Open Extensões de Ore e Álgebras de Weyl

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

❊①t❡♥sõ❡s ❞❡ ❖r❡ ❡ ➪❧❣❡❜r❛s ❞❡

❲❡②❧

♣♦r

P❡❞r♦ ❆❧❢r❡❞♦ ❊✉❣❡♥✐♦

s♦❜ ♦r✐❡♥t❛çã♦ ❞♦

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛

❉✐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✐❝❛✳

❛❜r✐❧✴✷✵✶✸ ❏♦ã♦ P❡ss♦❛ ✲ P❇

❊①t❡♥sõ❡s ❞❡ ❖r❡ ❡ ➪❧❣❡❜r❛s ❞❡

❲❡②❧

♣♦r

P❡❞r♦ ❆❧❢r❡❞♦ ❊✉❣❡♥✐♦

❉✐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✐❝❛✳

➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ➪❧❣❡❜r❛✳

❆♣r♦✈❛❞❛ ♣♦r✿

Pr♦❢✳ ❉r✳ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛ ✲❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮

Pr♦❢✳ ❉r✳ ❘♦❜❡rt♦ ❈❛❧❧❡❥❛s ❇❡❞r❡❣❛❧ ✲ ❯❋P❇

Pr♦❢✳ ❉r✳ ❆r♦♥ ❙✐♠✐s ✲ ❯❋P❊

❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛

❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

❈✉rs♦ ❞❡ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

❛❜r✐❧✴✷✵✶✸

◗✉❡r♦ ❛❣r❛❞❡❝❡r ❛ ❉❡✉s ❡♠ ♣r✐♠❡✐r♦ ❧✉❣❛r✳

➚ ♠✐♥❤❛ ♠ã❡✱ ❏♦s❡❢❛ ▼❛r✐❛ ❞❛ ❈♦♥❝❡✐çã♦ ✭✐♥ ♠❡♠♦r✐❛♥✮❡ ♠❡✉ ♣❛✐ ▼❛♥♦❡❧ ❆❧❢r❡❞♦ ❊✉✲ ❣❡♥✐♦✱ q✉❡ ♥ã♦ s❡ ♥❡❣❛r❛♠ ❡♠ ♠♦♠❡♥t♦ ❛❧❣✉♠ ❡♠ ❞✐s♣♦♥✐❜✐❧✐③❛r ❝♦♥❞✐çõ❡s ♣❛r❛ ♦s ♠❡✉s ❡st✉❞♦s✳

➚ ♠✐♥❤❛ ❡s♣♦s❛ ❡ ❝♦♠♣❛♥❤❡✐r❛ ✐♥❝♦♥❞✐❝✐♦♥❛❧ ●❧❛✉❝✐❡t❡ ▼❛r✐❛ ❞❛ ❙✐❧✈❛ ❊✉❣❡♥✐♦✱ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ♥❛s ♠✐♥❤❛s ❛✉sê♥❝✐❛s✳ ❆♦s ♠❡✉s ✐r♠ã♦ ❡ ♣❛r❡♥t❡s q✉❡✱ ♠❡s♠♦ ❞✐st❛♥t❡s ♥ã♦ ♣♦✉♣❛r❛♠ ♠❛♥✐❢❡st❛çã♦ ❞❡ ✐♥❝❡♥t✐✈♦✳

❆♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ◆❛♣♦❧❡ó♥ ❈❛r♦ ❚✉❡st❛✱ q✉❡ ❢♦✐ ❞❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♠♣♦rtâ♥❝✐❛ ♥♦ ❞❡✲ s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳

❆♦s ♣r♦❢❡ss♦r❡s ❆r♦♥ ❙✐♠✐s ❡ ❘♦❜❡rt♦ ❇❡❞r❡❣❛❧✱ ♣♦r ♣❛rt✐❝✐♣❛r❡♠ ❞❛ ❜❛♥❝❛✱ ❡ ❛ t♦❞♦s ♦s ♦✉tr♦s ♣r♦❢❡ss♦r❡s ❞♦ ❞❡♣❛rt❛♠❡♥t♦ ❞❡ ♠❛t❡♠át✐❝❛ ❞❛ ❯❋P❇✱ ♣❡❧❛ ❝♦♥tr✐❜✉çã♦ ♥❛ ♠✐♥❤❛ ❢♦r♠❛çã♦✳

P♦r ✜♠✱ ❛ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s✱ ❡♠ ❡s♣❡❝✐❛❧ ♦s ❞❡ t✉r♠❛ ❝♦♠ ♦s q✉❛✐s ❝♦♠♣❛rt✐❧❤❡✐✱ ❡st✉❞♦s✱ ♣r♦❜❧❡♠❛s ❡ ❛❧❡❣r✐❛s✳

❉❡❞✐❝❛tór✐❛

❆♦s ♠❡✉s ♣❛✐s✳

◆❡st❡ tr❛❜❛❧❤♦ ❡st✉❞❛r❡♠♦s ❛s ❞❡✜♥✐çõ❡s✱ ❡①❡♠♣❧♦s ❡ ♣r♦♣r✐❡❞❛❞❡s ❜ás✐❝❛s ❞❛s ❡①✲ t❡♥sõ❡s ❞❡ ❖r❡✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛♣r❡s❡♥t❛r❡♠♦s ✉♠ t✐♣♦ ❡s♣❡❝✐❛❧ ❞❡ ❡①t❡♥sõ❡s ❞❡ ❖r❡✱ ❛s á❧❣❡❜r❛s ❞❡ ❲❡②❧An(K)s♦❜r❡ ✉♠ ❝♦r♣♦K✳ ❱❡r❡♠♦s q✉❡An(K)é ✉♠ ❞♦♠í♥✐♦ ♥♦❡t❤❡r✐❛♥♦

s✐♠♣❧❡s✳ ❊st✉❞❛r❡♠♦s t❛♠❜é♠ ❛ ❞✐♠❡♥sã♦ d(M) ❞❡ ✉♠ An✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦

M ❡ ♣r♦✈❛r❡♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❇❡r♥st❡✐♥✱ n ≤d(M) ≤2n✳ ❋✐♥❛❧♠❡♥t❡ ❡st✉❞❛r❡♠♦s

♦s An(K)✲♠ó❞✉❧♦s ❤♦❧♦♥ô♠✐❝♦s✱ ✐st♦ é✱ ♦s An(K)✲♠ó❞✉❧♦s ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦s t❛✐s q✉❡

d(M) = n ✳

P❛❧❛✈r❛s ❝❤❛✈❡s ✿ ❊①t❡♥sõ❡s ❞❡ ❖r❡✱ ➪❧❣❡❜r❛s ❞❡ ❲❡②❧✱ ♠ó❞✉❧♦s ❤♦❧♦♥ô♠✐❝♦s✳

❆❜str❛❝t

■♥ t❤✐s ✇♦r❦ ✇❡ ✇✐❧❧ st✉❞② t❤❡ ❞❡✜♥✐t✐♦♥s✱ ❡①❛♠♣❧❡s ❛♥❞ ❜❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❖r❡ ❡①t❡♥✲ s✐♦♥s✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✇❡ ✇✐❧❧ ♣r❡s❡♥t ❛ s♣❡❝✐❛❧ ❝❛s❡ ♦❢ ❖r❡ ❡①t❡♥s✐♦♥s✱ t❤❡ ❲❡②❧ ❛❧❣❡❜r❛s

An(K) ♦✈❡r ❛ ✜❡❧❞ K✳ ❲❡ ✇✐❧❧ s❡❡ t❤❛t An(K) ✐s ❛ s✐♠♣❧❡ ♥♦❡t❤❡r✐❛♥ ❞♦♠❛✐♥✳ ❲❡ ✇✐❧❧

st✉❞② ❛❧s♦ t❤❡ ❞✐♠❡♥s✐♦♥ d(M) ♦❢ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ An(K)✲♠♦❞✉❧❡ ❛♥❞ ✇❡ ✇✐❧❧ ♣r♦✈❡

t❤❡ ❇❡r♥st❡✐♥✬s ✐♥❡q✉❛❧✐t②✱n ≤ d(M)≤ 2n✳ ❋✐♥❛❧❧② ✇❡ ✇✐❧❧ st✉❞② t❤❡ ❤♦❧♦♥♦♠✐❝ An(K)✲

♠♦❞✉❧❡s✱ t❤❛t ✐s✱ t❤❡ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞An(K)✲♠♦❞✉❧❡s s✉❝❤ t❤❛t d(M) = n.

■♥tr♦❞✉çã♦ ✐✐

✶ ❊①t❡♥sõ❡s ❞❡ ❖r❡ ✷

✶✳✶ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸ ❆♥❡✐s ❞❡ ♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛✉r❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✶✳✹ ❉♦♠í♥✐♦s ❞❡ ❖r❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✷ ➪❧❣❡❜r❛s ❞❡ ❲❡②❧ ✸✵

✷✳✶ ❉❡✜♥✐çõ❡s ❡ Pr♦♣r✐❡❞❛❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✷✳✷ ❚❡♦r❡♠❛ ❞❡ ❉✐✈✐sã♦ ❡♠An✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼

✷✳✸ ▼ó❞✉❧♦s s♦❜r❡ ➪❧❣❡❜r❛s ❞❡ ❲❡②❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

✸ An✲♠ó❞✉❧♦s ❣r❛❞✉❛❞♦s ❡ ✜❧tr❛❞♦s ✹✸

✸✳✶ ▼ó❞✉❧♦s ❣r❛❞✉❛❞♦s ❡ ✜❧tr❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷ B✲✜❧tr❛çõ❡s ❡♠An✲♠ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾

✸✳✸ ❋✐❧tr❛çõ❡s ❇♦❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷

✹ An✲♠ó❞✉❧♦s ❤♦❧♦♥ô♠✐❝♦s ✺✺

✹✳✶ ❖ P♦❧✐♥ô♠✐♦ ❞❡ ❍✐❧❜❡rt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✹✳✷ ❉✐♠❡♥sã♦ ❡ ▼✉❧t✐♣❧✐❝✐❞❛❞❡ ❞❡ An✲♠ó❞✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

✹✳✸ ▼ó❞✉❧♦s ❍♦❧♦♥ô♠✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✶ ✹✳✹ ▼❛✐s ❡①❡♠♣❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹

❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✻✼

■♥tr♦❞✉çã♦

❊♠ ✶✾✸✸ ♦ ♠❛t❡♠át✐❝♦ ♥♦r✉❡❣✉ês ❖②st❡✐♥ ❖r❡ ✐♥tr♦❞✉③✐✉ ✉♠ t✐♣♦ ❡s♣❡❝✐❛❧ ❞❡ ❡①t❡♥sõ❡s ❞❡ ❛♥❡✐s✳ ❊❧❡ ❝♦♥s✐❞❡r♦✉ ♣♦❧✐♥ô♠✐♦s s♦❜r❡ ✉♠ ❛♥❡❧R ♥✉♠❛ ✐♥❞❡t❡r♠✐♥❛❞❛x✱ ❛ q✉❛❧ ♥ã♦

❝♦♠✉t❛ ❝♦♠ ♦s ❡❧❡♠❡♥t♦s ❞❡ R✳ ❈♦♠♦ ♣♦❧✐♥ô♠✐♦s✱ é ❞❡s❡❥❛❞♦ q✉❡ ❝❛❞❛ ❡❧❡♠❡♥t♦ s❡❥❛

❡s❝r✐t♦ ❞❡ ♠❛♥❡✐r❛ ú♥✐❝❛ ♥❛ ❢♦r♠❛Prixi✱ ♣❛r❛ ❛❧❣✉♥s ri ∈R✳ ■st♦ t❡♠ q✉❡ s❡ ❛♣❧✐❝❛r✱ é

❝❧❛r♦✱ ❛♦ ❡❧❡♠❡♥t♦xr✱ ♣❛r❛ ❝❛❞❛r ∈R✳ ➱ ❞❡ ❡s♣❡r❛r q✉❡ t❛✐s ♣♦❧✐♥ô♠✐♦s t❡♥❤❛♠ ✉♠ ❜♦♠

❝♦♠♣♦rt❛♠❡♥t♦ ❝♦♠ r❡❧❛çã♦ ❛♦ ❣r❛✉✱ ✐st♦ é✱ grau(f g) ≤ grau(f) +grau(g)✱ ♣♦rt❛♥t♦ é

r❡q✉❡r✐❞♦ q✉❡xr ∈Rx+R✱ ✐st♦ é✱ xr =α(r)x+δ(r)✳ ◆❡st❛s ❝♦♥❞✐çõ❡s✱ é ❝❧❛r♦ q✉❡ α ❡ δ sã♦ ❡♥❞♦♠♦r✜s♠♦s ❞♦ ❣r✉♣♦ ❛❞✐t✐✈♦ ❞❡ R✳ ▼❛✐s ❛✐♥❞❛✱

x(rs) = α(rs)x+δ(rs) ❡ (xr)s= (α(r)α(s))x+ (δ(r)s+α(r)δ(s)).

