦✱ ▼❛r✐❛✱ ❈♦♥❝❡✐çã♦✱ ❍❡❧❞❡r✱ ●♦r❡t❡✱ ❏♦sé ❲✐❧s♦♥✱ ●r❛ç❛✱ ▲✐♥❞❡❜❡rt♦✱ ❩✐❧❡✐❞❡✱ ●r❛ç❛✳ ▼❡✉s ♣r✐♠♦s✿ ❈❛t❛r✐♥❛✱ ❏ú♥✐♦r✱ ❱✐tór✐❛✱ ❙✐❧❛s✱ ▲♦r❡♥❛✱ ❲✐❧❧✐❛♠✱ ▲❡❛♥❞r♦✱ ◆❛❧❞✐♥❤♦✳✳✳ ❆ t♦❞♦s ❡♥✜♠ ♣❡❧♦ ❛♠♦r ❡ ❝✉✐❞❛❞♦✳
❆ ❚♦♥②✱ ❛ ♠❡❧❤♦r ❝♦✐s❛ q✉❡ ❛ ♠❛t❡♠át✐❝❛ ♠❡ ❞❡✉✳
❆ ♠✐♥❤❛ ❢❛♠í❧✐❛ P❡❞r❡❣❛❧✱ s❡♠ ❛ q✉❛❧ ❡✉ ♥ã♦ t❡r✐❛ ❝♦♥s❡❣✉✐❞♦ t❡r♠✐♥❛r ❡st❡ tr❛❜❛❧❤♦✱ s❡❥❛ ♣❡❧❛ ❛❥✉❞❛ ❛❝❛❞ê♠✐❝❛ ❞❡ ❢❛t♦ ♦✉ só ♣❡❧❛s r✐s❛❞❛s ♥❛s ❤♦r❛s ♠❛✐s ❞✐❢í❝❡✐s✿ ▼ô♥✐❝❛✱ ❊✉❞❡s✱ ●érs✐❝❛✱ ▼②❧❡♥♥❛✱ ❲❛♥❞❡rs♦♥✱ ▲✉❛♥✱ ●✐♥❛❧❞♦✱ ❘❡♥❛t♦✱ ❱❡✈❡✱ ▲✐❧✐✳
❆♦s ❞❡♠❛✐s ❝♦❧❡❣❛s ❞♦ ♠❡str❛❞♦✱ ♣❡❧❛s ❝♦rr❡♥t❡s ❞❡ ✉♥✐ã♦ ❡ ❢é ❞✉r❛♥t❡ ❛s ❞✐s❝✐♣❧✐♥❛s✿ ❘❛❢❛❡❧✱ ❊❜❡rs♦♥✱ P❛✉❧♦✱ ❈❤✐❝ó✱ ❈❛r❧♦s✱ ❲❛❧❧❛❝❡✱ ❱✐✈✐✱ ▲✉✐s✱ ❊❞♥❛✱ ❏ú♥✐♦r✱ ❘✐❝❛r❞♦✱ ❊♥✐❡③❡✱ ❨❛♥❡✱ ❉❡st❡rr♦✱ ◆❛❝✐❜✱ ❘❡❣✐♥❛❧❞♦✱ ❯❡❧✐ss♦♥✱ ❋❡❧✐♣❡✱ ●✐❧s♦♥✱ ❉✐❡❣♦✱ ❑❡❧②❛♥❡✳✳✳
❆ ♠❡✉s ♣r♦❢❡ss♦r❡s q✉❡ t❛♥t♦ ✜③❡r❛♠ ♣❡❧♦ ♠❡✉ ❝r❡s❝✐♠❡♥t♦✿ ❇r✉♥♦✱ ❆❧❡①❛♥❞r❡✱ ❏❛❝q✉❡❧✐♥❡✱ ▲✐③❛♥❞r♦✱ ◆❛♣♦❧❡♦♥✱ ▼✐r✐❛♠✱ ❉❛♥✐❡❧✱ ❇❡❞r❡❣❛❧✱ ❏♦ã♦ ▼❛r❝♦s✱ ❆r♦♥✱ ❈❧❛✉❞✐❛♥♦r✳
❆ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❈❧❡t♦✱ ♣❡❧❛ ❝♦♥✜❛♥ç❛ ❞❡♣♦s✐t❛❞❛✳
❆ ♠❡✉s ❛♠✐❣♦s ❞❛ ❣r❛❞✉❛çã♦✿ ❲❛♥❞❡r❧❡②✱ ❙ér❣✐♦✱ ▼❛rí❧✐❛✱ P❛✉❧❛✱ ▼❛rt❛✱ ▼ár❝✐❛✱ ❈í❝❡r♦✱ P❛✉❧♦✱ ❊❞♥❡②✱ ❆❣❧❛❡r✱ P❡tr✐❝❦✱ ❚❤❛②❛♥❛✱ ❲✐❧❧✱ ❈❛r❧✐♥❤❛✱ ❉✐❛♥❛✱ ❈❛✐♦✳✳✳ ❊♠ ❡s♣❡❝✐❛❧ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❋❛❧❝ã♦✱ q✉❡ ♠❡ ❢❡③ ❛❝r❡❞✐t❛r q✉❡ t✉❞♦ ✐ss♦ ❡r❛ ♣♦ssí✈❡❧✳
❘❡s✉♠♦
◆❡st❡ tr❛❜❛❧❤♦✱ ❛♣ós ✐♥tr♦❞✉③✐r♠♦s ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❞❡ ➪❧❣❡❜r❛ ❈♦♠✉t❛t✐✈❛✱ ❝♦♠♦ ❞✐♠❡♥sã♦✱ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡s✱ ❡ ♠✉❧t✐♣❧✐❝✐❞❛❞❡✱ ♣r♦✈❛♠♦s ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ ❝❧❛ss❡ ❞❡ ♠ó❞✉❧♦s ❜❛st❛♥t❡ ❡s♣❡❝✐❛❧ s♦❜r❡ ❛♥é✐s ❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ♦s ❝❤❛♠❛❞♦s ♠ó❞✉❧♦s ❞❡ ❯❧r✐❝❤✳ ➱ s❛❜✐❞♦ q✉❡✱ s❡ M é ✉♠ A✲♠ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠❛①✐♠❛❧ s♦❜r❡ ✉♠ t❛❧ ❛♥❡❧✱
❡♥tã♦ µ(M) ≤ e(M)✳ ❖ ♦❜❥❡t✐✈♦ ❞♦ ♥♦ss♦ ❡st✉❞♦ é ❞❡♠♦♥str❛r ♦s ♣r✐♥❝✐♣❛✐s ❝❛s♦s ❡♠ q✉❡
✈❛❧❡µ(M) = e(M)✳
P❛❧❛✈r❛s ✲ ❝❤❛✈❡✿ ▼ó❞✉❧♦ ❞❡ ❯❧r✐❝❤✱ ▼ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠❛①✐♠❛❧✱ ◆ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡s✱ ▼✉❧t✐♣❧✐❝✐❞❛❞❡✳
❆❜str❛❝t
■♥ t❤✐s ✇♦r❦✱ ❛❢t❡r t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ s♦♠❡ ❝♦♥❝❡♣ts ♦❢ ❈♦♠♠✉t❛t✐✈❡ ❆❧❣❡❜r❛✱ ❢♦r ✐♥st❛♥❝❡ ❞✐♠❡♥s✐♦♥✱ ♠✐♥✐♠❛❧ ♥✉♠❜❡r ♦❢ ❣❡♥❡r❛t♦rs✱ ❛♥❞ ♠✉❧t✐♣❧✐❝✐t②✱ ✇❡ ♣r♦✈❡ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ✈❡r② s♣❡❝✐❛❧ ❝❧❛ss ♦❢ ♠♦❞✉❧❡s ♦✈❡r ❈♦❤❡♥✲▼❛❝❛✉❧❛② r✐♥❣s✱ t❤❡ s♦✲❝❛❧❧❡❞ ❯❧r✐❝❤ ♠♦❞✉❧❡s✳ ■t ✐s ❦♥♦✇♥ t❤❛t✱ ✐❢ M ✐s ❛ ♠❛①✐♠❛❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡ ♦✈❡r s✉❝❤ r✐♥❣✱ t❤❡♥ µ(M) ≤ e(M)✳ ❖✉r
❣♦❛❧ ✐♥ t❤✐s st✉❞② ✐s t♦ ♣r♦✈❡ t❤❡ ♠❛✐♥ ❝❛s❡s ✇❤❡r❡ t❤❡ ❡q✉❛❧✐t②µ(M)≤e(M) ❤♦❧❞s✳
❙✉♠ár✐♦
■♥tr♦❞✉çã♦ ✾
✶ Pr❡❧✐♠✐♥❛r❡s ✶✶
✶✳✶ ●❡r❛❞♦r❡s ❡ ♣♦st♦ ❞❡ ✉♠ ♠ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ ❉✐♠❡♥sã♦ ❞❡ ❑r✉❧❧ ❡ s✐st❡♠❛s ❞❡ ♣❛râ♠❡tr♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✸ ❙❡q✉ê♥❝✐❛s r❡❣✉❧❛r❡s ❡ ♠ó❞✉❧♦s ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ ❘❡s♦❧✉çã♦ ❧✐✈r❡ ❡ ❞✐♠❡♥sã♦ ❤♦♠♦❧ó❣✐❝❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✺ ▼ó❞✉❧♦s ❝❛♥ô♥✐❝♦s ❡ ❛♥é✐s ●♦r❡♥st❡✐♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✶✳✻ ▼✉❧t✐♣❧✐❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸
✷ ▼ó❞✉❧♦s ❞❡ ❯❧r✐❝❤ ✷✾
✷✳✶ ❖ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ♣♦r ❇✳ ❯❧r✐❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ❖ ❝❛s♦ ✵✲❞✐♠❡♥s✐♦♥❛❧ ❡ ❛❧❣✉♥s r❡s✉❧t❛❞♦s ❣❡r❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✸ ❘❡s♦❧✉çõ❡s ❧✐♥❡❛r❡s ❡ ♦ ❝❛s♦ ✶✲❞✐♠❡♥s✐♦♥❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✹ ❆♥é✐s ❝♦♠ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ♠✐♥✐♠❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✺ ❉♦♠í♥✐♦s ❤♦♠♦❣ê♥❡♦s ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✷✲❞✐♠❡♥s✐♦♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✷✳✻ ■♥t❡rs❡çõ❡s ❝♦♠♣❧❡t❛s ❡str✐t❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼
❆ ❋❛t♦r❛çã♦ ♠❛tr✐❝✐❛❧ ✺✷
❆✳✶ ❋❛t♦r❛çã♦ ♠❛tr✐❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✺✽
■♥tr♦❞✉çã♦
❊♠ ✶✾✽✹✱ ❯❧r✐❝❤ ♣✉❜❧✐❝❛ ●♦r❡♥st❡✐♥ r✐♥❣s ❛♥❞ ♠♦❞✉❧❡s ✇✐t❤ ❤✐❣❤ ♥✉♠❜❡rs ♦❢ ❣❡♥❡r❛t♦rs ❬✸✶❪✱ ♦♥❞❡✱ ❛♦ ♠♦str❛r ❛ ❞❡s✐❣✉❛❧❞❛❞❡
µ(M)≤e(M)
❢❛③ ❛ s❡❣✉✐♥t❡ ♣❡r❣✉♥t❛✿
❙❡ A é ✉♠ ❛♥❡❧ ❧♦❝❛❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞❡ ❞✐♠❡♥sã♦ ♣♦s✐t✐✈❛ ❡ ❝♦r♣♦ ❞❡ ❝❧❛ss❡s r❡s✐❞✉❛✐s
