❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❉✐♠❡♥s✐♦♥s ❏✐❛♥❝❤❛♦ ❲✉ ✭❥♦✐♥t ✇✐t❤ ■❧❛♥ ❍✐rs❤❜❡r❣✮ P❡♥♥ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❋✐❡❧❞s ■♥st✐t✉t❡✱ ❚♦r♦♥t♦✱ ❆✉❣✉st ✹✱ ✷✵✶✼ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✶ ✴ ✶✷
◆✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❛♥❞ t❤❡ ❊❧❧✐♦tt ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❣r❛♠ ❲✐♥t❡r ❛♥❞ ❩❛❝❤❛r✐❛s ❞❡✈❡❧♦♣❡❞ ❛ ❦✐♥❞ ♦❢ ❞✐♠❡♥s✐♦♥ t❤❡♦r② ❢♦r ✭♥✉❝❧❡❛r✮ C ∗ ✲❛❧❣❡❜r❛s✳ dim nuc : CStarAlg → Z ≥ 0 ∪ {∞} ✳ ❙♦♠❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s✿ X t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ⇒ dim nuc ( C 0 ( X )) = dim( X ) ✭❝♦✈❡r✐♥❣ ❞✐♠✳✮✳ X ❛ ♠❡tr✐❝ s♣❛❝❡ ⇒ dim nuc ( C ∗ u ( X )) ≤ asdim( X ) ✭❛s②♠♣t♦t✐❝ ❞✐♠✳✮✳ → ( ✜♥✳❞✐♠✳ C ∗ ✲❛❧❣ ) ✮✳ dim nuc ( A ) = 0 ⇐ ⇒ A ✐s ❆❋ ✭ = lim − A ❑✐r❝❤❜❡r❣ ❛❧❣❡❜r❛ ✭❡✳❣✳ O n ✮ = ⇒ dim nuc ( A ) = 1 ✳ ❋✐♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r t❛❦✐♥❣✿ ⊕ ✱ ⊗ ✱ q✉♦t✐❡♥ts✱ ❤❡r❡❞✐t❛r② s✉❜❛❧❣❡❜r❛s✱ ❞✐r❡❝t ❧✐♠✐ts✱ ❡①t❡♥s✐♦♥s✱ ❡t❝✳ ❚❤❡♦r❡♠ ✭●♦♥❣✲▲✐♥✲◆✐✉✱ ❊❧❧✐♦tt✲●♦♥❣✲▲✐♥✲◆✐✉✱ ❚✐❦✉✐s✐s✲❲❤✐t❡✲❲✐♥t❡r✱✳ ✳ ✳ ✱ ❑✐r❝❤❜❡r❣✲P❤✐❧❧✐♣s✱ ✳ ✳ ✳ ✮ ❚❤❡ ❝❧❛ss ♦❢ ✉♥✐t❛❧ s✐♠♣❧❡ s❡♣❛r❛❜❧❡ C ∗ ✲❛❧❣❡❜r❛s ✇✐t❤ ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✭❋❆❉✮ ❛♥❞ s❛t✐s❢②✐♥❣ ❯❈❚ ✐s ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✳ ❈r♦ss❡❞ ♣r♦❞✉❝ts ❛r❡ ❛ ♠❛❥♦r s♦✉r❝❡ ♦❢ ✐♥t❡r❡st✐♥❣ C ∗ ✲❛❧❣❡❜r❛s✳ ❲❡ ❛s❦✿ ◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s ❋❆❉ ♣❛ss t❤r♦✉❣❤ t❛❦✐♥❣ ❝r♦ss❡❞ ♣r♦❞✉❝ts❄ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ dim nuc ( A ) < ∞ ✫ G � A ✱ ✇❤❡♥ dim nuc ( A ⋊ G ) < ∞ ❄ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✷ ✴ ✶✷