P♦rt❛♥t♦αt❡♠ q✉❡ s❡r ✉♠ ❡♥❞♦♠♦r✜s♠♦ ❞♦ ❛♥❡❧R❡δ✉♠❛ ❛♣❧✐❝❛çã♦ ❛❞✐t✐✈❛ q✉❡ s❛t✐s❢❛③

❛ ❝♦♥❞✐çã♦

δ(rs) =δ(r)s+α(r)δ(s),

q✉❡ é ❛ ♣r♦♣r✐❡❞❛❞❡ q✉❡ ❞❡✜♥❡ ✉♠❛α✲❞❡r✐✈❛çã♦ s♦❜r❡ R.

◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦ ❞❡ ♥♦ss♦ tr❛❜❛❧❤♦ ♠♦str❛r❡♠♦s q✉❡ ♣❛r❛ t♦❞❛α✲❞❡r✐✈❛çã♦δs♦❜r❡

✉♠ ❛♥❡❧ R é ♣♦ssí✈❡❧ ❝♦♥str✉✐r ✉♠ ❛♥❡❧ S ❝♦♠ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s ✿

(i) R é ✉♠ s✉❜❛♥❡❧ ❞❡ S✱ (ii) ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ x ∈S t❛❧ q✉❡ S é ✉♠ R✲♠ó❞✉❧♦ ❧✐✈r❡

❝♦♠ ❜❛s❡ {1, x, x2_{, ...} ❡(iii) xr=α(r)x+δ(r) ♣❛r❛ t♦❞♦r ∈R.}

▼❛✐s ❛✐♥❞❛ ♣r♦✈❛r❡♠♦s q✉❡ t❛❧ ❛♥❡❧ s❛t✐s❢❛③ ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❡ ♥❡ss❡ s❡♥t✐❞♦ é ú♥✐❝♦✳ ❖ ❛♥❡❧ S é ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞♦ ❛♥❡❧ R ❡ é ❞❡♥♦t❛❞♦ ♣♦r S = R[x;α, δ]✳

❊①✐❜✐r❡♠♦s ❛❧❣✉♥s ❡①❡♠♣❧♦s ✐♥t❡r❡ss❛♥t❡s ❞❡ ❛♥❡✐s ❡ ♣r♦✈❛r❡♠♦s q✉❡ sã♦ ❡①t❡♥sõ❡s ❞❡ ❖r❡✳ ❚❛♠❜é♠ ❞❡♠♦♥str❛r❡♠♦s✱ ❛❧é♠ ❞❡ ♦✉tr❛s ♣r♦♣r✐❡❞❛❞❡s✱ ✉♠ r❡s✉❧t❛❞♦ s❡♠❡❧❤❛♥t❡ ❛♦ t❡♦r❡♠❛ ❞❛s ❜❛s❡s ❞❡ ❍✐❧❜❡rt ♣❛r❛ ❛♥❡✐s ❞❡ ♣♦❧✐♥ô♠✐♦s✱ s❡ ♦ ❛♥❡❧R é ✉♠ ❛♥❡❧ ♥♦❡t❤❡r✐❛♥♦

à ❡sq✉❡r❞❛ ✭à ❞✐r❡✐t❛✮ ❡ α é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ R✱ ❡♥tã♦ S é ✉♠ ❛♥❡❧ ♥♦❡t❤❡r✐❛♥♦ à

❡sq✉❡r❞❛ ✭à ❞✐r❡✐t❛✮✳

❞❡ ❍❡✐s❡♥❜❡r❣ ❡♠ ♠❡❝â♥✐❝❛ q✉â♥t✐❝❛✳ ❉❡✜♥✐r❡♠♦s✱ ♣❛r❛ ❝❛❞❛ n ∈ N✱ ❛ n✲és✐♠❛ á❧❣❡❜r❛

❞❡ ❲❡②❧An(K)❝♦♠♦ ♦ ❛♥❡❧ ❞❡ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s ❞♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦sK[x1, ..., xn]

❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥✉♠ ❝♦r♣♦K❡ ♣r♦✈❛r❡♠♦s q✉❡An(K)é ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❡ ♣♦rt❛♥t♦

❤❡r❞❛♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡♠♦♥str❛❞❛s ♥♦ ❝❛♣ít✉❧♦ ✶✱ ❡♠ ♣❛rt✐❝✉❧❛r ✈❡r❡♠♦s q✉❡An(K)é

✉♠ ❞♦♠í♥✐♦ ♥♦❡t❤❡r✐❛♥♦ s✐♠♣❧❡s✳

❖ ❝❛♣ít✉❧♦ ✸ ❡stá ❞❡❞✐❝❛❞♦ ❛♦ ❡st✉❞♦ ❞♦s An(K)✲♠ó❞✉❧♦s ❣r❛❞✉❛❞♦s ❡ ✜❧tr❛❞♦s✳ ❊♠

♣❛rt✐❝✉❧❛r ❡st✉❞❛r❡♠♦s ❛ ✜❧tr❛çã♦ ❞❡ ❇❡rst❡✐♥ ❞❡An(K)❡ ✈❡r❡♠♦s q✉❡ ♦ ❛♥❡❧ ❣r❛❞✉❛❞♦

❛ss♦❝✐❛❞♦ é ✐s♦♠♦r❢♦ ❛♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡♠ 2n ✐♥❞❡t❡r♠✐♥❛❞❛s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ♥♦

❝♦r♣♦ K✳

❋✐♥❛❧♠❡♥t❡✱ ♥♦ ❝❛♣ít✉❧♦ ✹ ❡st✉❞❛r❡♠♦s✱ ✉s❛♥❞♦ ♣♦❧✐♥ô♠✐♦s ❞❡ ❍✐❧❜❡rt✱ ❛ ❞✐♠❡♥sã♦

d(M) ❞❡ ✉♠ An(K)✲♠ó❞✉❧♦ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦ M✳ Pr♦✈❛r❡♠♦s q✉❡ n ≤ d(M) ≤ 2n✱

r❡s✉❧t❛❞♦ ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡ ❇❡r♥st❡✐♥✳ ❖sAn(K)✲♠ó❞✉❧♦s M t❛✐s q✉❡

d(M) = n sã♦ ♦s ♠ó❞✉❧♦s ❤♦❧♦♥ô♠✐❝♦s ❡ s❡rã♦ ❡st✉❞❛❞♦s ❝♦♠ ❛❧❣✉♥s ❞❡t❛❧❤❡s✳

❈❛♣ít✉❧♦ ✶

❊①t❡♥sõ❡s ❞❡ ❖r❡

✶✳✶ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s

◆♦ q✉❡ s❡❣✉❡✱ R ❞❡♥♦t❛rá ✉♠ ❛♥❡❧ ❡ α:R−→R ✉♠ ❡♥❞♦♠♦r✜s♠♦ ❞❡ R✳

❉❡✜♥✐çã♦ ✶✳✶ ❯♠❛ α✲❞❡r✐✈❛çã♦ ❞❡ R é ✉♠❛ ❛♣❧✐❝❛çã♦ ❛❞✐t✐✈❛ δ : R −→ R ❝♦♠ ❛

s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡✱

δ(rs) = δ(r)s+α(r)δ(s), ♣❛r❛ t♦❞♦ r, s∈R

◆♦t❡♠♦s q✉❡δ(1) =δ(1·1) =α(1)δ(1) +δ(1)1 =δ(1) +δ(1)✱ ❞♦♥❞❡δ(1) = 0✳

❙❡ α é ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ R✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ❛❝✐♠❛ é ❝♦♥❤❡❝✐❞❛ ❝♦♠♦ r❡❣r❛ ❞❡

▲❡✐❜♥✐③✳ ◆❡st❡ ❝❛s♦✱ ❛sα✲❞❡r✐✈❛çõ❡s sã♦ ❞❡r✐✈❛çõ❡s ❞❡ R✳

❊①❡♠♣❧♦ ✶✳✷ ❆ ❛♣❧✐❝❛çã♦ ♥✉❧❛δ= 0 é ✉♠❛ α✲❞❡r✐✈❛çã♦ ♣❛r❛ q✉❛❧q✉❡r ❡♥❞♦♠♦r✜s♠♦ α

❞❡R✳

❊①❡♠♣❧♦ ✶✳✸ ❙❡❥❛R ♦ ❛♥❡❧ ❞❛s ❢✉♥çõ❡s r❡❛✐s ❞❡ ❝❧❛ss❡C∞ _{✱ ✐st♦ é✱} C∞(R) ={f :R−→R;f é ❞❡ ❝❧❛ss❡ C∞}✱ ❡♥tã♦ ❛ ❞❡r✐✈❛❞❛ d

dx, é ✉♠❛ ❞❡r✐✈❛çã♦

s♦❜r❡R✳

❊①❡♠♣❧♦ ✶✳✹ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❡ s❡❥❛ R = K[x] (R = K[[x]]) ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s

✭❛♥❡❧ ❞❡ sér✐❡s ❞❡ ♣♦tê♥❝✐❛s ❢♦r♠❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ K✳ ❆ r❡❣r❛

δ(X

i

aixi) =

X

i

iaixi−1 ❞❡✜♥❡ ✉♠❛ ❞❡r✐✈❛çã♦ δ s♦❜r❡ R✱ q✉❡ é ❞❡♥♦t❛❞❛ ♣♦r

d dx✳

▼❛✐s ❣❡r❛❧♠❡♥t❡✱ s❡ R = K[x1, ..., xn](R = K[[x1, ..., xn]]) é ✉♠ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ❡♠

n✲✐♥❞❡t❡r♠✐♥❛❞❛s ✭❛♥❡❧ ❞❡ sér✐❡s ❞❡ ♣♦tê♥❝✐❛s ❢♦r♠❛✐s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮✱ ❝❛❞❛ ❞❡r✐✈❛❞❛

♣❛r❝✐❛❧ ∂

∂xi

, i= 1, ..., n ❞❡✜♥❡ ✉♠❛ ❞❡r✐✈❛çã♦ ❡♠R.