✐♥✜♥✐t♦✱ ❡♥tã♦ s❡♠♣r❡ ❡①✐st✐rá ✉♠ A✲♠ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② M ❞❡ ♣♦st♦ ♣♦s✐t✐✈♦ t❛❧ q✉❡ µ(M) =e(M)❄
❆q✉✐✱ µ(−)✱ e(−) ❡ rk(−) ❞❡♥♦t❛♠✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡s✱
♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❡ ♣♦st♦✳
❖❜s❡r✈❛♠♦s q✉❡✱ s❡M é ✉♠ ♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s♦❜r❡ ✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦ A✱ ❡ ❛❧é♠ ❞✐ss♦✱ ♣♦ss✉✐ ✉♠ ♣♦st♦ ♣♦s✐t✐✈♦✱ ❡♥tã♦ ✈❛❧❡ e(M) =e(A)rk(M)✳
❊♠ ✶✾✽✼✱ ❇r❡♥♥❛♥✱ ❍❡r③♦❣ ❡ ❯❧r✐❝❤ ♣✉❜❧✐❝❛♠ ▼❛①✐♠❛❧❧② ❣❡♥❡r❛t❡❞ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ❬✼❪✱ ♦♥❞❡ tr❛t❛♠ ❞❛ ❡①✐stê♥❝✐❛ ❡ ♣r♦♣r✐❡❞❛❞❡s ❞❡ t❛✐s A✲♠ó❞✉❧♦s ♠❛①✐♠❛✐s ❈♦❤❡♥✲
▼❛❝❛✉❧❛② ♠❛①✐♠❛❧♠❡♥t❡ ❣❡r❛❞♦s✳ ▼❛✐s t❛r❞❡✱ ❡♠ ▼❛①✐♠❛❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠♦❞✉❧❡s ♦✈❡r ●♦r❡♥st❡✐♥ r✐♥❣s ❛♥❞ ❇♦✉r❜❛❦✐ s❡q✉❡♥❝❡s ❬✶✻❪✱ ❍❡r③♦❣ ❡ ❑ü❤❧ ♣❛ss❛♠ ❛ ❝❤❛♠á✲❧♦s ▼ó❞✉❧♦s ❞❡ ❯❧r✐❝❤✳ ❖✉tr❛s ♥♦♠❡♥❝❧❛t✉r❛s t❛♠❜é♠ sã♦ ❡♥❝♦♥tr❛❞❛s ♥❛ ❧✐t❡r❛t✉r❛✱ ❝♦♠♦ ♠ó❞✉❧♦s ♠❛①✐♠❛✐s ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❧✐♥❡❛r❡s ❡ ▼ó❞✉❧♦s ❚♦♣✲❍❡❛✈②✳
❆ ♣❡r❣✉♥t❛ ❞❡ ❯❧r✐❝❤✱ ❛q✉✐ tr❛t❛❞❛✱ ❢♦✐ r❡s♣♦♥❞✐❞❛ ❞❡ ❢♦r♠❛ ❛✜r♠❛t✐✈❛ ♥♦s s❡❣✉✐♥t❡s ❝❛s♦s✿
✶✳ dim(A)≤1✱ ❡♠ ❬✼❪❀
✸✳ A é ❞♦♠í♥✐♦ ❤♦♠♦❣ê♥❡♦ ✷✲❞✐♠❡♥s✐♦♥❛❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❝♦♠ ❝♦r♣♦ ❞❡ ❝❧❛ss❡s r❡s✐❞✉❛✐s
✐♥✜♥✐t♦✱ ❡♠ ❬✼❪❀
✹✳ A é ✉♠❛ ✐♥t❡rs❡❝çã♦ ❝♦♠♣❧❡t❛ ❡str✐t❛✱ ❡♠ ❬✺❪✳
❖ ♦❜❥❡t✐✈♦ ❞♦ ♥♦ss♦ ❡st✉❞♦ s❡rá ❞❡♠♦♥str❛r ♦s q✉❛tr♦ ❝❛s♦s ❝✐t❛❞♦s ❛❝✐♠❛✳
■♥✐❝✐❛♠♦s ❝♦♠ ✉♠ ❝❛♣ít✉❧♦ ❞❡ ♣ré✲r❡q✉✐s✐t♦s✱ ♦♥❞❡ ❝♦♥st❛♠ ❛s ♣r✐♥❝✐♣❛✐s ♥♦çõ❡s ❡ r❡s✉❧t❛❞♦s ❛q✉✐ ✉s❛❞♦s✳ ❆❧❣✉♥s ❞❡❧❡s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ✉♠ ❝✉rs♦ ❜ás✐❝♦ ❞❡ ➪❧❣❡❜r❛ ❈♦♠✉t❛t✐✈❛❀ ♦✉tr♦s r❡q✉❡r❡♠ ✉♠ ♣♦✉❝♦ ♠❛✐s ❞❡ ❡①♣❡r✐ê♥❝✐❛ ♣♦r ♣❛rt❡ ❞♦ ❧❡✐t♦r✳ ▼✉✐t♦s r❡s✉❧t❛❞♦s ♥❡ss❡ ❝❛♣ít✉❧♦ ♣r❡❧✐♠✐♥❛r s❡rã♦ ❛❞♠✐t✐❞♦s s❡♠ ♠❛✐♦r❡s ❞✐s❝✉ssõ❡s✱ ✈✐st♦ q✉❡ s✉❛s ❞❡♠♦♥str❛çõ❡s t♦r♥❛r✐❛♠ ♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡①❝❡ss✐✈❛♠❡♥t❡ ❡①t❡♥s♦✳
◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ tr❛t❛♠♦s ❞❛ ❡①✐stê♥❝✐❛ ❞❡ ♠ó❞✉❧♦s ❞❡ ❯❧r✐❝❤ s♦❜r❡ ❛s q✉❛tr♦ ❝❧❛ss❡s ❞❡ ❛♥é✐s ❧✐st❛❞❛s ❛❝✐♠❛✳ P❛r❛ t❛❧✱ r❡❝♦rr❡♠♦s ❛ r❡s✉❧t❛❞♦s ❞♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ❛ ♦✉tr♦s q✉❡ s❡ ❡♥❝♦♥tr❛♠ ♥♦ ❛♣ê♥❞✐❝❡ ❡ ❛ ✈ár✐♦s ♦✉tr♦s q✉❡ ❞❡s❡♥✈♦❧✈❡♠♦s ♥❡st❡ ♠❡s♠♦ ❝❛♣ít✉❧♦✳
■♥❝❧✉í♠♦s ❛✐♥❞❛ ✉♠ ❛♣ê♥❞✐❝❡ ♦♥❞❡ ❛♣r❡s❡♥t❛♠♦s ♦s ❡❧❡♠❡♥t♦s ❜ás✐❝♦s ❞❛ t❡♦r✐❛ ❞❡ ❢❛t♦r❛çã♦ ♠❛tr✐❝✐❛❧ ❡ ♠ó❞✉❧♦s ❞❡ ❈❧✐✛♦r❞✱ q✉❡ ❢♦r❛♠ ✉t✐❧✐③❛❞♦s ❡♠ ♣❛rt❡ ❞♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✳ ▼❡♥❝✐♦♥❛♠♦s✱ ❡♥ ♣❛ss❛♥t✱ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ t❡♦r✐❛ ❞♦s ♠ó❞✉❧♦s ❞❡ ❯❧r✐❝❤ t❛♠❜é♠ ❡♠ ●❡♦♠❡tr✐❛ ❆❧❣é❜r✐❝❛✱ ❛tr❛✈és ❞♦s ❝❤❛♠❛❞♦s ✜❜r❛❞♦s ❞❡ ❯❧r✐❝❤✱ ✐♥✈❡st✐❣❛❞♦s r❡❝❡♥t❡♠❡♥t❡ ♣♦r ❘✳ ❍❛rts❤♦r♥❡ ❡ ♦✉tr♦s ❛✉t♦r❡s ✭✈✐❞❡✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✾❪ ❡ ❬✶✵❪✮✳
❊s♣❡r❛♠♦s ♣r♦♣♦r❝✐♦♥❛r ❛♦ ❧❡✐t♦r ✉♠ ♠❛t❡r✐❛❧ s✉✜❝✐❡♥t❡♠❡♥t❡ ✐♥t❡r❡ss❛♥t❡✱ ❡ q✉❡ s❡❥❛ ❝❛♣❛③ ❞❡ ❡st✐♠✉❧á✲❧♦ ❛ ❜✉s❝❛r ♠❛✐s s♦❜r❡ ❡st❡ ❛ss✉♥t♦ q✉❡ t❛♥t♦ ♥♦s ❝❛t✐✈♦✉✳
❈❛♣ít✉❧♦ ✶
Pr❡❧✐♠✐♥❛r❡s
◆❡st❡ ❝❛♣ít✉❧♦✱ ❞❡✜♥✐r❡♠♦s ❛❧❣✉♥s ❞♦s ❝♦♥❝❡✐t♦s q✉❡ s❡rã♦ ♥❡❝❡ssár✐♦s ❛♦ ❧♦♥❣♦ ❞❡st❡ tr❛❜❛❧❤♦✳ P❛r❛ ♠❛✐♦r❡s ❞❡t❛❧❤❡s s✉❣❡r✐♠♦s ❛♦ ❧❡✐t♦r ❝♦♥s✉❧t❛r ♦s ❧✐✈r♦s ❬✽❪ ❡ ❬✷✷❪✳
❈♦♥✈❡♥çã♦✿ ◆❡st❡ tr❛❜❛❧❤♦✱ ❛ ♠❡♥♦s ❞❡ ♠❡♥çã♦ ❡①♣❧í❝✐t❛ ❡♠ ❝♦♥trár✐♦✱ t♦❞♦s ♦s ❛♥é✐s s❡rã♦ ❝♦♠✉t❛t✐✈♦s ❡ ❝♦♠ ✶✳
✶✳✶ ●❡r❛❞♦r❡s ❡ ♣♦st♦ ❞❡ ✉♠ ♠ó❞✉❧♦s
❉❡✜♥✐çã♦ ✶✳✶ ❙❡❥❛ A ✉♠ ❛♥❡❧✳ ❯♠ A✲♠ó❞✉❧♦ M é ❞✐t♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ✭s♦❜r❡ A✮
s❡ ❡①✐st✐r ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ {m1, ..., mr} ⊂ M✱ ❝❤❛♠❛❞♦ ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s
❞❡ M✱ t❛❧ q✉❡ M =
r
X
i=1
Ami✱ ♦✉ s❡❥❛✱ ❝❛❞❛ m ∈ M s❡ ❡①♣r❡ss❛ ❝♦♠♦ ❝♦♠❜✐♥❛çã♦ A✲
❧✐♥❡❛r m = a1m1 +... +armr, ai ∈ A✳ ❚❛❧ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s s❡rá ❞✐t♦ ♠✐♥✐♠❛❧
s❡ mj ∈/
X
i6=j
Ami,∀j = 1, ..., r✳ ❙❡✱ ❛❧é♠ ❞✐ss♦✱ ♦ ❝♦♥❥✉♥t♦ {m1, ..., mr} ❢♦r ❧✐♥❡❛r♠❡♥t❡
✐♥❞❡♣❡♥❞❡♥t❡ s♦❜r❡ A✱ ✐st♦ é✱ s❡ m =
r
X
i=1
aimi = 0⇔ ai = 0 ∀i✱ ❞✐③❡♠♦s q✉❡ {m1, ..., mr}
é ✉♠❛ ❜❛s❡ ♣❛r❛ M✱ ❡ ♥❡st❡ ❝❛s♦M s❡rá ❝❤❛♠❛❞♦ ✉♠ A✲♠ó❞✉❧♦ ❧✐✈r❡✳
❚❡♦r❡♠❛ ✶✳✷ ❙❡❥❛♠ (A,m) ✉♠ ❛♥❡❧ ❧♦❝❛❧✱ k = A
m s❡✉ ❝♦r♣♦ r❡s✐❞✉❛❧ ❡ M ✉♠ A✲♠ó❞✉❧♦
✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❙❡❥❛♠ m1, ..., mr ∈ M t❛✐s q✉❡ {m¯1, ...,m¯r} é ✉♠❛ ❜❛s❡ ❞♦ k✲❡s♣❛ç♦
✈❡t♦r✐❛❧ M
mM✳ ❊♥tã♦✱ {m1, ..., mr} é ✉♠ ❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ ❞❡ ❣❡r❛❞♦r❡s ❞❡ M✳
Pr♦✈❛✿✳ ❉❡✜♥❛ N =
r
X
i=1
M π N/ i ? ?