➽✇❤❡♥❄ dim nuc ( A ) < ∞ = ⇒ dim nuc ( A ⋊ G ) < ∞ ❆ ♣r♦♠✐♥❡♥t ❝❛s❡ ✐s ✇❤❡♥ A = C ( X ) ❢♦r ♠❡tr✐❝ s♣❛❝❡ X ❛♥❞ G ✐s ♥♦♥❝♣t✳ ❚❤❡♦r❡♠ ✭❚♦♠s✲❲✐♥t❡r✱ ❍✐rs❤❜❡r❣✲❲✐♥t❡r✲❩❛❝❤❛r✐❛s✮ ■❢ Z � X ♠✐♥✐♠❛❧❧② ❛♥❞ dim( X ) < ∞ ✱ t❤❡♥ dim nuc ( C ( X ) ⋊ Z ) < ∞ ✳ ❍✐rs❤❜❡r❣✲❲✐♥t❡r✲❩❛❝❤❛r✐❛s ♣r♦✈✐❞❡❞ ❛ ♠♦r❡ ❝♦♥❝❡♣t✉❛❧ ❛♣♣r♦❛❝❤ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ✭♠♦r❡ ♦♥ t❤❛t ❧❛t❡r✮✳ ◆♦t❡✿ ■❢ X ✐s ✐♥✜♥✐t❡✱ ❛ ♠✐♥✐♠❛❧ Z ✲❛❝t✐♦♥ ✐s ❢r❡❡✳ ❚❤❡♦r❡♠ ✭❙③❛❜ó✮ ■❢ Z m � X ❢r❡❡❧② ❛♥❞ dim( X ) < ∞ ✱ t❤❡♥ dim nuc ( C ( X ) ⋊ Z m ) < ∞ ✳ ❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮ ■❢ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣ G � X ❢r❡❡❧② ❛♥❞ dim( X ) < ∞ ✱ t❤❡♥ dim nuc ( C ( X ) ⋊ G ) < ∞ ✳ ④❋✳❣✳ ✈✐r✳♥✐❧♣✳ ❣♣s⑥ ●r♦♠♦✈ = ④❢✳❣✳ ❣♣s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤ ⑥ ∋ ✜♥✐t❡ ❣♣s✱ �� 1 a c � � Z m ✱ t❤❡ ❞✐s❝r❡t❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ : a, b, c ∈ Z ✱ ❡t❝✳ 0 1 b 0 0 1 ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✸ ✴ ✶✷
❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮ r❡♣❡❛t❡❞ ❋✳❣✳ ✈✐r✳♥✐❧♣✳ G � X ❢r❡❡❧② ✫ dim( X ) < ∞ ⇒ dim nuc ( C ( X ) ⋊ G ) < ∞ ✳ ■♥❣r❡❞✐❡♥ts ✐♥ t❤❡ ♣r♦♦❢✿ ✶ ❚❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dim Rok ( α ) ✱ ❞❡✜♥❡❞ ❢♦r ❛ C ∗ ✲❞②♥❛♠✐❝❛❧ s②st❡♠ α : G � A ✱ ✇❤❡r❡ G ✐s ✜♥✐t❡ ✭❍✲❲✲❩✮✱ Z ✭❍✲❲✲❩✮✱ Z m ✭❙③❛❜ó✮✱ r❡s✐❞✉❛❧❧② ✜♥✐t❡ ✭❙✲❲✲❩✮✱ ❝♦♠♣❛❝t ✭❍✐rs❤❜❡r❣✲P❤✐❧❧✐♣s✱ ●❛r❞❡❧❧❛✮✱ R ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✮✱ ✳✳✳ ❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮ dim +1 nuc ( A ⋊ α,w G ) ≤ asdim +1 ( � G ) · dim +1 nuc ( A ) · dim +1 Rok ( α ) ✳ ✷ ❚❤❡ ♠❛r❦❡r ♣r♦♣❡rt② ✭❛♥❞ t❤❡ t♦♣♦❧♦❣✐❝❛❧ s♠❛❧❧ ❜♦✉♥❞❛r② ♣r♦♣❡rt②✮✱ st✉❞✐❡❞ ❜② ▲✐♥❞❡♥str❛✉ss✱ ●✉t♠❛♥✱ ❙③❛❜ó✱ ❛♥❞ ♦t❤❡rs✳ ❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮ α ❋✳❣✳ ✈✐r✳♥✐❧♣✳ G � X ❢r❡❡❧② ✫ dim( X ) < ∞ ⇒ dim Rok ( G � C ( X )) < ∞ ✳ ✸ ❇♦✉♥❞ asdim +1 ( � G ) ❢♦r ❢✳❣✳ ✈✐r✳♥✐❧♣ G ✭❙✲❲✲❩✱ ❉❡❧❛❜✐❡✲❚♦✐♥t♦♥✮✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✹ ✴ ✶✷