❊①❡♠♣❧♦ ✶✳✺ ❙❡❥❛ K[x] ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s s♦❜r❡ ✉♠ ❝♦r♣♦ K✳ ❙❡❥❛ q ∈ K t❛❧ q✉❡

q6= 0, q 6= 1❡ s❡❥❛ α ♦K✲❛✉t♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❞❡ K[x] t❛❧ q✉❡ α(x) =qx✳ ❊♥tã♦ ❛

r❡❣r❛δ(f(x)) = f(qx)−f(x)

qx−x ✱ ❞❡✜♥❡ ✉♠❛α✲❞❡r✐✈❛çã♦ ❞❡K[x]❝♦♥❤❡❝✐❞❛ ❝♦♠♦ ❞❡r✐✈❛❞❛

❊✉❧❡r✐❛♥❛ ♦✉ q✲♦♣❡r❛❞♦r ❞❡ ❞✐❢❡r❡♥ç❛✳

❉❛❞♦ ✉♠ ❛♥❡❧ R ❡ δ ✉♠❛ α✲❞❡r✐✈❛çã♦ ❞❡ R ❝♦♥str✉✐r❡♠♦s✱ ✉♠ ❛♥❡❧ S q✉❡ ❝♦♥té♠ R ❝♦♠♦ s✉❜❛♥❡❧✱ q✉❡ t❡♠ ❝♦♠♦ ❡❧❡♠❡♥t♦s✱ ✧♣♦❧✐♥ô♠✐♦s ✧♥✉♠❛ ✐♥❞❡t❡r♠✐♥❛❞❛ x ❝♦♠

❝♦❡✜❝✐❡♥t❡s ❛ ✧❡sq✉❡r❞❛✧❡♠ R ❡ ❝♦♠ ✉♠❛ ♠✉❧t✐♣❧✐❝❛çã♦ q✉❡ s❛t✐s❢❛③ ❛ r❡❧❛çã♦ xr=α(r)x+δ(r)✱ ♣❛r❛ t♦❞♦ r ∈R✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ t❡♠♦s ♦ s❡❣✉✐♥t❡ ✿

❚❡♦r❡♠❛ ✶✳✻ ❙❡❥❛ δ ✉♠❛ α✲❞❡r✐✈❛çã♦ ❞❡ R✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ❛♥❡❧ S ❝♦♠ ❛s s❡❣✉✐♥t❡s

♣r♦♣r✐❡❞❛❞❡s✿

(a) R é ✉♠ s✉❜❛♥❡❧ ❞❡ S✳

(b) ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ x ∈ S t❛❧ q✉❡ S é ✉♠ R✲♠ó❞✉❧♦ ❧✐✈r❡ à ❡sq✉❡r❞❛ ❝♦♠ ❜❛s❡ {1, x, x2_{, ...}✳}

(c) xr=α(r)x+δ(r)✱ ♣❛r❛ t♦❞♦ r∈R

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ E = EndZ(R[z])✱ ♦♥❞❡ R[z] é ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ✭♦r❞✐♥ár✐♦✮

♥❛ ✐♥❞❡t❡r♠✐♥❛❞❛ z ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ R✳ ❈♦♥s✐❞❡r❡♠♦s ♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥❡✐s λ : R −→ E t❛❧ q✉❡ ♣❛r❛ r ∈ R ❡ p(z) ∈ R[z]✱ λ(r)(p(z)) = rp(z)✳ ➱ ❝❧❛r♦ q✉❡ λ é

✐♥❥❡t✐✈♦✱ ❧♦❣♦ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛rR ❝♦♠ ♦ s✉❜❛♥❡❧ λ(R)⊂E✳

❆❣♦r❛✱ ❞❡✜♥✐♠♦s x∈E ♣❡❧❛ r❡❣r❛

x(X

i

rizi) =

X

i

(α(ri)zi+1+δ(ri)zi),

❡ s❡❥❛S ♦ s✉❜❛♥❡❧ ❞❡E ❣❡r❛❞♦ ♣♦r R∪ {x}✳

P❛r❛r ∈R ❡ p(z) = P_irizi ∈R[z]✱ t❡♠♦s q✉❡

(xr)(p(z)) =x(X

i

rrizi) =

X

i

(α(rri)zi+1+δ(rri)zi) =

X

i

α(r)α(ri)zi+1+

X

i

(α(r)δ(ri)+

δ(r)ri)zi =α(r)

X

i

(α(ri)zi+1+δ(ri)zi) +δ(r)

X

i

rizi = (α(r)x+δ(r))(p(z)).

P♦rt❛♥t♦✱ xr = α(r)x+δ(r) ♣❛r❛ t♦❞♦ r ∈ R✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ xR ⊆ Rx+R✳ ❉❡st❛

r❡❧❛çã♦✱ s❡❣✉❡ ♣♦r ✐♥❞✉çã♦ q✉❡

xi_{R ⊆Rx}i_+Rxi−1_{+...+Rx+R, ♣❛r❛ t♦❞♦i∈Z}+_,

❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡✱

(Rxi_)(Rxj_)⊆Rxi+j_+Rxi+j−1_+...+Rxj_{, ♣❛r❛ t♦❞♦ i, j ∈Z}+

❈❛♣ít✉❧♦ ✶✳ ❊①t❡♥sõ❡s ❞❡ ❖r❡ ✶✳✶✳ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s

■st♦ ✐♠♣❧✐❝❛ q✉❡

∞

X

i=0

Rxi _{é ✉♠ s✉❜❛♥❡❧ ❞❡E✳}

❙❡❥❛ S =

∞

X

i=0

Rxi✳ ❊♥tã♦ ♦ ❝♦♥❥✉♥t♦ {1, x, x2_{, ...} ❣❡r❛ S ❝♦♠♦ R✲♠ó❞✉❧♦ à ❡sq✉❡r❞❛✳}

❙ó ❢❛❧t❛ ♠♦str❛r q✉❡ {1, x, x2_{, ...} é ❧✐♥❡❛r♠❡♥t❡ ✐♥❞❡♣❡♥❞❡♥t❡ s♦❜r❡ R✳ P❛r❛ ✐st♦✱ s❡❥❛♠}

r0, r1, ..., rn ∈ R✱ t❛✐s q✉❡ r0+r1x+...+rnxn = 0✳ ❆♣❧✐❝❛♥❞♦ ♥♦ ♣♦❧✐♥ô♠✐♦ ❝♦♥st❛♥t❡

1✱ t❡♠♦s q✉❡ (r0 +r1x+...+rnxn)(1) = 0 ❡♠ R[z]✳ ▼❛s xi(1) = zi, ♣❛r❛ t♦❞♦ i ≥ 0✳

P♦rt❛♥t♦✱ r0+r1z+...+rnzn= 0 ❡♠ R[z]✱ ❞♦♥❞❡r0 =r1 =...=rn= 0✳

❉❡✜♥✐çã♦ ✶✳✼ ❖ ❛♥❡❧ S ❝♦♥str✉í❞♦ ❛❝✐♠❛ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞❡ R ❡ é ❞❡♥♦t❛❞♦ ♣♦r R[x;α, δ]✳

◆❛ ❧✐t❡r❛t✉r❛ ❡♠ ✐♥❣❧ês t❛❧ ❛♥❡❧ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ✉♠ ✧s❦❡✇ ♣♦❧②♥♦♠✐❛❧ r✐♥❣✧ ❞❡ R✳

❙❡ δ é ❛ ❞❡r✐✈❛çã♦ ♥✉❧❛✱ ❡s❝r❡✈❡r❡♠♦s R= [x;α] ♥♦ ❧✉❣❛r ❞❡ R[x;α,0]✳

❙❡ α= 1R é ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡ ❞❡ R✱ ❡s❝r❡✈❡r❡♠♦s R[x;δ] ♥♦ ❧✉❣❛r ❞❡R[x; 1R, δ]✳ ❖

❛♥❡❧ R[x;δ] é ❝♦♥❤❡❝✐❞♦ t❛♠❜é♠ ❝♦♠♦ ✉♠ ❛♥❡❧ ❞❡ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s✳

❆s ❡①t❡♥sõ❡s ❞❡ ❖r❡ ♣♦ss✉❡♠ ❛ s❡❣✉✐♥t❡ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✿

❚❡♦r❡♠❛ ✶✳✽ ❙❡❥❛ S =R[x;α, δ] ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞❡ R✳ ❙✉♣♦♥❤❛♠♦s q✉❡ T é ✉♠

❛♥❡❧✱ ϕ:R −→T é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥❡✐s ❡ y∈T t❛❧ q✉❡ yϕ(r) =ϕα(r)y+ϕδ(r)

♣❛r❛ t♦❞♦ r ∈ R✳ ❊♥tã♦✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥❡✐s Ψ : S −→ T t❛❧ q✉❡

Ψ|R =ϕ ❡ Ψ(x) =y✱ ♦✉ s❡❥❛✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ é ❝♦♠✉t❛t✐✈♦✿

S

∃!Ψ

"

"

R ? ϕ //

ι

O

O

T

❉❡♠♦♥str❛çã♦✿ ❉❡✜♥❛♠♦s ❛ ❛♣❧✐❝❛çã♦Ψ :S −→T ♣♦r Ψ(P_irixi) =

P

iϕ(ri)yi✳ ❊♥✲

tã♦✱Ψ|R=ϕ ❡ Ψ(x) =y✳ ❆❧é♠ ❞✐ss♦✱Ψé ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥❡✐s✳ ❉❡ ❢❛t♦✱ ♣r✐♠❡✐r♦

♥♦t❡ q✉❡ s❡t =P_jbjxj é ✉♠ ❡❧❡♠❡♥t♦ ❛r❜✐trár✐♦ ❞❡ S✱ ❡♥tã♦ Ψ(xt) = Ψ(

P

jα(bj)xj+1+

P

jδ(bj)x

j_{) =}P

jϕα(bj)y

j+1₊P

jϕδ(bj)y j ₌P

j(ϕα(bj)y+ϕδ(bj))y j ₌P

jyϕ(bj)y j ₌

yΨ(t)✳

❙❡❣✉❡✲s❡ ♣♦r ✐♥❞✉çã♦ q✉❡ Ψ(xi_{t) = y}i_{Ψ(t)✱ ♣❛r❛ t♦❞♦ i∈Z}+ _{❡ t∈S✳}

▼❛✐s ❛✐♥❞❛✱ s❡a ∈R✱ ❡♥tã♦

Ψ(at) =X

j

ϕ(abj)yj =

X

j

ϕ(a)ϕ(bj)yj =ϕ(a)Ψ(t).

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ ❞❛❞♦s=P_iaixi ❡♠ S✱ t❡♠♦s

Ψ(st) =X

i

Ψ(aixit) =

X

i

ϕ(ai)Ψ(xit) =

X

i

ϕ(ai)yiΨ(t) = Ψ(s)Ψ(t).

P♦r ♦✉tr♦ ❧❛❞♦✱ é ❝❧❛r♦ q✉❡ Ψ(s+t) = Ψ(s) + Ψ(t)✳ P♦rt❛♥t♦✱ Ψé ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡

❛♥❡✐s q✉❡ s❛t✐s❢❛③ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡s❡❥❛❞❛✳

❋✐♥❛❧♠❡♥t❡✱ s❡❥❛ Ψ′ : S −→ T ✉♠ ♦✉tr♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥❡✐s t❛❧ q✉❡ Ψ′|R = ϕ ❡

Ψ′(x) =y✳ ❊♥tã♦✱

Ψ′(X

i

rixi) =

X

i

Ψ′(ri)(Ψ

′

(x))i ₌X i

ϕ(ri)yi = Ψ(

X

i

rixi)

❖ q✉❡ ♣r♦✈❛ ❛ ✉♥✐❝✐❞❛❞❡ ❞❡ Ψ✳

❈♦r♦❧ár✐♦ ✶✳✾ ❙❡❥❛♠S =R[x;α, δ]❡S′ =R[x′;α, δ]❡①t❡♥sõ❡s ❞❡ ❖r❡ ❞❡ R✱ ❡♥tã♦ ❡①✐st❡

✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ ❞❡ ❛♥❡✐s Ψ :S −→S′ t❛❧ q✉❡ Ψ(x) = x′ ❡ Ψ|R= 1R✳

❉❡♠♦♥str❛çã♦✿ ❙❡ ❛♣❧✐❝❛r♠♦s ♦ t❡♦r❡♠❛ ❝♦♠ ϕ :R−→ S′ s❡♥❞♦ ❛ ✐♥❝❧✉sã♦✱ ♦❜t❡♠♦s

✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦Ψ :S −→S′ t❛❧ q✉❡ Ψ|R =ϕ ❡ Ψ(x) =x

′

✳ P♦rt❛♥t♦✱ Ψ|R = 1R

❡Ψ(x) =x′✳

P♦r s✐♠❡tr✐❛✱ ♦ t❡♦r❡♠❛ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ Ψ′ : S′ −→ S t❛❧ q✉❡ Ψ′(x′) = x ❡ Ψ′|R = 1R✳ ❆ ❛♣❧✐❝❛çã♦ 1S t❡♠ ❛ ♠❡s♠❛ ♣r♦♣r✐❡❞❛❞❡✱ ❡♥tã♦

♣❡❧❛ ✉♥✐❝✐❞❛❞❡ t❡♠♦s q✉❡ Ψ′Ψ = 1S✳ ❉❡ ♠❛♥❡✐r❛ s❡♠❡❧❤❛♥t❡✱ ♣r♦✈❛✲s❡ q✉❡ ΨΨ

′

= 1S′✳

❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱Ψ é ♦ ú♥✐❝♦ ✐s♦♠♦r✜s♠♦ ♣r♦❝✉r❛❞♦✳

❊①❡♠♣❧♦ ✶✳✶✵ ❙❡❥❛K✉♠ ❝♦r♣♦ ❡ s❡❥❛q∈K, q 6= 0✳ P♦r ❞❡✜♥✐çã♦✱ ♦ ❛♥❡❧ ❞❡ ❝♦♦r❞❡✲

♥❛❞❛s q✉❛♥t✐③❛❞♦ ❞❡ K2 _{♦✉ ♣❧❛♥♦ q✉â♥t✐❝♦ O}

q(K2) é ❛ K✲á❧❣❡❜r❛ ❛♣r❡s❡♥t❛❞❛ ♣❡❧♦s

❣❡r❛❞♦r❡su ❡v ❡ ❛ r❡❧❛çã♦uv =qvu✳ ■st♦ é✱ s❡ KhU, Vi é ❛ á❧❣❡❜r❛ ❧✐✈r❡ ❝♦♠ ❞✉❛s ❧❡tr❛s U ❡ V ❡ hU V −qV Ui ❞❡♥♦t❛ ♦ ✐❞❡❛❧ ❣❡r❛❞♦ ♣♦rU V −qV U✱ ❡♥tã♦

Oq(K2) =

KhU, Vi hU V −qV Ui.