f=π◦i
/
/ M
mM
❆ss✐♠✱ ❞❛❞♦ n ∈N ⇒n =
r
X
i=1
aimi ♣❛r❛ a1, ..., ar ∈A✳ ▲♦❣♦✱f(n) = r
X
i=1 ¯
aim¯i✳
◆♦t❡ q✉❡ f é s♦❜r❡❥❡t♦r ♣♦✐s ♦s m¯′
is ❣❡r❛♠ M
mM ❝♦♠♦ A
m✲❡s♣❛ç♦ ✈❡t♦r✐❛❧✳
❙❡❥❛ ❛❣♦r❛ n=
r
X
i=1
aimi ∈ker(f)⊆N ⇒f(n) = r
X
i=1 ¯
aim¯i = ¯0⇒¯ai = ¯0⇒ ai ∈m,∀i=
1, ..., r ⇒n∈mM ∩N ⇒ker(f) = mM∩N✳
P❡❧♦ t❡♦r❡♠❛ ❞♦ ✐s♦♠♦r✜s♠♦✱
N mM∩N ∼=
M mM
⇒ mM+N mM ∼=
M mM
⇒mM +N =M
❆ss✐♠✱ ♣♦r ◆❛❦❛②❛♠❛✱ M = N✱ ❡ ♣♦rt❛♥t♦ {m1, ..., mr} ❣❡r❛ M✱ ❡ ❛❧é♠ ❞✐ss♦ é ✉♠
❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ ❞❡ ❣❡r❛❞♦r❡s✳ ❉❡ ❢❛t♦✱ s✉♣♦♥❞♦ ♦ ❝♦♥trár✐♦✱ t❡rí❛♠♦s✱
mj ∈ r
X
i=1,i6=j
Ami✱ ♣❛r❛ ❛❧❣✉♠j = 1, ..., r ⇒m¯j =π( r
X
i=1,i6=j
aimi)⇒m¯j = r
X
i=1,i6=j
¯
aim¯i✳
❈♦♥tr❛❞✐çã♦✱ ♣♦✐s ♦s m¯′
is sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✳ ▲♦❣♦✱ {m1, ..., mr} é ✉♠
❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ ❞❡ ❣❡r❛❞♦r❡s✳
❉❡✜♥✐çã♦ ✶✳✸ ❙❡❥❛ (A,m) ✉♠ ❛♥❡❧ ❧♦❝❛❧✳ ❆ ❝❛r❞✐♥❛❧✐❞❛❞❡ ❞❡ ✉♠ ❝♦♥❥✉♥t♦ ♠✐♥✐♠❛❧ ❞❡
❣❡r❛❞♦r❡s ❞❡ ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ M ✭t❛❧ ♥ú♠❡r♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ❞❡✈✐❞♦ ❛♦
t❡♦r❡♠❛ ✶✳✷✮✱ s❡rá ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡s ❞❡ M✱ ❞❡♥♦t❛❞♦ ♣♦r µA(M)✱ ❡ ❞❛❞♦ ♣♦r
µA(M) = dimk(mMM )✳
❯♠ ❝❛s♦ ❡s♣❡❝✐❛❧ ❛ s❡r ❝♦♥s✐❞❡r❛❞♦ é q✉❛♥❞♦ ♦ ♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ é ♦ ♣ró♣r✐♦m❀ ♥❡st❡ ❝❛s♦✱ µA(m) = dimk(mm2)✱ q✉❡ s❡rá ❝❤❛♠❛❞❛ ❞✐♠❡♥sã♦ ❞❡ ✐♠❡rsã♦ ❞❡A✱ ❞❡♥♦t❛❞♦ ♣♦r
edim(A)✳
Pr♦♣♦s✐çã♦ ✶✳✹ ❙❡❥❛ M ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s♦❜r❡ A✳ ❊♥tã♦ M é ❧✐✈r❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ M ∼=Ar✳
Pr♦✈❛✿✳ ❙❡ M é ❧✐✈r❡✱ ❡♥tã♦M ♣♦ss✉✐ ✉♠❛ ❜❛s❡{m1, ..., mr} ⊂M✳ ❆ss✐♠ ❞❡✜♥❛✱
φ:Ar →M✱φ r
X
i=1
aiei
!
=
r
X
i=1
aimi✳
◆♦t❡ q✉❡φé s♦❜r❡❥❡t✐✈❛ ♣♦✐s ♦sm′
is❣❡r❛♠M✱ ❡ ❛❧é♠ ❞✐ss♦φé ✐♥❥❡t✐✈❛ ❥á q✉❡φ r
X
i=1
aiei
! = 0⇔ r X i=1
aimi = 0⇔ai = 0,∀i= 1, ..., r ⇔ r
X
i=1
aiei = 0✳
▲♦❣♦✱ φ é ✉♠ ✐s♦♠♦r✜s♠♦✳
❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛ q✉❡ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦
φ:Ar →M✳
❆ss✐♠✱ ❞❛❞♦ q✉❛❧q✉❡r m ∈ M t❡♠♦s✱ ♣❡❧❛ s♦❜r❡❥❡t✐✈✐❞❛❞❡ ❞❡ φ✱ q✉❡ m = φ(x) ♣❛r❛
❛❧❣✉♠ x = Pri=1aiei ∈ Ar ⇒ m = φ r
X
i=1
aiei
!
✱ ❡ ♣♦r ❧✐♥❡❛r✐❞❛❞❡✱ m =
r
X
i=1
aiφ(ei)
❞❡ ♠♦❞♦ q✉❡ φ(e1), ..., φ(er) ❣❡r❛♠ M✳ ❋✐♥❛❧♠❡♥t❡ ❝♦♠♦ φ é ✐♥❥❡t✐✈❛ ♣♦❞❡♠♦s ❡s❝r❡✈❡r✿
m = 0 ⇔ φ
r
X
i=1
aiei
!
= 0 ⇔ Pri=1aiei = 0 ⇔ai = 0,∀i = 1, ..., r✱ ❡ ❛ss✐♠ φ(e1), ..., φ(er)
sã♦ ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡s✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡{φ(e1), ..., φ(er)} é ✉♠❛ ❜❛s❡ ♣❛r❛ M✳
❉❡✜♥✐çã♦ ✶✳✺ ❉✐③❡♠♦s q✉❡ ✉♠A✲♠ó❞✉❧♦M t❡♠ ♣♦st♦✭❣❡♥ér✐❝♦✱ ❝♦♥st❛♥t❡✮rs❡✱ ♣❛r❛ t♦❞♦ P ∈Ass(A)✱ ♦sAP✲♠ó❞✉❧♦sMP ❡ArP ❢♦r❡♠ ✐s♦♠♦r❢♦s✳ ◆♦t❛çã♦✿ rk(M) = r✳ ❊♠ ♣❛rt✐❝✉❧❛r✱
s❡M ∼=Ar ✭✐st♦ é✱M é ❧✐✈r❡✮ ❡♥tã♦ rk(M) = r✳
✶✳✷ ❉✐♠❡♥sã♦ ❞❡ ❑r✉❧❧ ❡ s✐st❡♠❛s ❞❡ ♣❛râ♠❡tr♦s
❙❡A é ✉♠ ❛♥❡❧ ❞❡♥♦t❛r❡♠♦s ♣♦rSpec(A)✱ ❝♦♠♦ ❞❡ ❝♦st✉♠❡✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ✐❞❡❛✐s ♣r✐♠♦s
❞❡A ✭❝♦♠♦ s❛❜❡♠♦s Spec(A) é ✉♠ ❡s♣❛ç♦ t♦♣♦❧ó❣✐❝♦ ♠✉♥✐❞♦ ❞❛ t♦♣♦❧♦❣✐❛ ❞❡ ❩❛r✐s❦✐✮✳
❉❡✜♥✐çã♦ ✶✳✻ ❙❡❥❛P ∈Spec(A)✳ ❆ ❛❧t✉r❛ ❞❡P é ♦ s✉♣r❡♠♦ ❞♦s ❝♦♠♣r✐♠❡♥t♦st❞❡ ❝❛❞❡✐❛s
❡str✐t❛s✱
P0 ⊂P1 ⊂...⊂Pt=P✱
❞❡ ✐❞❡❛✐s ♣r✐♠♦s✳ ❚❛❧ ♥ú♠❡r♦ é ❞❡♥♦t❛❞♦ ♣♦rht(P)✳
P❛r❛ ✉♠ ✐❞❡❛❧ ❛r❜✐trár✐♦ I✱ t❡♠♦s
ht(I) = inf{ht(P) | P ∈Spec(A), P ⊃I}✳
❚❡♦r❡♠❛ ✶✳✼ ✭❚❡♦r❡♠❛ ❞♦ ✐❞❡❛❧ ♣r✐♠♦ ❞❡ ❑r✉❧❧✮ ❙❡❥❛♠A✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦ ❡I = (x1, ..., xn) ✉♠ ✐❞❡❛❧ ♣ró♣r✐♦✳ ❊♥tã♦ ht(P)≤n ♣❛r❛ t♦❞♦ ✐❞❡❛❧ ♣r✐♠♦ P q✉❡ é ♠í♥✐♠♦ ❡♥tr❡
♦s ✐❞❡❛✐s ♣r✐♠♦s ❝♦♥t❡♥❞♦ I✳
❈♦♠♦ ❝♦♥s❡q✉ê♥❝✐❛✱ t❡♠♦s q✉❡ t♦❞♦ ✐❞❡❛❧ ♣ró♣r✐♦ ❞❡ ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦ t❡♠ ❛❧t✉r❛ ✜♥✐t❛✳
❚❡♦r❡♠❛ ✶✳✽ ❙❡❥❛♠ A ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦ ❡ I ✉♠ ✐❞❡❛❧ ♣ró♣r✐♦ ❝♦♠ ❛❧t✉r❛ ♥✳ ❊♥tã♦
❡①✐st❡♠ x1, x2, ..., xn ∈I✱ t❛✐s q✉❡ ht(x1, ..., xi) = i ♣❛r❛ i= 1,2, ..., n✳
❉❡✜♥✐çã♦ ✶✳✾ ❆ ❞✐♠❡♥sã♦ ❞❡ ❑r✉❧❧ ❞❡ ✉♠ ❛♥❡❧ A é ♦ s✉♣r❡♠♦ ❞❛s ❛❧t✉r❛s ❞❡ s❡✉s ✐❞❡❛✐s
♣r✐♠♦s✱
dim(A) = sup{ht(P) | P ∈Spec(A)}✳
❊①❡♠♣❧♦ ✶✳✶✵ ◆♦ ❛♥❡❧ ❧♦❝❛❧ (AP, PP) t❡♠♦s q✉❡ dim(AP) = ht(P)✱ ♣♦✐s ❡①✐st❡ ✉♠❛
❝♦rr❡s♣♦♥❞ê♥❝✐❛ ❡♥tr❡Spec(AP) ❡ ♦ ❝♦♥❥✉♥t♦ ❢♦r♠❛❞♦ ♣❡❧♦s Q∈Spec(A) t❛✐s q✉❡ Q⊆P✳
❱❡❥❛ ❛✐♥❞❛ q✉❡ ♣❛r❛ t♦❞♦ ✐❞❡❛❧ ♣ró♣r✐♦ I ⊂A ✈❛❧❡ ❛ ❞❡s✐❣✉❛❧❞❛❞❡
ht(I) + dim AI≤dim(A)✳
❉❡ ❢❛t♦✱ s❡ dim(A) = ∞ ♥❛❞❛ ❤á ❛ ❞✐s❝✉t✐r✳ ❈❛s♦ ❝♦♥trár✐♦✱ s❡ dim(A) = n < ∞✱ s❡❥❛
P ∈Spec(A)t❛❧ q✉❡ ht(I) = ht(P) = k✱ ❛ss✐♠ ❡①✐st❡P0 ⊂P1 ⊂...⊂Pk =P✳ ❆ss✐♠✱ ❡♠ AI✱
♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ ❝❛❞❡✐❛ ❞❡ ♣r✐♠♦s P I =
Q0
I ⊂ Q1
I ⊂...⊂ Ql
I ✱ ♦♥❞❡ ❝❛❞❛ Qi ∈Spec(A)
❝♦♥té♠ I ❡l é ❛ ❞✐♠❡♥sã♦ ❞❡ A I✳
▲♦❣♦✱ P0 ⊂ P1 ⊂ ... ⊂ Pk = P = Q0 ⊂ Q1 ⊂ ... ⊂ Ql é ✉♠❛ ❝❛❞❡✐❛ ❞❡ ♣r✐♠♦s ❡♠ A ❡✱
♣♦rt❛♥❞♦✱k+l ≤n⇒ht(I) + dim A I
≤dim(A)✳
❊♠ ♣❛rt✐❝✉❧❛r✱ s❡❣✉❡ q✉❡
dim(A
I)≤dim(A),✳
❉❡✜♥✐çã♦ ✶✳✶✶ ❙❡❥❛ M ✉♠ A✲♠ó❞✉❧♦✳ ❊♥tã♦ dim(M) é ♦ s✉♣r❡♠♦ ❞♦s ❝♦♠♣r✐♠❡♥t♦s ❞❡