P❛r❛❧❧❡❧ ❛♣♣r♦❛❝❤❡s ❙✐♠✐❧❛r ❛♣♣r♦❛❝❤❡s ♠❛❦❡ ✉s❡ ♦❢ ♦t❤❡r ❞✐♠❡♥s✐♦♥s ❞❡✜♥❡❞ ❢♦r t♦♣♦❧♦❣✐❝❛❧ ❞②♥❛♠✐❝❛❧ s②st❡♠s✱ ❡✳❣✳✱ ❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ DAD( − ) ✭●✉❡♥t♥❡r✲❲✐❧❧❡tt✲❨✉✮✱ ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ dim am ( − ) ✭●✲❲✲❨✱ ❙✲❲✲❩✱ ❛❢t❡r ❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✮✱ ❛♥❞ ✭✜♥❡✮ t♦✇❡r ❞✐♠❡♥s✐♦♥ dim tow ( − ) ✭❑❡rr✮✳ ❚❤❡② ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞ t❤r♦✉❣❤ ✐♥t❡rt✇✐♥✐♥❣ ✐♥❡q✉❛❧✐t✐❡s s✉❝❤ ❛s✿ ❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮ dim +1 am ( α ) ≤ dim +1 Rok ( α ) ≤ dim +1 Rok ( α ) · asdim +1 ( � G ) ✳ ❘❡♠❛r❦❛❜❧②✱ t❤❡ ♦r✐❣✐♥❛❧ ♠♦t✐✈❛t✐♦♥s ❢♦r ✐♥tr♦❞✉❝✐♥❣ dim am ❛♥❞ DAD ✇❡r❡ t♦ ❢❛❝✐❧✐t❛t❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ K ✲t❤❡♦r② ❢♦r A ⋊ G ✱ ✐♥ ♦r❞❡r t♦ ♣r♦✈❡ K ✲t❤❡♦r❡t✐❝ ✐s♦♠♦r♣❤✐s♠ ❝♦♥❥❡❝t✉r❡s ✭t❤❡ ❇❛✉♠✲❈♦♥♥❡s ❝♦♥❥❡❝t✉r❡ ❛♥❞ t❤❡ ❋❛rr❡❧❧✲❏♦♥❡s ❝♦♥❥❡❝t✉r❡ ✮✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✺ ✴ ✶✷
❚❤❡ ❝❛s❡ ♦❢ ✢♦✇s ❲❤❡♥ G = R � X ❝♦♥t✐♥✉♦✉s❧②✱ ✇❡ ❛❧s♦ ❤❛✈❡ ❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✮ ■❢ R � X ❢r❡❡❧② ❛♥❞ dim( X ) < ∞ ✱ t❤❡♥ dim nuc ( C ( X ) ⋊ R ) < ∞ ✳ ■♥❣r❡❞✐❡♥ts ✐♥ t❤❡ ♣r♦♦❢✿ ✶ ❚❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dim Rok ( α ) ❞❡✜♥❡❞ ❢♦r ❛♥② C ∗ ✲✢♦✇ α : R � A ✳ ❚❤❡♦r❡♠ ✭❍✲❙✲❲✲❲✮ nuc ( A ) · dim +1 dim +1 nuc ( A ⋊ α R ) ≤ 2 · dim +1 Rok ( α ) ✳ ✷ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ✏❧♦♥❣ t❤✐♥ ❝♦✈❡rs✑ ♦♥ ✢♦✇ s♣❛❝❡s ✱❞✉❡ t♦ ❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤ ❛♥❞ ✐♠♣r♦✈❡❞ ❜② ❑❛s♣r♦✇s❦✐✲❘ü♣✐♥❣✳ ❚❤❡♦r❡♠ ✭❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✱ ❑❛s♣r♦✇s❦✐✲❘ü♣✐♥❣✱ ❍✲❙✲❲✲❲✮ R � X ❢r❡❡❧② ❛♥❞ dim( X ) < ∞ ⇒ dim Rok ( G � C ( X )) < ∞ ✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✻ ✴ ✶✷
Recommend
More recommend