❖s ❡❧❡♠❡♥t♦s u ❡v ♥❛ ❞❡✜♥✐çã♦ sã♦ ❡♥tã♦ ❛s ❝❧❛ss❡s ❞❡ U ❡V ♠ó❞✉❧♦ hU V −qV Ui ✳

❆✜r♠❛çã♦✿ ❖ ♣❧❛♥♦ q✉â♥t✐❝♦ Oq(K2) é ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s

♦r❞✐♥ár✐♦s K[y]✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ s❡ α é ♦ ❛✉t♦♠♦r✜s♠♦ ❞❡ K[y] t❛❧ q✉❡ α(y) = qy✱

❈❛♣ít✉❧♦ ✶✳ ❊①t❡♥sõ❡s ❞❡ ❖r❡ ✶✳✶✳ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s

♠♦str❛r❡♠♦s q✉❡ Oq(K2)∼=K[y][x;α]✳

P❛r❛ ✐st♦✱ s❡❥❛σ:KhU, Vi −→K[y][x;α]♦K✲❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s t❛❧ q✉❡σ(U) = x

❡σ(V) =y✳ ❊♥tã♦ σ(U V −qV U) = xy−qyx= 0 ❡ ♣♦rt❛♥t♦hU V −qV Ui ⊆kerσ✳ P❡❧♦

t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞♦s ❤♦♠♦♠♦r✜s♠♦s✱σ ✐♥❞✉③ ✉♠ K✲❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s

Φ :Oq(K2) =

KhU, Vi

hU V −qV Ui −→K[y][x, α]

t❛❧ q✉❡ Φ(u) = x❡ Φ(v) =y✱ ♦♥❞❡ u=U+hU V −qV Ui ❡ v =V +hU V −qV Ui

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦K[y] é ✉♠ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s s♦❜r❡K✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✲

✜s♠♦ ❞❡K✲á❧❣❡❜r❛s η:K[y]−→ Oq(K2) t❛❧ q✉❡ η(y) =v✳

▲♦❣♦✱ s❡ p = X

i

aiyi ∈ K[y]✱ ❡♥tã♦ η(p) =

X

i

aivi ❡ α(p) =

X

i

aiα(y)i =

X

i

aiqiyi✳

❉♦♥❞❡✱ η(α(p))u=X

i

aiqiviu=

X

i

aiuvi =u(

X

i

aivi) =uη(p)✱ ♦✉ s❡❥❛✱

η(α(p))u =uη(p)✳ ❆ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛s ❡①t❡♥sõ❡s ❞❡ ❖r❡ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡

✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡K✲á❧❣❡❜r❛s Ψ :K[y][x, α]−→ Oq(K2)

t❛❧ q✉❡ Ψ|K[y]=η ❡Ψ(x) =u✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ Ψ(y) =η(y) =v

❈♦♠♦ Φ◦Ψ(y) = Φ(v) = y✱ Φ◦Ψ(x) = Φ(u) = x✱ Ψ◦Φ(u) = Ψ(x) = u ❡ Ψ◦Φ(v) = Ψ(y) =v✳ ▲♦❣♦✱ Φ◦Ψ = 1 ❡ Ψ◦Φ = 1✳ P♦rt❛♥t♦✱ Φé ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ K✲á❧❣❡❜r❛s✱ ♦

q✉❡ ♣r♦✈❛ ♥♦ss❛ ❛✜r♠❛çã♦✳

❊①❡♠♣❧♦ ✶✳✶✶ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❡ A ❛ K✲á❧❣❡❜r❛ ❛♣r❡s❡♥t❛❞❛ ♣❡❧♦s ❞♦✐s ❣❡r❛❞♦r❡s u, v

❡ r❡❧❛çã♦ vu− uv = u✳ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ K[y][x, α] ❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞♦ ❛♥❡❧ ❞❡

♣♦❧✐♥ô♠✐♦sK[y]✱ ♦♥❞❡ αé ♦K✲❛✉t♦♠♦r✜s♠♦ ❞❡ K[y]t❛❧ q✉❡α(y) =y−1✳ ❊♠K[y][x, α]✱

t❡♠♦s q✉❡ xy = α(y)x✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ xy−yx = x✳ P♦rt❛♥t♦✱ ♣r♦❝❡❞❡♥❞♦ ❝♦♠♦ ♥♦

❡①❡♠♣❧♦ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ♣r♦✈❛r q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦K✲✐s♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s

ϕ:A−→K[y][x, α] t❛❧ q✉❡ ϕ(u) =x ❡ ϕ(v) = y✳

❊①❡♠♣❧♦ ✶✳✶✷ ❙❡❥❛g ❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ♥ã♦ ❛❜❡❧✐❛♥❛ ❞❡ ❞✐♠❡♥sã♦ 2 s♦❜r❡ ✉♠ ❝♦r♣♦K ❡

s❡❥❛{X, Y} ✉♠❛ ❜❛s❡ ❞❡ gt❛❧ q✉❡ [X, Y] =Y✳

❆ K✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ✭✉♥✐✈❡rs❛❧✮ ❞❡g é U(g) :=T (g)/I✱ ♦♥❞❡ T (g) é ❛ K✲á❧❣❡❜r❛

t❡♥s♦r✐❛❧ ❞❡g ❡ I é ♦ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡ T(g) ❣❡r❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞❛ ❢♦r♠❛

a⊗b−b⊗a−[a, b], a, b ∈g.

❆✜r♠❛çã♦✿ U(g)é ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦sK[y]✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱

s❡❥❛ δ ❛ ú♥✐❝❛ ❞❡r✐✈❛çã♦ ❞❡ K[y] t❛❧ q✉❡ δ(y) = y✱ ♦✉ s❡❥❛✱ δ(p(y)) = y d

dyp(y)✱ ♣❛r❛ p(y) ∈ K[y]✳ ◆♦ ❛♥❡❧ K[y][x;δ]✱ t❡♠♦s xy = yx + δ(y) = yx+ y✳ Pr♦✈❛r❡♠♦s q✉❡

U(g)∼=K[y][x;δ] ❝♦♠♦ K✲á❧❣❡❜r❛s✳

❈♦♠ ❡❢❡✐t♦✱ s❡❥❛η :g−→K[y][x;δ] ♦K✲❤♦♠♦♠♦r✜s♠♦ ❧✐♥❡❛r t❛❧ q✉❡η(X) = x❡η(Y) =

y✳ ❊♥tã♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛ á❧❣❡❜r❛ t❡♥s♦r✐❛❧ T(g)✱ ♣♦❞❡♠♦s ❡①t❡♥❞❡r η ❛

✉♠ ú♥✐❝♦ K✲❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s η˜ : T (g) −→ K[y][x;δ]✳ ❆❣♦r❛✱ ♥♦t❡♠♦s q✉❡

♣❛r❛ a, b ∈ g✱ ❡①✐st❡♠ ❡s❝❛❧❛r❡s λi, βj ∈ K✱ 1 ≤ i, j ≤ 2✱ t❛✐s q✉❡ a = λ1X +β1Y ❡

b = λ2X +β2Y✳ ❉♦♥❞❡✱ η˜(a⊗b −b⊗a −[a, b]) = (λ1β2 −λ2β1)(xy −yx−y) = 0✳

■st♦ ✐♠♣❧✐❝❛ q✉❡ ♦ ✐❞❡❛❧ I ❡stá ❝♦♥t✐❞♦ ❡♠ ker(˜η)✳ ❊♥tã♦✱ ♣❡❧♦ t❡♦r❡♠❛ ❢✉♥❞❛♠❡♥t❛❧ ❞♦s

❤♦♠♦♠♦r✜s♠♦s✱ ❡①✐st❡ ✉♠ ú♥✐❝♦K✲❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s

ϕ:U(g) =T (g)/I −→K[y][x;δ]

t❛❧ q✉❡ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

T (g) η˜ //

π

K[y][x, δ]

T(g)/I

∃!ϕ

8

8

❊♠ ♣❛rt✐❝✉❧❛r✱ ϕ(X+I) = ˜η(X) = x ❡ϕ(Y +I) = ˜η(Y) = y✳

P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ K[y] é ✉♠ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦

❞❡K✲á❧❣❡❜r❛s σ :K[y]−→T (g)/I t❛❧ q✉❡ σ(y) = Y +I✳

❙❡❥❛p(y) =P_iaiyi ∈K[y]✱ ❡♥tã♦ σ(p(y))(X+I) +σ(δ(p(y))) =

=P_iai(YiX+I) +σ(P_iiaiyi) =P_iai(YiX+I) +P_iiai(Yi+I) =

=P_iai(YiX+iYi+I) ✳✳✳✳✳✳✳✭✯✮

❈♦♠♦ XY +I = (Y X+Y) +I✱ ❡♥tã♦ ♣♦r ✐♥❞✉çã♦ XYi _{+I = (Y}i_{X +iY}i_{) +I✱ ♣❛r❛}

t♦❞♦i≥1✳ ▲♦❣♦✱ ❡♠ ✭✯✮✱

σ(p(y))(X+I) +σ(δ(p(y))) = P_iai(XYi+I) = (X+I)P_iai(Yi+I) = (X+I)σ(p(y))✳

❆❣♦r❛✱ ❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛s ❡①t❡♥sõ❡s ❞❡ ❖r❡✱ ❣❛r❛♥t❡ ❛ ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡K✲á❧❣❡❜r❛s Ψ :K[y][x, δ]−→U(g)t❛❧ q✉❡ Ψ|K[y]=σ ❡ Ψ(x) =X+I✳

■st♦ é✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

K[y][x;δ]

∃!Ψ

%

%

K[ ?y] σ //

ι

O

O

U(g)

❈❛♣ít✉❧♦ ✶✳ ❊①t❡♥sõ❡s ❞❡ ❖r❡ ✶✳✶✳ ❉❡✜♥✐çõ❡s ❡ ❡①❡♠♣❧♦s

❋✐♥❛❧♠❡♥t❡✱ ♥♦t❡♠♦s q✉❡ Ψ◦ϕ(X +I) = Ψ(x) = X +I✱ ϕ◦Ψ(y) = ϕ(Y +I) = y ❡ ϕ◦Ψ(x) =ϕ(X+I) =x✳ ■st♦ ✐♠♣❧✐❝❛ q✉❡ Ψ ❡ϕ sã♦ ♠✉t✉❛♠❡♥t❡ ✐♥✈❡rs♦s✱ ♦ q✉❡ ♣r♦✈❛

♥♦ss❛ ❛✜r♠❛çã♦✳

❊①❡♠♣❧♦ ✶✳✶✸ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❡ s❡❥❛ sln(K) = {A∈Mn(K) tal que T r(A) = 0} ❛

á❧❣❡❜r❛ ❞❡ ▲✐❡ ❡s♣❡❝✐❛❧ ❧✐♥❡❛r✱ ❝♦♠ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ [A, B] = AB −BA✳ ❊♠ ♣❛rt✐❝✉❧❛r✱

sl2(K) é ❞❡ ❞✐♠❡♥sã♦ 3 ❡ ✉♠❛ ❜❛s❡ ❞❡ sl2(K) ❡stá ❢♦r♠❛❞❛ ♣❡❧❛s ♠❛tr✐③❡s✿

E =



 0 1

0 0



_✱ F =



 0 0

1 0



_✱ H =



 1 0

0 −1

 