❝❛❞❡✐❛s ❡str✐t❛s✱
P0 ⊂P1 ⊂...⊂Pt✱ ❝♦♠ Pi ∈Supp(M)✱
♦♥❞❡Supp(M) = {P ∈Spec(A) |MP 6= 0}✳
◆♦ss♦ ❝❛s♦ ❞❡ ✐♥t❡r❡ss❡ é q✉❛♥❞♦M é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s♦❜r❡A✳ ◆❡st❛ s✐t✉❛çã♦ ✈❛❧❡
Supp(M) ={P ∈Spec(A) | P ⊃0 :M}✱ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱
dim(M) = dim A
0:M
✳
❊♠ ♣❛rt✐❝✉❧❛r✱ dim(M)≤dim(A)✳
❉❡✜♥✐çã♦ ✶✳✶✷ ❙❡❥❛♠ (A,m) ✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦ ❡ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡
❣❡r❛❞♦✳ ❯♠❛ s❡q✉ê♥❝✐❛ x = x1, x2, ..., xt ∈ A é ✉♠ s✐st❡♠❛ ❞❡ ♣❛râ♠❡tr♦s ❞❡ M✱ s❡ t é ♦
♠❡♥♦r ✐♥t❡✐r♦ t❛❧ q✉❡
dim( M
(x1,...,xt)M) = 0
♦✉ ❡q✉✐✈❛❧❡♥t❡♠❡♥t❡
Supp( M
(x1,...,xt)M) =m✳
❊♠ ♣❛rt✐❝✉❧❛r✱ (x1, ..., xt)M é ✉♠ s✉❜♠ó❞✉❧♦ m✲♣r✐♠ár✐♦✳ ❖ ✐♥t❡✐r♦ t✱ q✉❡ ❞❡♥♦t❛r❡♠♦s
♣♦rs(M)✱ é ✉♠ ✐♥✈❛r✐❛♥t❡ ❞♦ ♠ó❞✉❧♦M✳
❙❡Supp(M) =m✱ ❡♥tã♦ s(M) = 0✳
❚❡♦r❡♠❛ ✶✳✶✸ ✭❈❤❡✈❛❧❧❡②✲❑r✉❧❧✲❙❛♠✉❡❧✮ ❙❡ A é ✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦ ❡ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❡♥tã♦
s(M) = dim(M)✳
P❛r❛ ❡st❛ ❞❡♠♦♥str❛çã♦ ✈❡❥❛ ❬✸✵❪✳
✶✳✸ ❙❡q✉ê♥❝✐❛s r❡❣✉❧❛r❡s ❡ ♠ó❞✉❧♦s ❈♦❤❡♥✲▼❛❝❛✉❧❛②
❉❡✜♥✐çã♦ ✶✳✶✹ ❙❡❥❛M ✉♠ ♠ó❞✉❧♦ s♦❜r❡ ✉♠ ❛♥❡❧A✳ ❯♠ ❡❧❡♠❡♥t♦x∈Aé ❞✐t♦M✲r❡❣✉❧❛r✱
s❡x é ♥ã♦ ❞✐✈✐s♦r✲❞❡✲③❡r♦ ❞❡ M✱ ♦✉ s❡❥❛✱ s❡ xz = 0 ♣❛r❛ z ∈M✱ ❡♥tã♦ z = 0✳
◆♦t❛çã♦✳ Z(M) ❂ {❞✐✈✐s♦r❡s✲❞❡✲③❡r♦ ❞❡M}✳
❉❡✜♥✐çã♦ ✶✳✶✺ ❯♠❛ s❡q✉ê♥❝✐❛ x = x1, x2, ..., xn✱ ❝♦♠ xi ∈ A é ❞✐t❛ ✉♠❛ s❡q✉ê♥❝✐❛ M✲
r❡❣✉❧❛r ♦✉ ✉♠❛ M✲s❡q✉ê♥❝✐❛ s❡ s❛t✐s❢❛③ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿
✶✳ M
xM 6= 0✳
✷✳ xi é ✉♠ ❡❧❡♠❡♥t♦ (x1,...,xiM
−1)M✲r❡❣✉❧❛r✱ ♣❛r❛ i= 1, ...n✳
❯♠❛ s❡q✉ê♥❝✐❛ q✉❡ s❛t✐s❢❛③ ❛♣❡♥❛s ✷ é ❝❤❛♠❛❞❛ ✉♠❛ M✲s❡q✉ê♥❝✐❛ ❢r❛❝❛✳
◆♦t❡ q✉❡ ♥♦ ❝❛s♦ ❡♠ q✉❡ A é ✉♠ ❛♥❡❧ ❧♦❝❛❧✱ ❡ M 6= 0 ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱
❡♥tã♦ ♦ ❧❡♠❛ ❞❡ ◆❛❦❛②❛♠❛ ❣❛r❛♥t❡ q✉❡ ❛ ❝♦♥❞✐çã♦ ✶ s❡rá s❡♠♣r❡ s❛t✐s❢❡✐t❛✳
❊①❡♠♣❧♦ ✶✳✶✻ ❆ s❡q✉ê♥❝✐❛ x1, ..., xn ❞❛s ✐♥❞❡t❡r♠✐♥❛❞❛s ❡♠ ✉♠ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s A =
K[x1, ...xn]✱ ♦♥❞❡ K é ✉♠ ❝♦r♣♦✱ é ✉♠❛A✲s❡q✉ê♥❝✐❛✳
❖❜s❡r✈❛çõ❡s ✶✳✶✼ ✶✳ ❊♠ ❣❡r❛❧✱ s❡ ✉♠❛ s❡q✉ê♥❝✐❛ x = x1, ..., xn é ✉♠❛ s❡q✉ê♥❝✐❛ M✲
r❡❣✉❧❛r✱ ✉♠❛ ♣❡r♠✉t❛çã♦ ❞❡ s❡✉s ❡❧❡♠❡♥t♦s ♣♦❞❡ ♥ã♦ s❡r✳ ◆♦ ❡♥t❛♥t♦✱ s❡ (A,m) é
✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦t❤❡r✐❛♥♦✱ ❡ M é ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❡♥tã♦ q✉❛❧q✉❡r
♣❡r♠✉t❛çã♦ ❞❡ ✉♠❛ M✲s❡q✉ê♥❝✐❛ ❡♠ m ❛✐♥❞❛ s❡rá ✉♠❛ M✲s❡q✉ê♥❝✐❛✳
✷✳ ❙❡ (A,m) é ✉♠ ❛♥❡❧ ❧♦❝❛❧✱ ❡♥tã♦ t♦❞❛ s❡q✉ê♥❝✐❛ M✲r❡❣✉❧❛r s❡rá ♣❛rt❡ ❞❡ ✉♠ s✐st❡♠❛
❞❡ ♣❛râ♠❡tr♦s✳
❉❡✜♥✐çã♦ ✶✳✶✽ ❙❡❥❛♠I ✉♠ ✐❞❡❛❧ ❞❡ ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦A❡M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡
❣❡r❛❞♦ ❝♦♠ IM 6= M✳ ❯♠❛ M✲s❡q✉ê♥❝✐❛ ♠❛①✐♠❛❧ ❡♠ I é ✉♠❛ M✲s❡q✉ê♥❝✐❛ x = {x1, x2, ..., xn} ⊆I✱ t❛❧ q✉❡ {x1, ..., xn, x} ♥ã♦ éM✲s❡q✉ê♥❝✐❛ ♣❛r❛ q✉❛❧q✉❡r x∈I✳
◗✉❛♥❞♦ A é ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦✱ t♦❞❛ M✲s❡q✉ê♥❝✐❛ ♣♦❞❡ s❡r ❡st❡♥❞✐❞❛ ❛ ✉♠❛ M✲
s❡q✉ê♥❝✐❛ ♠❛①✐♠❛❧✳
▲❡♠❛ ✶✳✶✾ ❙❡❥❛♠ A ✉♠ ❛♥❡❧✱ ❡ M, N A✲♠ó❞✉❧♦s✳ ❚♦♠❡ I = 0 :N✳
✶✳ ❙❡ I ❝♦♥té♠ ✉♠ ❡❧❡♠❡♥t♦ M✲r❡❣✉❧❛r✱ ❡♥tã♦ HomA(N, M) = 0✳
✷✳ ■♥✈❡rs❛♠❡♥t❡✱ s❡Aé ◆♦❡t❤❡r✐❛♥♦ ❡M, N sã♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦s✱ ❝♦♠HomA(N, M) =
0✱ ❡♥tã♦ I ❝♦♥té♠ ✉♠ ❡❧❡♠❡♥t♦ M✲r❡❣✉❧❛r✳
Pr♦✈❛✿✳
✶✳ ❙❡❥❛ a ∈ I = 0 : N M✲r❡❣✉❧❛r✳ ❙❡ ❡①✐st❡ f ∈ HomA(N, M)✱ ♥ã♦ ♥✉❧♦✱ ❡♥tã♦
❡①✐st❡ n ∈ N t❛❧ q✉❡ f(n) = 06 ✳ ❆❣♦r❛ s❛❜❡♠♦s q✉❡ an = 0, ∀n ∈ N✱ ❧♦❣♦ f(an) = 0 ⇒ af(n) = 0 ⇒ a ∈ Z(M)✱ ♦ q✉❡ ❝♦♥tr❛❞✐③ ❛ ❤✐♣ót❡s❡ ❞❡ q✉❡ a é M✲
r❡❣✉❧❛r✳
✷✳ ❋❛ç❛♠♦s ♣♦r ❝♦♥tr❛✲♣♦s✐t✐✈❛✳
❙✉♣♦♥❤❛♠♦s q✉❡I ⊆ Z(M)✳ ❙❡♥❞♦A◆♦❡t❤❡r✐❛♥♦✱ t❡♠♦s q✉❡Z(M) = SP∈Ass(M)P ❡✱
♣❡❧♦ ❧❡♠❛ ❞❛ ❡sq✉✐✈❛✱ I ⊆P ♣❛r❛ ❛❧❣✉♠ P ∈ Ass(M)✱ ✐st♦ é✱ P = 0 : (m) ♣❛r❛ ❛❧❣✉♠
m 6= 0∈M✳ ❆ss✐♠ ❛ ❛♣❧✐❝❛çã♦ ϕ : A
P −→M ❞❛❞❛ ♣♦rϕ(a+P) = am
❡stá ❜❡♠ ❞❡✜♥✐❞❛ ❡ ❛✐♥❞❛ é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦✳
❈♦♠♦ N é ✉♠A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❡♥tã♦✱ ❞❛❞♦n ∈N✱ ❡①✐st❡♠a1, ..., ar ∈A
❡ n1, ..., nr∈N t❛❧ q✉❡ n = r
X
i=1
aini✱ ✐♥❞✉③✐♥❞♦ ♦ s❡❣✉✐♥t❡ ❤♦♠♦♠♦r✜s♠♦✱
ψ :N → A P✱ψ(
r
X
i=1
aini) =a1+P✱
q✉❡ é s♦❜r❡❥❡t✐✈♦✳
P♦rt❛♥t♦✱ φ=ϕ◦ψ ∈Hom(N, M) ❡ φ6= 0✳ ❖ q✉❡ t❡r♠✐♥❛ ❛ ❞❡♠♦♥str❛çã♦✳
▲❡♠❛ ✶✳✷✵ ❙❡❥❛♠A ✉♠ ❛♥❡❧✱M, N A✲♠ó❞✉❧♦s✱ ❡ x=x1, ..., xn ✉♠❛M✲s❡q✉ê♥❝✐❛ ❢r❛❝❛ ❡♠
0 :N✳ ❊♥tã♦
HomA N,xMM
∼
= ExtnA(N, M)✳
Pr♦✈❛✿✳ ❋❛ç❛♠♦s ♣♦r ✐♥❞✉çã♦ s♦❜r❡n✳
✶✳ n = 0
❙❛❜❡♠♦s✱ ❞❛ ❝♦♥str✉çã♦ ❞♦ ❢✉♥t♦r ❊①t✱ q✉❡ HomA(N, M)∼= Ext0A(N, M)✳
✷✳ n ≥1
❈♦♠♦ x′ =x1, x2, ..., xn−1 é ✉♠❛ s❡q✉ê♥❝✐❛ M✲r❡❣✉❧❛r ❢r❛❝❛✱ ♣♦r ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ HomA N,xM′M
∼
= ExtnA−1(N, M)✳
❆❣♦r❛✱ ❝♦♠♦xn ∈ Z/ xM′M
❡xn ∈0 :N✱ ❡♥tã♦HomA N,xM′M
= 0⇒ExtnA−1(N, M) =
0✳
❈♦♥s✐❞❡r❡ ❛ s❡q✉ê♥❝✐❛ ❡①❛t❛✱
0−→M ·x1
−→M −→ M
x1M −→0✳
P❡❧❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❧♦♥❣❛ ❞♦ ❊①t t❡♠♦s
0 = Extn−1
A (N, M)−→Ext n−1
A (N, M x1M)
∆
−→Extn
A(N, M) ϕ
−→Extn
A(N, M)−→
ExtnA(N,xM1M)−→...
❆ss✐♠✱ s❡♥❞♦ ϕ ✐♥❞✉③✐❞❛ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r x1 ∈ 0 : N t❡♠♦s q✉❡ ϕ ≡ 0✱ ❧♦❣♦✱ Extn−1
A (N, M x1M)
∼
= ExtnA(N, M) ✈✐❛∆✳
❈♦♠♦ x2, ..., xn é M✲r❡❣✉❧❛r ❢r❛❝❛✱ ❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦ ExtnA−1(N, M x1M)
∼
=
HomA N,xMM
❣❛r❛♥t❡ q✉❡ Extn
A(N, M)∼= HomA N,xMM
✳
❚❡♦r❡♠❛ ✶✳✷✶ ✭❘❡❡s✮ ❙❡❥❛ A ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦✱ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱
❡ I ✉♠ ✐❞❡❛❧ ❞❡ A t❛❧ q✉❡ IM 6=M✳ ❊♥tã♦ t♦❞❛ M✲s❡q✉ê♥❝✐❛ ♠❛①✐♠❛❧ ❡♠ I t❡♠ ♦ ♠❡s♠♦
❝♦♠♣r✐♠❡♥t♦✱
n=mini | ExtAi (AI, M)6= 0 ✳
Pr♦✈❛✿✳ ❙❡❥❛ x=x1, ..., xn ✉♠❛ M✲s❡q✉ê♥❝✐❛ ♠❛①✐♠❛❧ ❡♠ I✳ ❈♦♠♦ I ❝♦♥té♠ ✉♠ ❡❧❡♠❡♥t♦ M
(x1,...,xi−1)M✲r❡❣✉❧❛r✱ i= 1, ..., n✱ t❡♠♦s
Exti−1
A A I, M
∼
= HomA
A I,
M
(x1,...,xi−1)M
= 0✳
▲♦❣♦✱
ExtjA A I, M
= 0 ∀j ≤n−1✳
❊ ♠❛✐s✱
ExtnA AI, M
∼
= HomA AI,xMM
6
= 0✳
Pr♦✈❛♥❞♦ ♦ t❡♦r❡♠❛✳
❊st❡ r❡s✉❧t❛❞♦ ♥♦s ♣❡r♠✐t❡ ✐♥tr♦❞✉③✐r ❛ s❡❣✉✐♥t❡ ❞❡✜♥✐çã♦✳
❉❡✜♥✐çã♦ ✶✳✷✷ ❙❡❥❛A ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦✱ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❡ I ✉♠
✐❞❡❛❧ ❞❡ A t❛❧ q✉❡IM 6=M✳ ❊♥tã♦✱ ♦ ❝♦♠♣r✐♠❡♥t♦ ❝♦♠✉♠ ❞❛s M✲s❡q✉ê♥❝✐❛s ♠❛①✐♠❛✐s ❡♠ I s❡rá ❝❤❛♠❛❞♦ ❣r❛❞❡ ❞❡ I ❡♠ M✱
grade(I, M)✳
◆♦ ❝❛s♦ ❡s♣❡❝✐❛❧ ❡♠ q✉❡ M =A ❡s❝r❡✈❡r❡♠♦s grade(I)❛♦ ✐♥✈és ❞❡ grade(I, A)✳
❉❡✜♥✐çã♦ ✶✳✷✸ ❙❡❥❛ (A,m, k) ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦ ❧♦❝❛❧✱ ❡ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t♠❡♥t❡
❣❡r❛❞♦✳ ❆ ♣r♦❢✉♥❞✐❞❛❞❡ ❞❡M é ♦ ♥ú♠❡r♦
prof(M) = grade(m, M)✳
❉❡✜♥✐çã♦ ✶✳✷✹ ❙❡❥❛A✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦✳ ❯♠A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦M 6= 0
é ✉♠ ♠ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② s❡prof(M) = dim(M)✳ ❙❡A ❢♦r ❈♦❤❡♥✲▼❛❝❛✉❧❛② ✈✐st♦ ❝♦♠♦
♠ó❞✉❧♦ s♦❜r❡ ❡❧❡ ♣r♦♣r✐♦✱ é ❞✐t♦ ✉♠ ❛♥❡❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳
❉❡✜♥✐çã♦ ✶✳✷✺ ❯♠ ♠ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ▼❛①✐♠❛❧ é ✉♠ ♠ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② M
t❛❧ q✉❡ dim(M) = dim(A)✳
❋✐♥❛❧✐③❛♠♦s ❡st❛ s❡çã♦ ❝♦♠ ♠❛✐s ❛❧❣✉♠❛s ♥♦çõ❡s ❡①tr❡♠❛♠❡♥t❡ ✐♠♣♦rt❛♥t❡s✳
❉❡✜♥✐çã♦ ✶✳✷✻ ❯♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦(A,m) é r❡❣✉❧❛r s❡ m ♣♦❞❡ s❡r ❣❡r❛❞♦ ♣♦r ✉♠❛
A✲s❡q✉ê♥❝✐❛✱ ❝❤❛♠❛❞❛ s✐st❡♠❛ r❡❣✉❧❛r ❞❡ ♣❛râ♠❡tr♦s✳
❊①❡♠♣❧♦ ✶✳✷✼ ❯♠ ❛♥❡❧ ❧♦❝❛❧ r❡❣✉❧❛r (A,m) é ✉♠ ❛♥❡❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❉❡ ❢❛t♦✱ ✉♠
s✐st❡♠❛ r❡❣✉❧❛r ❞❡ ♣❛râ♠❡tr♦s s❡rá ✉♠❛ s❡q✉ê♥❝✐❛ r❡❣✉❧❛r ♠❛①✐♠❛❧ ❡♠m✳
❉❡✜♥✐çã♦ ✶✳✷✽ ❯♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦B✱ s❡rá ✉♠ ❛♥❡❧ ✐♥t❡rs❡çã♦ ❝♦♠♣❧❡t❛ s❡B ∼= AI✱
♦♥❞❡A é ✉♠ ❛♥❡❧ ❧♦❝❛❧ r❡❣✉❧❛r ❡I ⊂A é ✉♠ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦r ✉♠❛ A✲s❡q✉ê♥❝✐❛✳
❉❡✜♥✐çã♦ ✶✳✷✾ ❯♠ ❛♥❡❧ q✉♦❝✐❡♥t❡B =A/I ❞❡ ✉♠ ❛♥❡❧ r❡❣✉❧❛r ❧♦❝❛❧(A,m)é ❝❤❛♠❛❞♦ ✉♠❛
✐♥t❡rs❡❝çã♦ ❝♦♠♣❧❡t❛ ❡str✐t❛ s❡ ♦ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❛ss♦❝✐❛❞♦grm
I(B)é ✉♠❛ ✐♥t❡rs❡çã♦ ❝♦♠♣❧❡t❛✳
▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s❡grm
I(B) =
grmA
(f∗
1,...,fn∗)✱ ♦♥❞❡f
∗
1, ..., fn∗ é ✉♠❛grm(A)✲s❡q✉ê♥❝✐❛✳ ◆❡st❡ ❝❛s♦✱
f1, ..., fn é ✉♠❛A✲s❡q✉ê♥❝✐❛✱ ❡ B = (f1,...,fnA )✳
✶✳✹ ❘❡s♦❧✉çã♦ ❧✐✈r❡ ❡ ❞✐♠❡♥sã♦ ❤♦♠♦❧ó❣✐❝❛
❙❡❥❛ A ✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦✳ ❉❛❞♦ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ♣♦r m1, ..., mn✱ ❡①✐st❡ ✉♠❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ♥❛t✉r❛❧✱
0→Ker(ϕ0)−→An ϕ0
−→M →0✱
♦♥❞❡ ❛ ❛♣❧✐❝❛çã♦A✲❧✐♥❡❛r ϕ0 é ❞❛❞❛ ♣♦r ϕ0(a1, ..., an) = a1m1+...+anmn✳
❆ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❝✉rt❛ ❛❝✐♠❛ é ❝❤❛♠❛❞❛ ❛♣r❡s❡♥t❛çã♦ ❧✐✈r❡ ❞❡ M✱ ❡ ♦ ♠ó❞✉❧♦ ker(ϕ0) = {(a1, ..., an)∈An |a1m1+...+anmn = 0} é ❞✐t♦ ♦ ♣r✐♠❡✐r♦ ♠ó❞✉❧♦ ❞❡ s✐③✐❣✐❛s
❞❡ M✱ ❞❡♥♦t❛❞♦ ♣♦rSyz(M)✳
◆♦t❡ q✉❡✱ s❡♥❞♦A◆♦❡t❤❡r✐❛♥♦✱ t❡♠♦s q✉❡Syz(M)é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❞✐❣❛♠♦s✱ ❣❡r❛❞♦
♣♦r=n1✳ ❆ss✐♠ é ♣♦ssí✈❡❧ ❡♥❝♦♥tr❛r ✉♠ ❛♣r❡s❡♥t❛çã♦ ❧✐✈r❡ ❞❡ Syz(M)✱ 0→Syz(Syz(M)) = Syz2(M)−→An1 −→Syz(M)→0✱ ♦♥❞❡ ♣♦r ❝♦♠♣♦s✐çã♦ ❣❛♥❤❛♠♦s✱
❊ss❡ ♣r♦❝❡ss♦ ♣♦❞❡ s❡r ❝♦♥t✐♥✉❛❞♦✱ ✐♥❞✉③✐♥❞♦ ✉♠❛ s❡q✉ê♥❝✐❛
...−→ϕi+1 Ani −→...−→An1 −→ϕ1 An −→ϕ0 M →0✳
❱❡❥❛ q✉❡ ker(ϕi) = Im(ϕi+1)✱ ❡ ♣♦rt❛♥t♦✱ ❛ s❡q✉ê♥❝✐❛ ❛❝✐♠❛ s❡rá ✉♠❛ s❡q✉ê♥❝✐❛ ❡①❛t❛
❧♦♥❣❛ ❞❡ ♠ó❞✉❧♦s ❧✐✈r❡s✱ ♦✉ ❛✐♥❞❛✱ ✉♠❛ r❡s♦❧✉çã♦ ❧✐✈r❡ ❞❡ M✳ ❊✈❡♥t✉❛❧♠❡♥t❡ ❡❧❛ ♣♦❞❡ s❡r
✐♥✜♥✐t❛ ❡ ♣♦❞❡♠♦s tr✉♥❝á✲❧❛ ❡♠ q✉❛❧q✉❡r ❡t❛♣❛✱
0→Syzn(M) = ker(ϕn−1)−→Ani−1 −→...−→An1 ϕ1
−→An ϕ0
−→M →0✱
❞❡✐①❛♥❞♦ ❞❡ s❡r ✉♠❛ r❡s♦❧✉çã♦ ❧✐✈r❡ ❛ ♠❡♥♦s q✉❡ Syzn(M)s❡❥❛ ✉♠ ♠ó❞✉❧♦ ❧✐✈r❡✳
❚♦♠❛♥❞♦ n = µ(M) ❡ ni = µ(Syzi(M))✱ ♣❛r❛ ❝❛❞❛ i ≥ 1✱ t❛❧ r❡s♦❧✉çã♦ ❧✐✈r❡ é ♠✐♥✐♠❛❧✳
◆❡st❡ ❝❛s♦ ♦ ♥ú♠❡r♦ ♠í♥✐♠♦ ❞❡ ❣❡r❛❞♦r❡sni =µ(Syzi(M))é ❝❤❛♠❛❞♦ ♦ ✐✲és✐♠♦ ♥ú♠❡r♦ ❞❡
❇❡tt✐ ❞❡M✱ ❞❡♥♦t❛❞♦ ♣♦rβi(M)✳ ❆❧é♠ ❞✐ss♦✱Syzi(M)é ú♥✐❝♦ ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦✱ ❡ t❛❧
r❡s♦❧✉çã♦ ❧✐✈r❡ ♠✐♥✐♠❛❧ é ❞❡t❡r♠✐♥❛❞❛ ♣♦r M ❛ ♠❡♥♦s ❞❡ ✐s♦♠♦r✜s♠♦s ❞❡ ❝♦♠♣❧❡①♦s✳ ❈♦♠