❖s ♣r♦❞✉t♦s ❞❡ ▲✐❡ ❞❡ t❛✐s ♠❛tr✐③❡s sã♦✿ [E, F] =H✱ [H, E] = 2E✱[H, F] =−2F · · · ✭✯✮

❆ K✲á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ ❞❡ sl2(K)é U(sl2(K)) :=T(sl2(K))/I ❛K✱ ♦♥❞❡ T(sl2(K))é ❛

K✲á❧❣❡❜r❛ t❡♥s♦r✐❛❧ ❞❡sl2(K) ❡ I é ♦ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡ T(sl2(K)) ❣❡r❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s

❞❛ ❢♦r♠❛

a⊗ b − b ⊗a − [a, b]✱ a, b ∈ sl2(K). ❙❡❥❛♠ e := E + I✱ f := F + I ❡ h := H +I✳

❯s❛♥❞♦ ✭✯✮✱ t❡♠♦s ♥❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡U(sl2(K)❛s r❡❧❛çõ❡s ef−f e=h✱ he−eh= 2e✱

hf−f h=−2f✳ ❊q✉✐✈❛❧❡♥t❡♠❡♥t❡✱ eh= (h−2)e✱ f h= (h+ 2)f✱ f e=ef −h · · · ✭✯✯✮

❆❣♦r❛ s❡❥❛ K[y] ♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ♥❛ ✐♥❞❡t❡r♠✐♥❛❞❛ y ❡ ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ K✳

❙❡❥❛σ :K[y]−→U(sl2(K))♦ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦ ❞❡ K✲á❧❣❡❜r❛s t❛❧ q✉❡σ(y) = h❡ s❡❥❛

α : K[y] −→ K[y] ♦ K✲❛✉t♦♠♦r✜s♠♦ ❞❡ K[y] t❛❧ q✉❡ α(y) = y−2✳ ❯s❛♥❞♦ ❛s r❡❧❛çõ❡s (∗∗)✱ t❡♠♦s ❡♠ U(sl2(K)) q✉❡ eσ(y) = eh= (h−2)e=σ(y−2)e=σ(α(y))e.

P♦rt❛♥t♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛s ❡①t❡♥sõ❡s ❞❡ ❖r❡✱ ❡①✐st❡ ✉♠ ú♥✐❝♦K✲❤♦♠♦♠♦r✜s♠♦

❞❡ á❧❣❡❜r❛s σ˜ :K[y][x;α] −→U(sl2(K)) t❛❧ q✉❡ σ|˜K[y]= σ ❡ σ˜(x) =e✱ ♦✉ s❡❥❛ ♦ s❡❣✉✐♥t❡

❞✐❛❣r❛♠❛ ❝♦♠✉t❛

K[y][x;α]

∃!˜σ

&

&

K[ ?y] σ //

ι

O

O

U(sl2(K)

❆❣♦r❛ s❡❥❛β ♦ ú♥✐❝♦ K✲❛✉t♦♠♦r✜s♠♦ ❞❡K[y][x, α] t❛❧ q✉❡β(y) = y+ 2 ❡β(x) = x✳ ❙❡❥❛

t❛♠❜é♠ δ ❛ ú♥✐❝❛ β✲❞❡r✐✈❛çã♦ ❞❡ K[y][x;α] t❛❧ q✉❡ δ(y) = 0 ❡ δ(x) = −y✳

❯s❛♥❞♦ ❛s ❡q✉❛çõ❡s ✭✯✯✮✱ t❡♠♦s ❛s s❡❣✉✐♥t❡s r❡❧❛çõ❡s ❡♠ U(sl2(K)) ✿

fσ˜(y) = f h = (h+ 2)f = ˜σ(y+ 2)f = ˜σ(β(y))f + ˜σδ(y) ❡ fσ˜(x) = f e = ef −h = ˜

σ(β(x))f + ˜σδ(x)✳ P♦rt❛♥t♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛s ❡①t❡♥sõ❡s ❞❡ ❖r❡✱ ❡①✐st❡ ✉♠

ú♥✐❝♦K✲❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s

ϕ:K[y][x;α][z;β, δ]−→U(sl2(K))

q✉❡ ❡①t❡♥❞❡σ˜ ❡ϕ(z) =f✳ ❖✉ s❡❥❛✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

K[y][x;α][z;β, δ]

∃!ϕ

(

(

K[y][ ?x;α] σ˜ //

ι

O

O

U(sl2(K))

❊♠ ♣❛rt✐❝✉❧❛r✱ ϕ(y) = ˜σ(y) = h✱ ϕ(x) = ˜σ(x) =e ❡ ϕ(z) = f✳

❆✜r♠❛çã♦✿ ϕ é ✉♠ K✲✐s♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ❡ ♣♦rt❛♥t♦✱ ❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡

U(sl2(K)) é ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ✭✐t❡r❛❞❛✮ ❞♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦sK[y]✳

❈♦♠ ❡❢❡✐t♦✱ ❝♦♥s✐❞❡r❡ ♦ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦K✲❧✐♥❡❛r φ:sl2(K)−→K[y][x;α][z;β, δ]

t❛❧ q✉❡ φ(H) = y✱ φ(E) = x ❡ φ(F) = z✳ ❊♥tã♦✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❛ á❧❣❡❜r❛

t❡♥s♦r✐❛❧✱ ❛ ❛♣❧✐❝❛çã♦ K✲❧✐♥❡❛r φ ♣♦❞❡ s❡r ❡①t❡♥❞✐❞❛ ♣❛r❛ ✉♠ ú♥✐❝♦ K✲❤♦♠♦♠♦r✜s♠♦

ˆ

φ:T(sl2(K)−→K[y][x;α][z;β, δ]✳ ❖✉ s❡❥❛✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

sl2(K) ι //

φ

T(sl2(K))

∃! ˆφ

v

v

K[y][x;α][z;β, δ]

❆s ❡q✉❛çõ❡s ✭✯✮ ❡ ✭✯✯✮ ✐♠♣❧✐❝❛♠ q✉❡ E, F, H ∈ kerφˆ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ I ⊂ kerφˆ✳

P♦rt❛♥t♦ ♣♦❞❡♠♦s ❢❛t♦r❛r_φˆ_{✱ ✐st♦ é✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ K✲ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s} Φ :T(sl2(K))/I −→K[y][x;α][z;β, δ]

t❛❧ q✉❡ Φ◦π = ˆφ✱ ♦✉ s❡❥❛✱ ♦ s❡❣✉✐♥t❡ ❞✐❛❣r❛♠❛ ❝♦♠✉t❛

T(sl2(K)

ˆ

φ

/

/

π

K[y][x, α][z, b, δ]

T(sl2(K)/I

∃!Φ

6

6

❊♠ ♣❛rt✐❝✉❧❛r✱ Φ(h) = ˜φ(H) =y✱Φ(e) = ˜φ(E) =x ❡ Φ(f) = ˜φ(F) = z✳

❆❣♦r❛ é ❝❧❛r♦ q✉❡ φ ❡ ϕ sã♦K✲❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s ♠✉t✉❛♠❡♥t❡ ✐♥✈❡rs♦s✳ ❊♠

❝♦♥❝❧✉sã♦✿

❈❛♣ít✉❧♦ ✶✳ ❊①t❡♥sõ❡s ❞❡ ❖r❡ ✶✳✷✳ Pr♦♣r✐❡❞❛❞❡s

U(sl2(K) ∼= K[y][x;α][z;β, δ] ❝♦♠♦ K✲á❧❣❡❜r❛s ❡ ♣♦rt❛♥t♦ ❛ á❧❣❡❜r❛ ✉♥✐✈❡rs❛❧ ❡♥✈♦❧✲

✈❡♥t❡ ❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❡s♣❡❝✐❛❧ ❧✐♥❡❛rsl2(K)é ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s

K[y]✳

◆❛ s❡❣✉✐♥t❡ s❡çã♦ ❡st✉❞❛r❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❛s ❡①t❡♥sõ❡s ❞❡ ❖r❡✳

✶✳✷ Pr♦♣r✐❡❞❛❞❡s

❉❡✜♥✐çã♦ ✶✳✶✹ ❙❡❥❛ R[x;α, δ] ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞❡ R ❡ s❡❥❛ p ∈ R[x;α, δ] ✉♠ ❡❧❡✲

♠❡♥t♦ ❞✐❢❡r❡♥t❡ ❞❡ ③❡r♦✱❡♥tã♦ p ♣♦❞❡ s❡r ❡s❝r✐t♦ ❞❡ ♠❛♥❡✐r❛ ú♥✐❝❛ ❝♦♠♦

p=rnxn+rn−1xn−1 +...+r1x+r0,

♣❛r❛ ❛❧❣✉♠ n ∈ N ❡ ❛❧❣✉♥s ri ∈ R, i = 0, ..., n t❛❧ q✉❡ rn 6= 0✳ ❖ ✐♥t❡✐r♦ n é ❝❤❛♠❛❞♦

❣r❛✉ ❞❡ ♣ ❡ é ❞❡♥♦t❛❞♦ ♣♦r n =gr(p)✳ ❖ ❡❧❡♠❡♥t♦ rn é ❝❤❛♠❛❞♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡

p✳ P♦r ❝♦♥✈❡♥çã♦✱ ❞✐③❡♠♦s q✉❡ ♦ ❣r❛✉ ❞♦ ❡❧❡♠❡♥t♦ ③❡r♦ ❞❡ R[x;α, δ] é −∞✳

❖❜s❡r✈❛çã♦ ✶✳✶✺ ◆♦ ❝❛s♦ ❞❡ ✉♠ ❛♥❡❧ ❞❡ ♦♣❡r❛❞♦r❡s ❞✐❢❡r❡♥❝✐❛✐s✱ ♦✉ s❡❥❛✱R[x;δ]✱ ❞✐r❡✲

♠♦s q✉❡ n é ❛ ♦r❞❡♠ ❞❡ p✳

▲❡♠❛ ✶✳✶✻ ❙❡❥❛R[x;α, δ] ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❡ s❡❥❛♠ r∈R ❡ n ∈N✱ ❡♥tã♦

xnr =αn(r)xn+an−1xn−1+...+a1x+δn(r)

♣❛r❛ ❛❧❣✉♥s an−1, ..., a1 ∈ R✳ P♦rt❛♥t♦✱ s❡ r 6= 0 ❡ α é ✐♥❥❡t✐✈❛✱ ❡♥tã♦ xnr t❡♠ ❣r❛✉ n ❡

❝♦❡✜❝✐❡♥t❡ ❧í❞❡r αn_(r)✳

❉❡♠♦♥str❛çã♦✿ Pr♦✈❡♠♦s ❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ♣♦r ✐♥❞✉çã♦ s♦❜r❡ n✳

❙❡ n = 0 ♥ã♦ ❤á ♥❛❞❛ ❛ ♣r♦✈❛r✳ ❖ ❝❛s♦ n = 1 é s✐♠♣❧❡s♠❡♥t❡ ❛ r❡❣r❛ ❞❛ ♠✉❧t✐♣❧✐❝❛çã♦

❡♠ R[x;α, δ]✳

❙✉♣♦♥❤❛ q✉❡ ♣❛r❛n≥1✱ ❡①✐st❡♠ an−1, ..., a1 ∈R t❛✐s q✉❡

xn_r=αn_(r)xn_+a

n−1xn−1+· · ·+a1x+δn(r)✳ ❊♥tã♦✱

xn+1_{r =xα}n_(r)xn_+a

n−1xn−1+xan−1xn−1+· · ·+xa1x+xδn(r) =

(αn+1_(r)x+δ(αn_(r)))xn_{+ (α(a}

n−1)x+δ(an−1))xn−1 +· · ·+δn+1(r) =

αn+1_(r)xn+1_{+ (δ(α}n_{(r)) +α(a}

n−1))xn+· · ·+δn+1(r)✳

❆ss✐♠✱ xn+1_{r ♣♦❞❡ s❡r ❡s❝r✐t♦ ♥❛ ❢♦r♠❛ ❞❡s❡❥❛❞❛✳ ❆ s❡❣✉♥❞❛ ♣❛rt❡ s❡❣✉❡✲s❡ ❞♦ ❢❛t♦ q✉❡}

s❡α é ✐♥❥❡t✐✈❛✱ αn _{é ✐♥❥❡t✐✈❛✳ P♦rt❛♥t♦✱ s❡ r6= 0✱ ❡♥tã♦ α}n_{(r)6= 0✳}

Pr♦♣♦s✐çã♦ ✶✳✶✼ ❙❡❥❛♠ R ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡ ❡ α✉♠ ❡♥❞♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦ ❞❡ R✳ ❊♥tã♦✱ gr(pq) =gr(p) +gr(q)♣❛r❛ t♦❞♦ p, q ∈R[x;α, δ]✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡ R[x;α, δ]

é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳

❉❡♠♦♥str❛çã♦✿ ❙❡ p ♦✉ q é ♦ ❡❧❡♠❡♥t♦ ③❡r♦✱ ❛ ✐❣✉❛❧❞❛❞❡ s❡❣✉❡✲s❡ ❞❛ ❝♦♥✈❡♥çã♦ q✉❡ −∞+n =−∞ ♣❛r❛ t♦❞♦n ∈N ❡−∞+−∞=−∞✳

❙✉♣♦♥❤❛♠♦s q✉❡p6= 0 ❡ q6= 0✳ ❊♥tã♦✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r ❞❡ ♠❛♥❡✐r❛ ú♥✐❝❛

p=rnxn+rn−1xn−1+· · ·+r1x+r0, ♦♥❞❡ n =gr(p)

q =smxm+sm−1xm−1+· · ·+s1x+s0, ♦♥❞❡ m=gr(q)

❆❣♦r❛✱ (rnxn)(smxm) = rn(xnsm)xm ❡ ✉s❛♥❞♦ ♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ♣♦❞❡♠♦s ❡s❝r❡✈❡r

xn_s

m=αn(sm)xn+an−1xn−1+· · ·+a1x+δn(sm), ♣❛r❛ ❛❧❣✉♥s an−1, ..., a1 ∈R

▲♦❣♦✱ (rnxn)(smxm) =rnαn(sm)xm+n+✭t❡r♠♦s ❞❡ ♠❡♥♦r ❣r❛✉✮✳

■st♦ ✐♠♣❧✐❝❛ q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ pq é rnαn(sm) ❡ ❝♦♠♦ rn 6= 0✱ α é ✉♠ ❡♥❞♦✲

♠♦r✜s♠♦ ✐♥❥❡t✐✈♦ ❡ R é ✉♠ ❞♦♠í♥✐♦✱ rnαn(sm) 6= 0✳ P♦rt❛♥t♦✱ gr(pq) = m+n✳ ❖ ❢❛t♦

❛♥t❡r✐♦r ❣❛r❛♥t❡ q✉❡ s❡ p 6= 0 ❡ q 6= 0✱ ❡♥tã♦ pq 6= 0✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ R[x;α, δ] é ✉♠

❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳

▲❡♠❛ ✶✳✶✽ ❙❡❥❛R[x;α, δ] ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞❡ R ♦♥❞❡ α é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ R✳

❊♥tã♦α−1 _{é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞♦ ❛♥❡❧ ♦♣♦st♦ R}op _{❡ −δα}−1 _{é ✉♠❛ α}−1_{✲❞❡r✐✈❛çã♦ ❞❡ R}op_✳

▼❛✐s ❛✐♥❞❛✱ R[x;α, δ]op_=Rop_[x;α−1_,−δα−1_].

❉❡♠♦♥str❛çã♦✿ Pr♦✈❡♠♦s ♣r✐♠❡✐r♦ q✉❡ α−1 _{é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ R}op_{✳ ❉❡♥♦t❡♠♦s}

❝♦♠∗❛ ♦♣❡r❛çã♦ ❡♠Rop_{✱ ✐st♦ é✱ s❡a, b∈R}op_{✱a∗b=ba✳ ❈♦♠♦α}−1 _{é ✉♠ ❤♦♠♦♠♦r✜s♠♦}

❞❡R✱ t❡♠♦s q✉❡ α−1_{(a∗b) = α}−1_{(ba) =α}−1_(b)α−1_{(a) =α}−1_(a)∗α−1_(b)✳

▼♦str❡♠♦s ❛❣♦r❛ q✉❡ −δα−1 _{é ✉♠❛ α}−1_{✲❞❡r✐✈❛çã♦ ❡♠ R}op_{✳ ❈♦♠ ❡❢❡✐t♦✱ −δα}−1_{(r∗s) =}

−δ(α−1_{(sr)) =−δ(α}−1_(s)α−1_{(r)) =−(δ(α}−1_(s))α−1_{(r) +α}−1_{(s) +δ(α}−1_{(r)) =}

−(α−1_(r)∗δ(α−1_{(s)) +δ(α}−1_(r))∗α−1_{(s)) =−(α}−1 _∗δα−1_{(s) +δα}−1_(r)∗α−1_(s))✳

❋✐♥❛❧♠❡♥t❡✱ ♣r♦✈❡♠♦s q✉❡ R[x;α, δ]op _{= R}op_[x;α−1_,−δα−1_{]✳ ❈♦♠ ❡❢❡✐t♦✱ ❡♠ R[x;α, δ]✱}

t❡♠♦s q✉❡ xr = α(r)x+δ(r) ♣❛r❛ t♦❞♦ r ∈ R✳ ▲♦❣♦✱ xα−1_{(r) = rx+δ(α}−1_{(r))✱ ❡ ❞❛í}

rx=xα−1_(r)−δ(α−1_{(r))✳ ❆ss✐♠ ❡♠R[x;α, δ]}op_{✱ t❡♠♦s q✉❡x∗r =α}−1_{(r)∗x−δ(α}−1_(r))✳

▼❛s ❡st❛ é ❛ ♠❡s♠❛ r❡❣r❛ ❞❡ ♦♣❡r❛çã♦ ❡♠Rop_[x;α−1_,−δα−1_{]✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱}

R[x;α, δ]op _=Rop_[x;α−1_,−δα−1_]✳

❈❛♣ít✉❧♦ ✶✳ ❊①t❡♥sõ❡s ❞❡ ❖r❡ ✶✳✷✳ Pr♦♣r✐❡❞❛❞❡s

❖ ❚❡♦r❡♠❛ ❞❛s ❇❛s❡ ❞❡ ❍✐❧❜❡rt ❞❡♠♦♥str❛ q✉❡ s❡ R é ♥♦❡t❤❡r✐❛♥♦ à ❞✐r❡✐t❛ ✭à ❡s✲

q✉❡r❞❛✮✱ ❡♥tã♦ ✉♠ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ✭♦r❞✐♥ár✐♦✮ é t❛♠❜é♠ ♥♦❡t❤❡r✐❛♥♦ à ❞✐r❡✐t❛ ✭à ❡sq✉❡r❞❛✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✮✳ ◆ós ♣r♦✈❛r❡♠♦s ✉♠ r❡s✉❧t❛❞♦ s❡♠❡❧❤❛♥t❡ ♣❛r❛ ❡①t❡♥sõ❡s ❞❡ ❖r❡✳

❚❡♦r❡♠❛ ✶✳✶✾ ❙❡❥❛S =R[x;α, δ]✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞❡R✱ ♦♥❞❡αé ✉♠ ❛✉t♦♠♦r✜s♠♦

❞❡R✳ ❙❡ R é ✉♠ ❛♥❡❧ ♥♦❡t❤❡r✐❛♥♦ à ❞✐r❡✐t❛ ✭❡sq✉❡r❞❛✮✱ ❡♥tã♦S é ✉♠ ❛♥❡❧ ♥♦❡t❤❡r✐❛♥♦ à

❞✐r❡✐t❛✭❡sq✉❡r❞❛✮✳

❉❡♠♦♥str❛çã♦✿

❈❛s♦ ■✿ ❙✉♣♦♥❤❛♠♦s ♣r✐♠❡✐r♦ q✉❡ R é ♥♦❡t❤❡r✐❛♥♦ à ❞✐r❡✐t❛✳ Pr♦✈❛r❡♠♦s ♥❡st❡ ❝❛s♦

q✉❡ q✉❛❧q✉❡r ✐❞❡❛❧ ♥ã♦✲♥✉❧♦I à ❞✐r❡✐t❛ ❞❡S é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❙❡❣✉✐r❡♠♦s ♦s s❡❣✉✐♥t❡s

♣❛ss♦s✿

P❛ss♦ ✶✿ ❙❡❥❛J ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝♦❡✜❝✐❡♥t❡s ❧í❞❡r❡s ❞❡ I ❥✉♥t❛♠❡♥t❡ ❝♦♠ 0✱ ✐st♦ é✱

J =r∈R / rxd_+r

d−1xd−1+...+r0 ∈I, ♣❛r❛ ❛❧❣✉♥s rd−1, ..., r0 ∈R .

➱ ❝❧❛r♦ q✉❡Jé ✉♠ s✉❜❣r✉♣♦ ❛❞✐t✐✈♦ ❞❡R✳ ❆❣♦r❛ ❝♦♥s✐❞❡r❡ ❡❧❡♠❡♥t♦sr∈J❡a ∈R✱ ❡♥tã♦

❡①✐st❡ ❛❧❣✉♠p∈I ❞❛ ❢♦r♠❛p=rxd_{+[t❡r♠♦s ❞❡ ♠❡♥♦r ❣r❛✉]❡♠I✳ ❊♥tã♦pa∈I✳ ❈♦♠♦}

pa=rαd_(a)xd_{+ [t❡r♠♦s ❞❡ ♠❡♥♦r ❣r❛✉]✱ ❝♦♥❝❧✉í♠♦s q✉❡ rα}d_{(a)∈J✳ P❛r❛ ♦❜t❡r♠♦s ra✱}

❞❡✈❡♠♦s s✉❜st✐t✉✐ra ♣♦r α−d_{(a)✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱ t❡♠♦s pα}−d_{(a)∈I ❡}

pα−d(a) = raxd+ [t❡r♠♦s ❞❡ ♠❡♥♦r ❣r❛✉].

P♦rt❛♥t♦ra∈J✳ ■st♦ ♠♦str❛ q✉❡ J é ✉♠ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❞❡R✳

P❛ss♦ ✷✿ ❈♦♠♦ R é ♥♦❡t❤❡r✐❛♥♦ à ❞✐r❡✐t❛✱ J é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳ ❙✉♣♦♥❤❛♠♦s q✉❡ r1, ..., rk é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ❣❡r❛❞♦r❡s ♣❛r❛ J✱ ❡♥tã♦ ❡①✐st❡♠ p1, ..., pk ∈ I t❛❧ q✉❡ ❝❛❞❛ pi

t❡♠ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ri ❡ ❣r❛✉ ni✳ ❙❡❥❛ n =max{n1, ..., nk}✳ ◆♦t❡♠♦s q✉❡ pixn−ni é ✉♠

❡❧❡♠❡♥t♦ ❞❡I ❝♦♠ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡rri ♠❛s ❝♦♠ ❣r❛✉n✳ ❆ss✐♠✱ s❡♠ ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡✱

♣♦❞❡♠♦s ❛ss✉♠✐r q✉❡ t♦❞♦s ♦spi ♣♦ss✉❡♠ ♦ ♠❡s♠♦ ❣r❛✉ n✱ ✐st♦ é✱

pi =rixn+ [t❡r♠♦s ❞❡ ♠❡♥♦r ❣r❛✉].