✐st♦ ♣♦❞❡♠♦s ❞❡✜♥✐r✿
❉❡✜♥✐çã♦ ✶✳✸✵ ❖ ❝♦♠♣r✐♠❡♥t♦ ❞❡ ✉♠❛ r❡s♦❧✉çã♦ ❧✐✈r❡ ♠✐♥✐♠❛❧ ❞❡M s♦❜r❡ ♦ ❛♥❡❧ ❧♦❝❛❧Aé
❛ ❞✐♠❡♥sã♦ ❤♦♠♦❧ó❣✐❝❛ ❞❡M✳ ❚❛❧ ♥ú♠❡r♦ é ❞❡♥♦t❛❞♦ ♣♦r hdA(M)✱ ♦✉ s✐♠♣❧❡s♠❡♥t❡hd(M)
q✉❛♥❞♦ ♥ã♦ ❤♦✉✈❡r r✐s❝♦ ❞❡ ❛♠❜✐❣✉✐❞❛❞❡✳
❉❡✜♥✐çã♦ ✶✳✸✶ ❙❡❥❛A✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦✳ ❯♠A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦M 6= 0
é ❞✐t♦ ♣❡r❢❡✐t♦ s❡
hd(M) = grade(0 :M)✳
❯♠ ✐❞❡❛❧I ⊂A s❡rá ♣❡r❢❡✐t♦ s❡ AI ❢♦r ✉♠ ♠ó❞✉❧♦ ♣❡r❢❡✐t♦✱ ♦✉ s❡❥❛✱ hd(AI) = grade(I)✳
✶✳✺ ▼ó❞✉❧♦s ❝❛♥ô♥✐❝♦s ❡ ❛♥é✐s ●♦r❡♥st❡✐♥
❙❡❥❛(A,m, k)✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦✳
❉❡✜♥✐çã♦ ✶✳✸✷ ❆ ❞✐♠❡♥sã♦ ❞♦k✲❡s♣❛ç♦ ✈❡t♦r✐❛❧
Extn(k, M)6= 0✱ ♦♥❞❡n = prof(M)✱
s❡rá ♦ t✐♣♦ ❞❡ M✱ ♦✉ s❡❥❛✱
tipo(M) = dimk(Extn(k, M))✳
❉❡✜♥✐çã♦ ✶✳✸✸ ❯♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦(A,m)é ❞✐t♦ ●♦r❡♥st❡✐♥ s❡
Supi| Exti(k, A)6= 0 <∞✱
♦♥❞❡Supi |Exti(k, A)6= 0 é ❛ ❞✐♠❡♥sã♦ ✐♥❥❡t✐✈❛ ❞❡A✭injdim(A)✮✳ ❯♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦
A é ●♦r❡♥st❡✐♥✱ s❡ ♣❛r❛ t♦❞♦ ✐❞❡❛❧ ♠❛①✐♠❛❧ m⊂A✱ ❛ ❧♦❝❛❧✐③❛çã♦Am é ✉♠ ❛♥❡❧ ●♦r❡♥st❡✐♥✳
◆♦t❡ q✉❡ prof(M)≤injdim(A)✳
❉❡✜♥✐çã♦ ✶✳✸✹ ❯♠ ✐❞❡❛❧ I ⊂ A✱ ♦♥❞❡ A é ✉♠ ❛♥❡❧ ❧♦❝❛❧ ❧♦❝❛❧ r❡❣✉❧❛r ✭♦✉ ✉♠ ❛♥❡❧ ❞❡
♣♦❧✐♥ô♠✐♦s s♦❜r❡ ✉♠ ❝♦r♣♦ k✮✱ s❡rá ✉♠ ✐❞❡❛❧ ●♦r❡♥st❡✐♥ s❡ I é ♣❡r❢❡✐t♦ ❡ ExtgA(A I, A) ∼=
A I✱
♦♥❞❡g = grade(I)✳
❉❡✜♥✐çã♦ ✶✳✸✺ ❙❡❥❛M ✉♠A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s♦❜r❡ ✉♠ ❛♥❡❧ ❧♦❝❛❧(A,m)✳ ❊♥tã♦✱ Soc(M) = (0 :M m)∼= Hom(k, M)✳
é ♦ s♦❝❧❡ ❞❡ M✳
Pr♦♣♦s✐çã♦ ✶✳✸✻ ❙❡❥❛♠ ✉♠ ❛♥❡❧ ❧♦❝❛❧(A,m)✱ M ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ s♦❜r❡A❡x=x1, ..., xn
✉♠❛ M✲s❡q✉ê♥❝✐❛ ♠❛①✐♠❛❧ ❡♠ m✳ ❊♥tã♦ tipo(M) = dimkSoc xMM
✳
❚❡♦r❡♠❛ ✶✳✸✼ A é ●♦r❡♥st❡✐♥ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A é ❈♦❤❡♥✲▼❛❝❛✉❧❛② ❞❡ t✐♣♦ ✶✳
❉❡✜♥✐çã♦ ✶✳✸✽ ❙❡❥❛ (A,m) ✉♠ ❛♥❡❧ ❧♦❝❛❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❯♠ ♠ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛②
♠❛①✐♠❛❧ M✱ ❞❡ t✐♣♦ ✶ ❡ ❞❡ ❞✐♠❡♥sã♦ ✐♥❥❡t✐✈❛ ✜♥✐t❛✱ é ❝❤❛♠❛❞♦ ♠ó❞✉❧♦ ❝❛♥ô♥✐❝♦✱ ❡ s❡rá
❞❡♥♦t❛❞♦ ♣♦rωA✳
✶✳✻ ▼✉❧t✐♣❧✐❝✐❞❛❞❡
❉❡✜♥✐çã♦ ✶✳✸✾ ❯♠ ❛♥❡❧ A 6= 0 é ❞✐t♦ ❣r❛❞✉❛❞♦ ✭N✲❣r❛❞✉❛❞♦✮✱ s❡ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛
{An}n∈N ❞❡ s✉❜❣r✉♣♦s ❛❞✐t✐✈♦sAn ⊂A✱ s❛t✐s❢❛③❡♥❞♦✿
✶✳ A =Ln≥0An❀
✷✳ AiAj ⊆Ai+j,∀i, j ∈N✳
An é ❞✐t❛ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❤♦♠♦❣ê♥❡❛ ❞❡ ❣r❛✉ n ❞❡ A✳ ❈❛❞❛ ❡❧❡♠❡♥t♦ ❞❡ An é ✉♠
❡❧❡♠❡♥t♦ ❤♦♠♦❣ê♥❡♦ ❞❡ ❣r❛✉ n✳
❙❡❣✉❡♠✲s❡ ❛s s❡❣✉✐♥t❡s ♦❜s❡r✈❛çõ❡s✿
✶✳ 0∈An,∀n∈N✱ ♣♦✐sAn é s✉❜❣r✉♣♦ ❛❞✐t✐✈♦✱ ❧♦❣♦ 0t❡♠ t♦❞♦s ♦s ❣r❛✉s✳
✷✳ ❚♦❞♦ ❡❧❡♠❡♥t♦ ❞❡A é ✉♠❛ s♦♠❛ ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s✳
✸✳ A0 é s✉❜❛♥❡❧ ❞❡A✱ ❧♦❣♦ An éA0✲♠ó❞✉❧♦ ∀n ∈N✳
❉❡✜♥✐çã♦ ✶✳✹✵ ❉✐③❡♠♦s q✉❡ ✉♠ A✲♠ó❞✉❧♦ M é ❣r❛❞✉❛❞♦✱ s❡ ❡①✐st❡ ✉♠❛ ❢❛♠í❧✐❛ ❞❡
s✉❜❣r✉♣♦s ❛❞✐t✐✈♦s {Mn}n∈ λ t❛❧ q✉❡✿
✶✳ M =Ln≥0Mn❀
✷✳ AiMj ⊆Mi+j,∀i, j ≥0✳
❯♠ ❡❧❡♠❡♥t♦x∈M é ❞✐t♦ ❤♦♠♦❣ê♥❡♦ s❡x∈Mn✱ ♣❛r❛ ❛❧❣✉♠n✳ ❈❛❞❛Mn é ✉♠A0✲♠ó❞✉❧♦✱
❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡ M s❡rá ✉♠❛ s♦♠❛ ✜♥✐t❛ ❞❡ ❡❧❡♠❡♥t♦s ❤♦♠♦❣ê♥❡♦s✳
❉❡✜♥✐çã♦ ✶✳✹✶ ❉✐③❡♠♦s q✉❡ ✉♠ A✲♠ó❞✉❧♦ M 6= 0 é ✉♠ ♠ó❞✉❧♦ s✐♠♣❧❡s✱ s❡ s❡✉s ú♥✐❝♦s
A✲s✉❜♠ó❞✉❧♦s sã♦0 ❡ ♦ ♣ró♣r✐♦ M✳
❉❡✜♥✐çã♦ ✶✳✹✷ ❙❡ M éA✲♠ó❞✉❧♦✱ ✉♠❛ ❝❛❞❡✐❛
0 =M0 ⊂M1 ⊂...⊂Mr=M
❞❡ s✉❜♠ó❞✉❧♦s ❞❡ M é ❝❤❛♠❛❞❛ ✉♠❛ sér✐❡ ❞❡ ❝♦♠♣♦s✐çã♦ ❞❡ M q✉❛♥❞♦ MiMi
+1 é s✐♠♣❧❡s✱ ∀i= 1, ..., r✳
❙❡ ✉♠❛ sér✐❡ ❞❡ ❝♦♠♣♦s✐çã♦ ❞❡M ❡①✐st❡✱ ❡♥tã♦ ♦ s❡✉ ❝♦♠♣r✐♠❡♥t♦ é ✉♠ ✐♥✈❛r✐❛♥t❡ ❞❡M
✭✐♥❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❛ sér✐❡ ❞❡ ❝♦♠♣♦s✐çã♦✮ ❝❤❛♠❛❞♦ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡M✱ ❞❡♥♦t❛❞♦ ♣♦r l(M)✳
❆ss✉♠✐r❡♠♦s ❛❣♦r❛ q✉❡ A0 é ✉♠ ❛♥❡❧ ❧♦❝❛❧ ❆rt✐♥✐❛♥♦ ❡ q✉❡ A s❡rá ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦
❝♦♠♦A0−♠ó❞✉❧♦✳
❉❡✜♥✐çã♦ ✶✳✹✸ ❙❡❥❛ A ❝♦♠♦ ❛❝✐♠❛ ❡ M ✉♠ A✲♠ó❞✉❧♦ ❣r❛❞✉❛❞♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳
❉❡✜♥✐♠♦s✿
H(M,−) :N→N✱ H(M, n) = l(Mn)✱
❝♦♠♦ s❡♥❞♦ ❛ ❢✉♥çã♦ ❞❡ ❍✐❧❜❡rt ❞❡ M✳
◆♦t❡ q✉❡ ❝❛❞❛ ❝♦♠♣♦♥❡♥t❡ ❤♦♠♦❣ê♥❡❛ ❞❡ M✱ Mn✱ s❡rá ✉♠ A0−♠ó❞✉❧♦ ✜♥✐t♦ ❡ ❛ss✐♠
t❡rá ❝♦♠♣r✐♠❡♥t♦ ✜♥✐t♦✳
❆ ♣❛rt✐r ❞❡ ❛❣♦r❛ ❛ss✉♠✐r❡♠♦s q✉❡A é ❣❡r❛❞♦✱ s♦❜r❡A0, ♣♦r ❡❧❡♠❡♥t♦s ❞❡ ❣r❛✉ 1✱ ✐st♦ é
A=A0[A1]✳ ❆❞♠✐t✐r❡♠♦s ♦ s❡❣✉✐♥t❡ r❡s✉❧t❛❞♦✿
❚❡♦r❡♠❛ ✶✳✹✹ ❙❡❥❛ M ✉♠ A✲♠ó❞✉❧♦ ❣r❛❞✉❛❞♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ ❞❡ ❞✐♠❡♥sã♦ d✳ ❊♥tã♦ H(M,−) é ❞❡ t✐♣♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❣r❛✉ d−1✱ ✐st♦ é✱ H(M, n) é ✉♠❛ ❢✉♥çã♦ ♣♦❧✐♥♦♠✐❛❧ ♣❛r❛
n≫0✳
P❛r❛ ❡st❛ ❞❡♠♦♥str❛çã♦ ✈✐❞❡ [✽]
❉❡✜♥✐çã♦ ✶✳✹✺ ❖ ♣♦❧✐♥ô♠✐♦ p(x) = pM(x) ∈ Q[x] t❛❧ q✉❡ H(M, n) = p(n) ♣❛r❛ n ≫ 0 é
❝❤❛♠❛❞♦ ♣♦❧✐♥ô♠✐♦ ❞❡ ❍✐❧❜❡rt✱
pM(x) = d−1
X
i=0
(−1)d−1−ie d−1−i
x+i i
✱
♦♥❞❡ei ∈Z✳
❉❡✜♥✐çã♦ ✶✳✹✻ ❙❡❥❛ M ✉♠ A✲♠ó❞✉❧♦ ❣r❛❞✉❛❞♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❆ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ M✱ ❞❡♥♦t❛❞❛ ♣♦r eA(M)✱ s❡rá ❞❡✜♥✐❞❛ ♣♦r
eA(M) =
e0 , se dim(M)>0
l(M) , se dim(M) = 0.
❖❜s❡r✈❛çã♦ ✶✳✹✼ ◗✉❛♥❞♦ ♥ã♦ ❤♦✉✈❡r r✐s❝♦ ❞❡ ❛♠❜✐❣✉✐❞❛❞❡ ❞❡♥♦t❛r❡♠♦s ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❛♣❡♥❛s ♣♦re(M)✳
◆♦t❡ q✉❡ ❞❡✜♥✐♠♦s ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ♣❛r❛ ❛♥é✐s ❡ ♠ó❞✉❧♦s ❣r❛❞✉❛❞♦s✱ ♠❛s ♣r❡❝✐s❛♠♦s ❡①♣❛♥❞✐r t❛❧ ❝♦♥❝❡✐t♦ ♣❛r❛ ♦ ❝❛s♦ ♥ã♦✲❣r❛❞✉❛❞♦✳
❉❡✜♥✐çã♦ ✶✳✹✽ ❙❡❥❛ A ✉♠ ❛♥❡❧✳ ❯♠❛ ✜❧tr❛çã♦ F s♦❜r❡ A é ✉♠❛ ❝❛❞❡✐❛ ❞❡s❝❡♥❞❡♥t❡ A = I0 ⊃ I1 ⊃ I2 ⊃ ... ❞❡ ✐❞❡❛✐s t❛❧ q✉❡ IiIj ⊂ Ii+j ♣❛r❛ t♦❞♦ i ❡ j✳ ❯♠ ❛♥❡❧ ✜❧tr❛❞♦ é
✉♠ ♣❛r(A,F)♦♥❞❡ A é ✉♠ ❛♥❡❧ ❡ F é ✉♠❛ ✜❧tr❛çã♦✳
❆ ✜❧tr❛çã♦ ❞❛❞❛ ♣❡❧❛s ♣♦tê♥❝✐❛s ❞❡ ✉♠ ✐❞❡❛❧ I é ❝❤❛♠❛❞❛ ✜❧tr❛çã♦ I✲á❞✐❝❛✳
❙❡❥❛A ✉♠ ❛♥❡❧ ✜❧tr❛❞♦ ❝♦♠ ❛ ✜❧tr❛çã♦ F = (Ii)i≥0✳ ◆ós
❞❡✜♥✐♠♦s ♦ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❛ss♦❝✐❛❞♦ ❛ A ❝♦♠ r❡s♣❡✐t♦ ❛F ♣♦r
grF(A) =
L∞
i=0
Ii Ii+1✳
P♦❞❡♠♦s ❛❣♦r❛ ❛❞❛♣t❛r ❛ ❞❡✜♥✐çã♦ ❛❝✐♠❛ ♣❛r❛ A✲♠ó❞✉❧♦s✳ ❉❛❞♦ ♦ A✲♠ó❞✉❧♦ M ❡ ✉♠❛
✜❧tr❛çã♦ F✱ t❡♠♦s q✉❡ grF(M) =
L∞
i=0
IiM
Ii+1M s❡rá ✉♠ grF(A)✲♠ó❞✉❧♦✳ ❈❛s♦ F s❡❥❛ ✉♠❛ ✜❧tr❛çã♦ I✲á❞✐❝❛✱ ♥♦ss❛ ♥♦t❛çã♦ s❡rá s✉❜st✐t✉í❞❛ ♣♦r grI(M)✳
❆❣♦r❛ ❝♦♥s✐❞❡r❡ ♦ ❛♥❡❧ ❧♦❝❛❧ ❣r❛❞✉❛❞♦ ❛ss♦❝✐❛❞♦ ❛ A ❝♦♠ r❡s♣❡✐t♦ ❛ F✱ ♦♥❞❡ F é ❛ ✜❧tr❛çã♦ m✲á❞✐❝❛✳ ❊♥tã♦✱ grm(A) s❡rá ✉♠ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❡ ❛✐♥❞❛ grm(M) s❡rá ✉♠ grm(A)✲
♠ó❞✉❧♦ ❣r❛❞✉❛❞♦✳
❋✐♥❛❧♠❡♥t❡ ❡st❛♠♦s ❛♣t♦s ❛ ❞❡✜♥✐r ♠✉❧t✐♣❧✐❝✐❞❛❞❡ q✉❛♥❞♦M é ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡
❣❡r❛❞♦ ♥♦ ❝♦♥t❡①t♦ ❧♦❝❛❧✳
❉❡✜♥✐çã♦ ✶✳✹✾ ❙❡❥❛(A,m) ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦ ❧♦❝❛❧ ❡ M 6= 0 ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡
❣❡r❛❞♦✳ ❉❡✜♥✐♠♦s ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ M ♣♦r
e(M) = e(grm(M))
▼❛✐s ❣❡r❛❧♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ✉♠ ✐❞❡❛❧ ❞❡ ❞❡✜♥✐çã♦ ❞❡ M✱ ♦✉ s❡❥❛✱I ⊆mt❛❧ q✉❡ mnM ⊂ IM✱ ❜❡♠ ❝♦♠♦ s❡✉ r❡s♣❡❝t✐✈♦ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❛ss♦❝✐❛❞♦ gr
I(A)✱ ✭q✉❡ é ✉♠❛ á❧❣❡❜r❛
❤♦♠♦❣ê♥❡❛✮ ❡ ♦grI(A)✲♠ó❞✉❧♦ ❣r❛❞✉❛❞♦grI(M)✳ ◆❡st❡ ❝♦♥t❡①t♦ ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡M ❝♦♠
r❡s♣❡✐t♦ ❛ I é ❞❡✜♥✐❞❛ ♣♦r e(I, M) =e(grI(M))✳
❉❡✜♥✐çã♦ ✶✳✺✵ ❆ ❢✉♥çã♦ ❞❛❞❛ ♣♦r
χI
M(n) =H1(grI(M), n) =
Pn
i=0H(grI(M), i) =
Pn
i=0l
IiM Ii+1M
=l M
In+1M
s❡rá ❝❤❛♠❛❞❛ ❢✉♥çã♦ ❞❡ ❍✐❧❜❡rt✲❙❛♠✉❡❧ ❞❡ M r❡❧❛t✐✈❛♠❡♥t❡ ❛♦ ✐❞❡❛❧ I✳
Pr♦♣♦s✐çã♦ ✶✳✺✶ ❙❡❥❛ (A,m) ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦ ❧♦❝❛❧ ❝♦♠ dim(A) = d✱ M 6= 0 ✉♠
A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❡ I ✉♠ ✐❞❡❛❧ ❞❡ ❞❡✜♥✐çã♦ ❞❡ M✳ ❊♥tã♦
✭❛✮ ❛ ❢✉♥çã♦ ❞❡ ❍✐❧❜❡rt✲❙❛♠✉❡❧ χI
M(n) é ❞❡ t✐♣♦ ♣♦❧✐♥♦♠✐❛❧ ❞❡ ❣r❛✉ ❞❀
✭❜✮ e(I, M) = lim
n→∞
d!
ndl M In+1M
✳
❉❡st❛ ♣r♦♣♦s✐çã♦ s❡❣✉❡♠ ✐♠❡❞✐❛t❛♠❡♥t❡ ❛s s❡❣✉✐♥t❡ ♦❜s❡r✈❛çõ❡s✿
❖❜s❡r✈❛çõ❡s ✶✳✺✷
✶✳ ❙❡ d= 0 ❡♥tã♦ e(I, M) = l(M)❀
✷✳ e(I, M)>0s❡ dimM =d✱ ❡e(I, M) = 0 s❡dimM < d❀
✸✳ e(Ir, M) =e(I, M)rd❀
✹✳ ❙❡ I ❡I′ sã♦ ✐❞❡❛✐s ❞❡ ❞❡✜♥✐çã♦ ❞❡ M t❛✐s q✉❡ I ⊃I′ ❡♥tã♦ e(I, M)≤e(I′, M)✳
❉❡✜♥✐çã♦ ✶✳✺✸ ❉❡✜♥✐♠♦s e(I) := e(I, A) ❝♦♠♦ s❡♥❞♦ ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ I✱ ❡ ❡s❝r❡✈❡♠♦s e(A)♣❛r❛ ✐♥❞✐❝❛r ❛ ♠✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ m✳
▲❡♠❛ ✶✳✺✹ ✭❆rt✐♥✲❘❡❡s✮ ❙❡❥❛♠ A ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦✱ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡
❣❡r❛❞♦✱ N ⊂ M s✉❜♠ó❞✉❧♦✱ ❡ I ✉♠ ✐❞❡❛❧ ❞❡ A✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ c t❛❧ q✉❡
♣❛r❛ t♦❞♦n > c✱ t❡♠♦s
InM ∩N =In−c(IcM ∩N)✳
❆❣♦r❛ s❡❥❛ (A,m) ✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦✳
❚❡♦r❡♠❛ ✶✳✺✺ ❙❡❥❛ 0 → M′ → M → M′′ → 0 ✉♠❛ s❡q✉ê♥❝✐❛ ❡①❛t❛ ❞❡ A✲♠ó❞✉❧♦s
✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦s✳ ❊♥tã♦✱ ❞❛❞♦ ✉♠I ✐❞❡❛❧ ❞❡ ❞❡✜♥✐çã♦ ❞❡ M✱ e(I, M) = e(I, M′) +e(I, M′′)✳
❊♠ ♣❛rt✐❝✉❧❛r✱ e(M) =e(M′) +e(M′′)
Pr♦✈❛✿✳ P❡❧❛ ✐♥❥❡t✐✈✐❞❛❞❡ ❞❛ ❛♣❧✐❝❛çã♦✱ ♣♦❞❡♠♦s ✈❡r M′ ❝♦♠♦ s✉❜♠ó❞✉❧♦ ❞❡M✳ ❊♥tã♦✱
l M InM
=l M′′
InM′′
+l M′
M′∩InM
❡InM′ ⊂M′∩InM✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ♣❡❧♦ ▲❡♠♠❛ ❞❡ ❆rt✐♥✲❘❡❡s✱ ❡①✐st❡ c >0 t❛❧ q✉❡
M′∩InM ⊂In−cM′✱ ♣❛r❛ t♦❞♦ n > c✳
❆ss✐♠✱
l M′
In−cM′
≤l M′
M′∩InM
≤l M′
InM′
❆❣♦r❛ ❞❛ ♣r♦♣♦s✐çã♦ ✶✳✺✶ t❡r❡♠♦s q✉❡
e(I, M)−e(I, M′′) = lim
n→∞
d!
ndl M′
M′∩InM
=e(I, M′)✳
❉❡✜♥✐çã♦ ✶✳✺✻ ❙❡❥❛♠ (A,m) ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦ ❧♦❝❛❧✱ ❡ I ✉♠ ✐❞❡❛❧ ❞❡ ❞❡✜♥✐çã♦ ❞❡ A✱
✐st♦ é✱ ✉♠ ✐❞❡❛❧m−♣r✐♠ár✐♦✳ ❋✐①❛❞♦ ✉♠ ✐♥t❡✐r♦q✱ ♣❛r❛ t♦❞♦A✲♠ó❞✉❧♦ ✜♥✐t♦M ❞❡ ❞✐♠❡♥sã♦
♠❡♥♦r ♦✉ ✐❣✉❛❧ ❛q✱ ❞❡✜♥✐♠♦s eq(I, M) =
e(I, M) , se dim(M) = q,
0 , se dim(M)< q.