P❛ss♦ ✸✿ ❙❡❥❛ N = R +Rx +...+Rxn−1_{✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s ❞❡ S ❝♦♠ ❣r❛✉}

♠❡♥♦r q✉❡ n✳ P❡❧♦ ❧❡♠❛ 1.16✱ N = R+xR+...+xn−1_{R✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ N é ✉♠}

R✲s✉❜♠ó❞✉❧♦ à ❞✐r❡✐t❛ ❞❡ S✳ ❱✐st♦ ❝♦♠♦ R✲♠ó❞✉❧♦✱ N é ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✱ ❡ ♣♦rt❛♥t♦

é ♥♦❡t❤❡r✐❛♥♦✳ ❆ss✐♠✱ s❡✉ s✉❜♠ó❞✉❧♦I∩N é ✉♠R✲♠ó❞✉❧♦ à ❞✐r❡✐t❛ ✜♥✐t❛♠❡♥t❡ ❣❡r❛❞♦✳

❉✐❣❛♠♦s q✉❡q1, ..., qt ❣❡r❛♠ I∩N✳

P❛ss♦ ✹ ❙❡❥❛ I0 ♦ ✐❞❡❛❧ à ❞✐r❡✐t❛ ❞❡ S ❣❡r❛❞♦ ♣♦r p1, ..., pk, q1, ..., qt✱ ❡♥tã♦✱ I0 ⊆ I✳ ❆

✐♥❝❧✉sã♦ ❛♥t❡r✐♦r ♥❛ ✈❡r❞❛❞❡ é ✉♠❛ ✐❣✉❛❧❞❛❞❡✳ ❈♦♠ ❡❢❡✐t♦✱ s❡❥❛ p ∈ I ❝♦♠ ❣r❛✉ ♠❡♥♦r

q✉❡n✱ ❡♥tã♦ p∈I ∩N ❡ p=q1a1+...+qtat ♣❛r❛ ❛❧❣✉♥s aj ∈R ❡ ♣♦rt❛♥t♦ p∈I0✳

P❛ss♦ ✺ ❆❣♦r❛ ❝♦♥s✐❞❡r❡ ❛❧❣✉♠p∈I ❝♦♠ ❣r❛✉m≥n✱ ❡ s✉♣♦♥❤❛ q✉❡ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s

❞❡I ❝♦♠ ❣r❛✉ ♠❡♥♦r q✉❡m ❡stã♦ ❡♠ I0✳ s❡❥❛r ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ p❀ ❛ss✐♠

p=rxm+ [t❡r♠♦s ❞❡ ♠❡♥♦r ❣r❛✉].

❈♦♠♦ p ∈I✱ s❡✉ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r r ❡stá ❡♠ J✱ ❡ ❛ss✐♠ r = r1a1 +...+rkak ♣❛r❛ ❛❧❣✉♥s

ai ∈R✳ ❉❡s❡❥❛♠♦s ❝♦♥str✉✐r ✉♠ ❡❧❡♠❡♥t♦ ❞❡I0 q✉❡ t❛♠❜é♠ t❡♠ ❣r❛✉m❡ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r

r✳ ❉❡✈❡♠♦s ❛♣❧✐❝❛r ❛s ♣♦tê♥❝✐❛s ♥❡❣❛t✐✈❛s ❛♣r♦♣r✐❛❞❛s ❞❡ α ❡♠ ai✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡✱

♦❜s❡r✈❡♠♦s q✉❡

piα−n(ai) =riaixn+ [t❡r♠♦s ❞❡ ♠❡♥♦r ❣r❛✉]

♣❛r❛ t♦❞♦i✳ ❈♦♥s❡q✉❡♥t❡♠❡♥t❡✱ s❡ q= (p1α−n(a1) +...+pkα−n(ak))xm−n✱ ❡♥tã♦q ∈I0 ❡

q=rxm+ [t❡r♠♦s ❞❡ ♠❡♥♦r ❣r❛✉].

❆❣♦r❛ p− q é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ I ❝♦♠ ❣r❛✉ ♠❡♥♦r q✉❡ m✳ P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ p−q ∈ I0✱ ❡ ❛ss✐♠ p ∈ I0✳ ❊st❛ ✐♥❞✉çã♦ ♠♦str❛ q✉❡ I = I0✳ ❆ss✐♠✱ I é ✜♥✐t❛♠❡♥t❡

❣❡r❛❞♦✳ P♦rt❛♥t♦✱S é ♥♦❡t❤❡r✐❛♥♦✳

❈❛s♦ ■■✳ ❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ R é ♥♦❡t❤❡r✐❛♥♦ à ❡sq✉❡r❞❛✳ ❊♥tã♦ ♦ ❛♥❡❧ ♦♣♦st♦ Rop _{é ♥♦❡t❤❡r✐❛♥♦ à ❞✐r❡✐t❛✳ P❡❧♦ ❧❡♠❛ 1.18 ✱ α}−1 _{é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ R}op _{❡ −δα}−1

é ✉♠❛ α−1 _{❞❡r✐✈❛çã♦ ❞❡ R}op_{✳ ▲♦❣♦✱ ♣❡❧♦ ❝❛s♦ ■ ❛❝✐♠❛✱ R}op_[x;α−1

,−δα−1_{] é ♥♦❡t❤❡r✐❛♥♦}

à ❞✐r❡✐t❛✳ ▼❛s✱ ♣❡❧♦ ♠❡s♠♦ ❧❡♠❛✱ Rop_[x;α−1_,−δα−1_{] = R[x;α, δ]}op_{✳ ❊♠ ❝♦♥s❡q✉❡♥❝✐❛✱}

R[x;α, δ] é ♥♦❡t❤❡r✐❛♥♦ à ❡sq✉❡r❞❛✳

❈♦r♦❧ár✐♦ ✶✳✷✵ ❖ ♣❧❛♥♦ q✉â♥t✐❝♦ Oq(K2)✱ ❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ U(g) ❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡

♥ã♦ ❛❜❡❧✐❛♥❛g ❞❡ ❞✐♠❡♥sã♦ ✷ ❡ ❛ á❧❣❡❜r❛ ❡♥✈♦❧✈❡♥t❡ U(sl2(K))❞❛ á❧❣❡❜r❛ ❞❡ ▲✐❡ ❡s♣❡❝✐❛❧

sl2(K) sã♦ ❛♥❡✐s ♥♦❡t❤❡r✐❛♥♦s✳

❉❡♠♦♥str❛çã♦✿ ❙❡❣✉❡✲s❡ ❞♦s ❡①❡♠♣❧♦s 1.9✱ 1.11✱ 1.12 ❡ ❞♦ t❡♦r❡♠❛1.19✳

❈❛♣ít✉❧♦ ✶✳ ❊①t❡♥sõ❡s ❞❡ ❖r❡ ✶✳✷✳ Pr♦♣r✐❡❞❛❞❡s

❈♦r♦❧ár✐♦ ✶✳✷✶ ❙❡❥❛ S =R[x1;α1, δ1][x2;α2, δ2]· · ·[xn;αn, δn] ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ✐t❡✲

r❛❞❛✱ ♦♥❞❡ ❝❛❞❛αi é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞♦ ❛♥❡❧R[x1;α1, δ1]· · ·R[xi−1;αi−1, δi−1], ♣❛r❛ ❝❛❞❛

i= 1, ..., n✳ ❙❡ R é ✉♠ ❛♥❡❧ ♥♦❡t❤❡r✐❛♥♦ à ❞✐r❡✐t❛ ✭❡sq✉❡r❞❛✮✱ ❡♥tã♦S é ✉♠ ❛♥❡❧ ♥♦❡t❤❡✲

r✐❛♥♦ à ❞✐r❡✐t❛ ✭❡sq✉❡r❞❛✮✳

❚❡♦r❡♠❛ ✶✳✷✷ ❙❡❥❛ R é ✉♠ ❛♥❡❧ ❞❡ ❞✐✈✐sã♦ ❡ s❡❥❛ S =R[x;α, δ] ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡

❞❡ R✳ ❊♥tã♦ S é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s à ❡sq✉❡r❞❛✳ ❙❡ ❛❧é♠ ❞✐ss♦✱ α é ✉♠

❛✉t♦♠♦r✜s♠♦ ❞❡R✱ ❡♥tã♦ S é t❛♠❜é♠ ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s à ❞✐r❡✐t❛✳

❉❡♠♦♥str❛çã♦✿ ❈♦♠♦Ré ✉♠ ❛♥❡❧ ❞❡ ❞✐✈✐sã♦✱αé ✉♠ ❡♥❞♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦ ❡ ♣♦rt❛♥t♦ S é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐♥t❡❣r✐❞❛❞❡✳

❆❣♦r❛✱ ❞❛❞♦ ✉♠ ✐❞❡❛❧J à ❡sq✉❡r❞❛ ♥ã♦ ♥✉❧♦ ❞❡S✱ s❡❥❛ m♦ ♠❡♥♦r ❣r❛✉ ❞♦s ❡❧❡♠❡♥t♦s

♥ã♦✲♥✉❧♦s ❞❡J✱ ❡ ❡s❝♦❧❤❛p∈J ❝♦♠ ❣r❛✉m✳ ❙❡r é ♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡ p✱ ❡♥tã♦p ♣♦❞❡

s❡r s✉❜st✐t✉í❞♦ ♣♦r r−1_{p✱ ❡ ❛ss✐♠ ♥ã♦ ❤á ♣❡r❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞❡ ❡♠ ❛ss✉♠✐r q✉❡ p t❡♠}

❝♦❡✜❝✐❡♥t❡ ❧í❞❡r1✳

❆✜r♠❛çã♦✿ J =Sp✳ ❈♦♠ ❡❢❡✐t♦✱ é ❝❧❛r♦ q✉❡ Sp⊆J✳ Pr♦✈❡♠♦s ❛ ✐♥❝❧✉sã♦ r❡❝í♣r♦❝❛ ♣♦r

✐♥❞✉çã♦ s♦❜r❡k✳ ❖ ú♥✐❝♦ ❡❧❡♠❡♥t♦ ❞❡J ❝♦♠ ❣r❛✉ ♠❡♥♦r q✉❡m é0 ❡ ❝❡rt❛♠❡♥t❡0∈Sp✳

❆❣♦r❛ ❛ss✉♠❛ q✉❡ ♣❛r❛ ✉♠ ✐♥t❡✐r♦k ≥m✱ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s ❞❡J ❝♦♠ ❣r❛✉ ♠❡♥♦r q✉❡k

❡stã♦ ❡♠Sp✳ ❙❡❥❛q ✉♠ ❡❧❡♠❡♥t♦ ❞❡J ❝♦♠ ❣r❛✉ k✱ ❡ s❡❥❛ a♦ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r ❞❡q✳ ❆❣♦r❛ axk−m_{p t❡♠ ❣r❛✉ k ❡ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r a✱ ♣♦rt❛♥t♦ q−ax}k−m_{p é ✉♠ ❡❧❡♠❡♥t♦ ❞❡ J ❝♦♠}

❣r❛✉ ♠❡♥♦r q✉❡ k✳ P❡❧❛ ❤✐♣ót❡s❡ ❞❡ ✐♥❞✉çã♦✱ q−axk−m_{p ❡stá ❡♠ Sp✱ ❡ ❛ss✐♠✱ q ∈ Sp✳}

■st♦ ♣r♦✈❛ q✉❡ J =Sp✳ P♦rt❛♥t♦✱ S é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s à ❡sq✉❡r❞❛✳

❙✉♣♦♥❤❛♠♦s ❛❣♦r❛ q✉❡ α é ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞❡ R✳ ❊♥tã♦✱ ♣❡❧❛ ♣r✐♠❡✐r❛ ♣❛rt❡ ❡

♦ ❧❡♠❛ 1.18✱ Rop_[x;α−1_,−δα−1_{] é ✉♠ ❞♦♠í♥✐♦ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s à ❡sq✉❡r❞❛✳ ❆ss✐♠✱}

R[x;α, δ]op _{= R}op_[x;α−1_,−δα−1_{] é ✉♠ ❛♥❡❧ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s à ❡sq✉❡r❞❛✳ P♦rt❛♥t♦✱}

R[x;α, δ] é ✉♠ ❛♥❡❧ ❞❡ ✐❞❡❛✐s ♣r✐♥❝✐♣❛✐s à ❞✐r❡✐t❛✳

❖❜s❡r✈❛çã♦ ✶✳✷✸ P❛r❛ ❝❛❞❛a∈R✱ ❛ r❡❣r❛δa(r) =ar−ra✱ r∈R ❞❡✜♥❡ ✉♠❛ ❞❡r✐✈❛çã♦

δa ❡♠ R✳

❉❡✜♥✐çã♦ ✶✳✷✹ ❯♠❛ ❞❡r✐✈❛çã♦δ❡♠ Ré ✉♠❛ ❞❡r✐✈❛çã♦ ✐♥t❡r✐♦r s❡δ=δa♣❛r❛ ❛❧❣✉♠

a∈R✳ ❈❛s♦ ❝♦♥trár✐♦✱ ❞✐r❡♠♦s q✉❡ δ é ✉♠❛ ❞❡r✐✈❛çã♦ ❡①t❡r✐♦r✳

❉❡✜♥✐çã♦ ✶✳✷✺ ❙❡❥❛δ ✉♠❛ ❞❡r✐✈❛çã♦ ❞❡ R ❡ s❡❥❛ I ✉♠ ✐❞❡❛❧ ❞❡ R✳ ❉✐③❡♠♦s q✉❡ I é ✉♠ δ✲✐❞❡❛❧ s❡ δ(I)⊂I.