▲❡♠❛ ✶✳✺✼ ✭❋ór♠✉❧❛ ❞❡ ❆ss♦❝✐❛t✐✈✐❞❛❞❡✮ ❙❡❥❛♠ (A,m) ✉♠ ❛♥❡❧ ◆♦❡t❤❡r✐❛♥♦ ❧♦❝❛❧✱ I
✉♠ ✐❞❡❛❧ ❞❡ ❞❡✜♥✐çã♦ ❞❡ A✱ ❡ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ t❛❧ q✉❡ dim(M) ≤ q✳
❊♥tã♦
eq(I, M) =
X
p
l(Mp)eq
I,A
p
✱
♦♥❞❡ p ♣❡r❝♦rr❡ t♦❞♦s ♦s ✐❞❡❛✐s ♣r✐♠♦s ❝♦♠ dim(A p) = q✳
Pr♦✈❛✿✳ ❱❡❥❛ ♦ ❈♦r♦❧ár✐♦ 4.7.8 ❞❡ ❬✽❪✳
❚❡♦r❡♠❛ ✶✳✺✽ ❙❡❥❛♠ (A,m) ✉♠ ❛♥❡❧ ❧♦❝❛❧ ◆♦❡t❤❡r✐❛♥♦✱ M ✉♠ A✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡
❣❡r❛❞♦ ❞❡ ♣♦st♦ ♣♦s✐t✐✈♦✱ ❡ I ✉♠ ✐❞❡❛❧ m✲♣r✐♠ár✐♦ ❞❡ A✳ ❊♥tã♦ e(I, M) = e(I, A)rk(M)✳
❊♠ ♣❛rt✐❝✉❧❛r✱ e(M) =e(A)rk(M)✳
Pr♦✈❛✿✳ ❙❡❥❛ r = rk(M)✳ ❚❡♠♦s q✉❡✱ ♣❛r❛ t♦❞♦p t❛❧ q✉❡ dim(A
p) = d = dim(A)✱ Mp é ✉♠
Ap✲♠ó❞✉❧♦ ❧✐✈r❡ ❞❡ ♣♦st♦ r✱ ❛ss✐♠✱ Mp ∼= Arp✳ ❊♠ ♣❛rt✐❝✉❧❛r M t❡♠ ❞✐♠❡♥sã♦ ♠❛①✐♠❛❧ ❡
❛ss✐♠✱e(I, M) = ed(I, M)✳ ▲♦❣♦ ♣❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱
e(I, M) = X
p
l(Mp)e
I,A
p
=X
p
r·l(Ap)e
I,A
p
=e(I, A)rk(M).
❖ q✉❡ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦✳
❈❛♣ít✉❧♦ ✷
▼ó❞✉❧♦s ❞❡ ❯❧r✐❝❤
❆ t❡♦r✐❛ ❞♦s ❝❤❛♠❛❞♦s ♠ó❞✉❧♦s ❞❡ ❯❧r✐❝❤ ✈❡♠ ❣❛♥❤❛♥❞♦ ❞❡st❛q✉❡ ♥❛ ➪❧❣❡❜r❛ ❈♦♠✉t❛t✐✈❛ ♠♦❞❡r♥❛✱ ❞❡s❞❡ ✶✾✽✹ q✉❛♥❞♦ ❛ q✉❡stã♦ ❢♦✐ ❧❡✈❛♥t❛❞❛ ♣♦r ❇✳❯❧r✐❝❤ ❡♠ ❬✸✶❪✳ ■♥tr♦❞✉③✐r❡♠♦s ❡st❡ ❝♦♥❝❡✐t♦ ♥❛ ❞❡✜♥✐çã♦ ✷✳✸✳
❆ tít✉❧♦ ❞❡ ✐♥❢♦r♠❛çã♦✱ s❛❧✐❡♥t❛♠♦s q✉❡ t❛❧ t❡♦r✐❛ t❡♠ r❡❝❡❜✐❞♦ ♥♦tá✈❡❧ tr❛t❛♠❡♥t♦ ❣❡♦♠étr✐❝♦ ❛tr❛✈és ❞♦s ❝❤❛♠❛❞♦s ✜❜r❛❞♦s ❞❡ ❯❧r✐❝❤ ✭✧❯❧r✐❝❤ ❜✉♥❞❧❡s✧✮✱ ❝♦♠♦ t❡♠ s✐❞♦ ❡st✉❞❛❞♦✱ r❡❝❡♥t❡♠❡♥t❡✱ ♣♦r ❘✳ ❍❛rts❤♦r♥❡ ❡ ♦✉tr♦s ❛✉t♦r❡s ✭✈❡❥❛✱ ♣♦r ❡①❡♠♣❧♦✱ ❬✾❪ ❡ ❬✶✵❪✮✳
❱❡❥❛♠♦s ❛❧❣✉♠❛s ♠♦t✐✈❛çõ❡s ✐♠♣♦rt❛♥t❡s ♣❛r❛ ♥♦ss♦ ❡st✉❞♦ s♦❜r❡ ❡st❛ ❝❧❛ss❡ ❞❡ ♠ó❞✉❧♦s✭✈✐❞❡ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ❬✺❪✮✿
✶✳ ❯♠❛ ❝❧❛ss❡ ❞❡ ❛♥é✐s ♠✉✐t♦ ❢✉♥❞❛♠❡♥t❛❧ ❡♠ ➪❧❣❡❜r❛ ❈♦♠✉t❛t✐✈❛ é ❛ ❞♦s ❛♥é✐s ●♦r❡♥st❡✐♥ ❡ s❛❜❡♠♦s q✉❡ ✉♠❛ ♠❛♥❡✐r❛ ❞❡ t❡st❛r t❛❧ ♣r♦♣r✐❡❞❛❞❡ é ❛ s❡❣✉✐♥t❡✿
A é ✉♠ ❛♥❡❧ ●♦r❡♥st❡✐♥ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ ExtiA(M, A) = 0✱ ♣❛r❛ i = 1, ...,dim(A) ❡
♣❛r❛ t♦❞♦ A✲♠ó❞✉❧♦ ♠❛①✐♠❛❧ ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳
❯s❛♥❞♦ ♠ó❞✉❧♦s ❞❡ ❯❧r✐❝❤ ♦❜t❡♠♦s ✉♠ t❡st❡ ♠❛✐s s✐♠♣❧❡s✿
A é ✉♠ ❛♥❡❧ ●♦r❡♥st❡✐♥ s❡ ❡①✐st❡ ✉♠ ♠ó❞✉❧♦ ❞❡ ❯❧r✐❝❤ M t❛❧ q✉❡ Exti
A(M, A) = 0✱
♣❛r❛ i= 1, ...,dim(A)
✷✳ ▼❛✐s ❛❞✐❛♥t❡ ♥❡st❡ tr❛❜❛❧❤♦ ❢❛❧❛r❡♠♦s s♦❜r❡ ❛ r❡❧❛çã♦ ❡♥tr❡ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ✉♠ A✲
♠ó❞✉❧♦ M s❡r ❞❡ ❯❧r✐❝❤ ❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠❛ r❡s♦❧✉çã♦ ❧✐♥❡❛r ❛♣r♦♣r✐❛❞❛ ❛ss♦❝✐❛❞❛ ❛ M✳ ❉♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❡st❛ s❡çã♦ s❡❣✉✐rá q✉❡ prof(M) = prof(grm(M))✱ ❡ ❛ss✐♠
❞❡ ❯❧r✐❝❤ M t❛♠❜é♠ s❡rá ✉♠ ♠ó❞✉❧♦ ❞❡ ❯❧r✐❝❤❀ ♣♦ré♠✱ ❛ r❡❝✐♣r♦❝❛ é✱ ❡♠ ❣❡r❛❧✱ ❢❛❧s❛✳
❆❧é♠ ❞✐ss♦✱ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♠ó❞✉❧♦ ❞❡ ❯❧r✐❝❤ ♥♦s ❧❡✈❛ ❛♦ s❡❣✉✐♥t❡ ❝r✐tér✐♦✿
❙❡ ✉♠ ❛♥❡❧ ❤♦♠♦❣ê♥❡♦A é ♦ ❛♥❡❧ ❣r❛❞✉❛❞♦ ❛ss♦❝✐❛❞♦ ❞❡ ✉♠ ❛♥❡❧ ❧♦❝❛❧
❈♦❤❡♥✲▼❛❝❛✉❧❛②✱ ❡♥tã♦ A t❡♠✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡✱ ✉♠ ♠ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛②
♠❛①✐♠❛❧ ❣r❛❞✉❛❞♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳
✸✳ ❆ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ♠ó❞✉❧♦ ❞❡ ❯❧r✐❝❤ s♦❜r❡ ✉♠ ❛♥❡❧ ❞❡ ✉♠❛ ❤✐♣❡rs✉♣❡r❢í❝✐❡ ❣❛r❛♥t❡✱ ❛✐♥❞❛✱ q✉❡ ♣❛r❛ t♦❞♦ ♣♦❧✐♥ô♠✐♦ ❤♦♠♦❣ê♥❡♦ f ∈B =k[X1, ..., Xn] ❡①✐st❡ ✉♠❛ ♣♦tê♥❝✐❛
❛❞❡q✉❛❞❛ fm q✉❡ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦ ❞❡t❡r♠✐♥❛♥t❡ ❞❡ ✉♠❛ ♠❛tr✐③ ❝✉❥❛s ❛s ❡♥tr❛❞❛s
sã♦ ❢♦r♠❛s ❧✐♥❡❛r❡s ❡♠ B✳
❆ss✐♠✱ ♥♦ss♦ ♦❜❥❡t✐✈♦✱ ♥❡st❡ ❝❛♣ít✉❧♦✱ s❡rá ❡st✉❞❛r ❛❧❣✉♠❛s ❝❧❛ss❡s ✐♠♣♦rt❛♥t❡s ❞❡ ❛♥é✐s ❈♦❤❡♥✲▼❛❝❛✉❧❛② s♦❜r❡ ♦s q✉❛✐s ❡①✐st❡ ❞❡ ♠ó❞✉❧♦s ❞❡ ❯❧r✐❝❤✳
❉❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡✱ t♦❞♦s ♦s ❛♥é✐s s❡rã♦ ❛❞♠✐t✐❞♦s ❧♦❝❛✐s ◆♦❡t❤❡r✐❛♥♦s ❈♦❤❡♥✲▼❛❝❛✉❧❛②✳ ❖ ✐❞❡❛❧ ♠❛①✐♠❛❧ ❞❡ ✉♠ ❛♥❡❧ ❧♦❝❛❧As❡rá ❞❡♥♦t❛❞♦mA=m❡ ♦ s❡✉ ❝♦r♣♦ r❡s✐❞✉❛❧k = Am q✉❡
s✉♣♦r❡♠♦s ✐♥✜♥✐t♦✳
✷✳✶ ❖ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦ ♣♦r ❇✳ ❯❧r✐❝❤
❖ r❡s✉❧t❛❞♦ ❣❡r❛❧ ❛ s❡❣✉✐r é ❜ás✐❝♦✿
Pr♦♣♦s✐çã♦ ✷✳✶ ❙❡M é ✉♠ A✲♠ó❞✉❧♦ ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠❛①✐♠❛❧✱ ❡♥tã♦ µ(M)≤e(M)✳
Pr♦✈❛✿✳ ❆♣ós ✉♠❛ ❡①t❡♥sã♦ tr❛♥s❝❡♥❞❡♥t❡ ♣✉r❛ ❞♦ ❝♦r♣♦ r❡s✐❞✉❛❧ ❞❡ A ♣♦❞❡♠♦s ❛ss✉♠✐r
q✉❡ m ♣♦ss✉✐ ✉♠❛ r❡❞✉çã♦ ♠✐♥✐♠❛❧ ❣❡r❛❞❛ ♣♦r ✉♠❛ M✲s❡q✉ê♥❝✐❛ ♠á①✐♠❛ x ✭♣❛r❛ ✉♠❛ ♣r♦✈❛ ❞❡st❛ ❛✜r♠❛çã♦ ✈✐❞❡ ❬✷✺❪✮✳ ❊♥tã♦✱ ❝♦♠♦ M é ❈♦❤❡♥✲▼❛❝❛✉❧❛② ♠❛①✐♠❛❧ t❡♠♦s q✉❡ e(M) = l( M
xM) ✭✈❡❥❛ ❬✶✹❪✮✳ ❆ss✐♠✱
µ(M) = dimk(mMM ) =l(mMM )≤l(xMM) =e(M)
❖❜s❡r✈❛çã♦ ✷✳✷ ❙❡ M t❡♠ ♣♦st♦ ❜❡♠ ❞❡✜♥✐❞♦✱ ❡♥tã♦ e(M) = e(A)rk(M)✳ ■st♦ s❡❣✉❡ ❞♦
❚❡♦r❡♠❛ ✶✳✺✽✳