❖ ❛♥❡❧ R é ❝❤❛♠❛❞♦ δ✲s✐♠♣❧❡s s❡ ♦s ú♥✐❝♦s δ✲✐❞❡❛✐s ❞❡R sã♦ 0 ❡ R✳

▲❡♠❛ ✶✳✷✻ ❙❡❥❛δ ✉♠❛ ❞❡r✐✈❛çã♦ ❞❡ R ❡ s❡❥❛ S=R[x;δ] ✉♠❛ ❡①t❡♥sã♦ ❞❡ ❖r❡ ❞❡ R✳

(a) ❙❡ δ=δa ♣❛r❛ ❛❧❣✉♠ a∈R✱ ❡♥tã♦ S =R[x−a]✱ ✉♠ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ♦r❞✐♥ár✐♦✳

P♦rt❛♥t♦✱ S(x−a) é ✉♠ ✐❞❡❛❧ ♣ró♣r✐♦ ♥ã♦✲♥✉❧♦ ❞❡ S✳

(b) ❙❡ I é ✉♠ δ✲✐❞❡❛❧ ❞❡ R✱ ❡♥tã♦ IS =SI✳ ❆ss✐♠ IS é ✉♠ ✐❞❡❛❧ ❞❡ S✳ ▼❛✐s ❛✐♥❞❛✱ s❡ I 6=R✱ ❡♥tã♦ IS 6=S ❡ s❡ I 6= 0✱ ❡♥tã♦ IS 6= 0✳

❉❡♠♦♥str❛çã♦✿

(a) ◆♦t❡ q✉❡ xr=rx+δa(r)✳ ❆ss✐♠✱xr=rx+ar−ra✳ ❉❡ss❡ ♠♦❞♦✱xr−ar =rx−ra

❡✱ ♣♦rt❛♥t♦✱ (x−a)r =r(x−a)♣❛r❛ t♦❞♦ r ∈ R✳ ▲♦❣♦✱ S =R[x−a] ✱ ♦✉ s❡❥❛✱ S

é ✈✐st♦ ❝♦♠♦ ❛♥❡❧ ❞❡ ♣♦❧✐♥ô♠✐♦s ♥❛ ✈❛r✐á✈❡❧ x−a✳ P♦rt❛♥t♦✱ S(x−a) é ✉♠ ✐❞❡❛❧

♣ró♣r✐♦ ♥ã♦✲♥✉❧♦✳

(b) Pr♦✈❡♠♦s q✉❡ IS = SI✳ ❙❡❥❛ j ∈ I✳ ❈♦♠♦ I é ✉♠ δ✲✐❞❡❛❧✱ rxj = r(xj) =

r(jx+δ(j)) = rjx+rδ(j) ∈ IS✳ ❘❡❝✐♣r♦❝❛♠❡♥t❡✱ jrx = xrj −δ(jr) ∈ SI✳ ❖

r❡s✉❧t❛❞♦ s❡❣✉❡✱ ❝❧❛r❛♠❡♥t❡✳

❉❡✜♥✐çã♦ ✶✳✷✼ ❯♠ ❛♥❡❧S é ❞✐t♦ s✐♠♣❧❡s s❡ ♦s ú♥✐❝♦s ✐❞❡❛✐s ❜✐❧❛t❡r❛✐s ❞❡ S sã♦ 0 ❡ S✳

Pr♦♣♦s✐çã♦ ✶✳✷✽ ❙❡❥❛ R ✉♠❛ Q✲á❧❣❡❜r❛ ❡ s❡❥❛ δ ✉♠❛ ❞❡r✐✈❛çã♦ ❞❡ R✳ ❊♥tã♦ R[x;δ] é

✉♠ ❛♥❡❧ s✐♠♣❧❡s s❡✱ ❡ s♦♠❡♥t❡ s❡✱ R é δ✲s✐♠♣❧❡s ❡ δ é ✉♠❛ ❞❡r✐✈❛çã♦ ❡①t❡r✐♦r✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ S = R[x;δ]✳ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r s❡ R ♥ã♦ é δ✲s✐♠♣❧❡s ♦✉ s❡ δ é

✐♥t❡r✐♦r✱ ❡♥tã♦ S ♥ã♦ é s✐♠♣❧❡s✳

❘❡❝✐♣r♦❝❛♠❡♥t❡✱ s✉♣♦♥❤❛♠♦s q✉❡ R éδ✲s✐♠♣❧❡s ❡ q✉❡ δ é ✉♠❛ ❞❡r✐✈❛çã♦ ❡①t❡r✐♦r✳ ❙❡❥❛ I

✉♠ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡S ❡ n ♦ ♠❡♥♦r ❣r❛✉ ❞♦s ❡❧❡♠❡♥t♦s ♥ã♦✲♥✉❧♦s ❞❡I✳ ❙❡❥❛ J ♦ s✉❜❝♦♥✲

❥✉♥t♦ ❞❡R❝♦♥s✐st✐♥❞♦ ❞❡0✱ ❥✉♥t❛♠❡♥t❡ ❝♦♠ ♦s ❝♦❡✜❝✐❡♥t❡s ❧í❞❡r❡s ❞❛q✉❡❧❡s ❡❧❡♠❡♥t♦s ❞❡

I q✉❡ tê♠ ❣r❛✉n✳ ◆ã♦ é ❞✐❢í❝✐❧ ✈❡r q✉❡ J é ✉♠ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡ R✳

❆✜r♠❛çã♦✿ J é ✉♠ δ✲✐❞❡❛❧ ❞❡ R✳ ❈♦♠ ❡❢❡✐t♦✱ t♦❞♦ ❡❧❡♠❡♥t♦ ♥ã♦✲♥✉❧♦ r ∈J é ♦ ❝♦❡✜❝✐✲

❡♥t❡ ❧í❞❡r ❞❡ ❛❧❣✉♠p∈ I ❝♦♠ ❣r❛✉ n✱ ✐st♦ é✱ p=rxn_+r′_xn−1_{+❬t❡r♠♦s ❞❡ ❣r❛✉ ♠❡♥♦r❪✳}

❈❛♣ít✉❧♦ ✶✳ ❊①t❡♥sõ❡s ❞❡ ❖r❡ ✶✳✸✳ ❆♥❡✐s ❞❡ ♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛✉r❡♥t

❖❜s❡r✈❡♠♦s q✉❡xp−px∈I ❡ q✉❡ xp−px=δ(r)xn_{+❬t❡r♠♦s ❞❡ ❣r❛✉ ♠❡♥♦r❪)✳}

P♦rt❛♥t♦✱ δ(r)∈J✱ ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ J é ✉♠ δ✲✐❞❡❛❧✳

❈♦♠♦Réδ✲s✐♠♣❧❡s ❡J 6= 0✱ ♥❡❝❡ss❛r✐❛♠❡♥t❡J =R✳ P♦rt❛♥t♦✱I ❝♦♥té♠ ✉♠ ❡❧❡♠❡♥t♦ q ❝♦♠ ❣r❛✉ n ❡ ❝♦❡✜❝✐❡♥t❡ ❧í❞❡r 1✳ ❙❡ n = 0✱ ❡♥tã♦ q = 1 ❡ I = S✳ ▼♦str❛r❡♠♦s q✉❡ s❡

❛ss✉♠✐r♠♦sn > 0✱ t❡r❡♠♦s ✉♠❛ ❝♦♥tr❛❞✐çã♦✳ ❈♦♠ ❡❢❡✐t♦✱

❊s❝r❡✈❛♠♦s q = xn_+axn−1_{+❬t❡r♠♦s ❞❡ ❣r❛✉ ♠❡♥♦r❪✱ ♣❛r❛ ❛❧❣✉♠ a ∈ R✳ P♦r ♦✉tr♦}

❧❛❞♦✱ ♣❛r❛ ❝❛❞❛r ∈R✱ ♦❜s❡r✈❡♠♦s q✉❡ rq−qr ∈I ❡ q✉❡

rq − qr = (rxn _{+ rax}n−1_{) + [t❡r♠♦s ❞❡ ❣r❛✉ ♠❡♥♦r] − (rx}n _{+ nδ(r)x}n−1 _+arxn−1_{) +}

[t❡r♠♦s ❞❡ ❣r❛✉ ♠❡♥♦r]) = (ra−nδ(r)−ar)xn−1_{+❬t❡r♠♦s ❞❡ ❣r❛✉ ♠❡♥♦r❪✳}

P❡❧❛ ♠✐♥✐♠❛❧✐❞❛❞❡ ❞❡n✱ ❞❡✈❡♠♦s t❡r rq−qr = 0✱ ❡ ♣♦rt❛♥t♦ ra−nδ(r)−ar = 0✳ ❈♦♠♦

n >0 ❡ R é ✉♠❛ Q✲á❧❣❡❜r❛✱ ♦❜t❡♠♦s δ(r) = (−a/n)r−r(−a/n)♣❛r❛ t♦❞♦ r ∈R✱ ♦ q✉❡

❝♦♥tr❛❞✐③ ♦ ❢❛t♦ ❞❡ q✉❡δ é ❡①t❡r✐♦r✳

❆ss✐♠✱ n = 0 ❡ I =S✳ P♦rt❛♥t♦✱ S é ✉♠ ❛♥❡❧ s✐♠♣❧❡s✳

◆♦ ❝❛♣ít✉❧♦ 2 ✉s❛r❡♠♦s ❡st❡ r❡s✉❧t❛❞♦ ♣❛r❛ ♣r♦✈❛r q✉❡ ❛s á❧❣❡❜r❛s ❞❡ ❲❡②❧ s♦❜r❡ ✉♠

❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ③❡r♦ sã♦ ❛♥❡✐s s✐♠♣❧❡s✳

✶✳✸ ❆♥❡✐s ❞❡ ♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛✉r❡♥t

❉❛❞♦ ✉♠ ❛♥❡❧R ❡ ✉♠ ❛✉t♦♠♦r✜s♠♦α❞❡R❝♦♥str✉✐r❡♠♦s ✉♠ ❛♥❡❧ q✉❡ t❡♠ ❝♦♠♦ ❡❧❡✲

♠❡♥t♦s ✧♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛✉r❡♥t✧♥✉♠❛ ✐♥❞❡t❡r♠✐♥❛❞❛ x❝♦♠ ❝♦❡✜❝✐❡♥t❡s à ✧❡sq✉❡r❞❛✧❡♠ R✱ ✐st♦ é ♦♥❞❡ ❛ ✐♥❞❡t❡r♠✐♥❛❞❛ x é ✐♥✈❡rtí✈❡❧✳ ▼❛✐s ♣r❡❝✐s❛♠❡♥t❡ t❡♠♦s ♦ s❡❣✉✐♥t❡✿

❚❡♦r❡♠❛ ✶✳✷✾ ❙❡❥❛ α ✉♠ ❛✉t♦♠♦r✜s♠♦ ❞♦ ❛♥❡❧ R✳ ❊♥tã♦ ❡①✐st❡ ✉♠ ❛♥❡❧ T ❝♦♠ ❛s

s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

(a) R é ✉♠ s✉❜❛♥❡❧ ❞❡ T✳

(b) ❊①✐st❡ ✉♠ ❡❧❡♠❡♥t♦ ✐♥✈❡rtí✈❡❧ x ∈ T t❛❧ q✉❡ T é ✉♠ R✲♠ó❞✉❧♦ ❧✐✈r❡ à ❡sq✉❡r❞❛ ❝♦♠

❜❛s❡ {1, x, x−1_{, x}2_{, x}−2_{, ...}✳}

(c) xr=α(r)x✱ ♣❛r❛ t♦❞♦ r∈R✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡♠♦s ♦ s❡❣✉✐♥t❡ ❝♦♥❥✉♥t♦✿

T :=

(

a= (ai)∈

Y

i∈Z

R/ ❡①✐st❡ n =n(a)∈N t❛❧ q✉❡ ai = 0 ♣❛r❛ t♦❞♦ i∈Z ❝♦♠ |i|> n(a)

)