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❘♦❦❤❧✐♥ ❉✐♠❡♥s✐♦♥ ❇❡②♦♥❞ ❘❡s✐❞✉❛❧❧② ❋✐♥✐t❡ ●r♦✉♣s

❏✐❛♥❝❤❛♦ ❲✉

P❡♥♥ ❙t❛t❡ ❯♥✐✈❡rs✐t②

❯❆❇✱ ❏✉❧② ✶✷✱ ✷✵✶✼

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✶ ✴ ✾

◆✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❛♥❞ t❤❡ ❊❧❧✐♦tt ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❣r❛♠

❲✐♥t❡r ❛♥❞ ❩❛❝❤❛r✐❛s ❞❡✈❡❧♦♣❡❞ ❛ ❦✐♥❞ ♦❢ ❞✐♠❡♥s✐♦♥ t❤❡♦r② ❢♦r ✭♥✉❝❧❡❛r✮ C∗✲❛❧❣❡❜r❛s✳ dimnuc : CStarAlg → Z≥0 ∪ {∞}✳ ❙♦♠❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s✿ X t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ⇒ dimnuc(C0(X)) = dim(X) ✭❝♦✈❡r✐♥❣ ❞✐♠✳✮✳ X ❛ ♠❡tr✐❝ s♣❛❝❡ ⇒ dimnuc(C∗

u(X)) ≤ asdim(X) ✭❛s②♠♣t♦t✐❝ ❞✐♠✳✮✳

dimnuc(A) = 0 ⇐ ⇒ A ✐s ❆❋ ✭= lim − →(✜♥✳❞✐♠✳ C∗✲❛❧❣)✮✳ A ❑✐r❝❤❜❡r❣ ❛❧❣❡❜r❛ ✭❡✳❣✳ On✮ = ⇒ dimnuc(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✳ ❙✐♥❝❡ ♠❛♥② ❢✉♥ ❡①❛♠♣❧❡s ❛r❡ ❝♦♥str✉❝t❡❞ ❛s ❝r♦ss❡❞ ♣r♦❞✉❝ts✱ ✇❡ ❛s❦✿

◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s ❋❆❉ ♣❛ss t❤r♦✉❣❤ t❛❦✐♥❣ ❝r♦ss❡❞ ♣r♦❞✉❝ts❄

▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ dimnuc(A) < ∞ ✫ G A✱ ✇❤❡♥ dimnuc(A ⋊ G) < ∞❄

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✷ ✴ ✾

dimnuc(A) < ∞

➽✇❤❡♥❄

= ⇒ dimnuc(A ⋊ G) < ∞

■♥ ❣❡♥❡r❛❧✱ t❤✐s ✐s ❞✐✣❝✉❧t✳ ❇✉t t❤❡r❡ ❛r❡ s♦♠❡ s✉❝❝❡ss❢✉❧ str❛t❡❣✐❡s✱ ❡✳❣✳✱ ✉s✐♥❣ ❞✐♠❡♥s✐♦♥s ❞❡✜♥❡❞ ❢♦r ❛❝t✐♦♥s✳ ❋✐rst ❝❛s❡✿ ■❢ A = C(X)✱ t❤❡r❡ ❛r❡ ❛ ♥✉♠❜❡r ♦❢ ❝❧♦s❡❧②✲r❡❧❛t❡❞ ❞✐♠❡♥s✐♦♥s t❤❛t ✇❡ ❝❛♥ ✉s❡✱ ❡✳❣✳✱ ❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ ✭●✉❡♥t♥❡r✲❲✐❧❧❡tt✲❨✉✮✱ ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ ✭●✲❲✲❨✮✱ ✭✜♥❡✮ t♦✇❡r ❞✐♠❡♥s✐♦♥ ✭❑❡rr✮✳ ❲❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t♦✇❡r ❞✐♠❡♥s✐♦♥ ✐♥ t❤✐s t❛❧❦✳ ▲❡t X ❜❡ ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ❛♥❞ α: G X ❛♥ ❛❝t✐♦♥✳

❉❡✜♥✐t✐♦♥ ✭❚♦✇❡r ❞✐♠❡♥s✐♦♥✮

❆ t♦✇❡r ✐s ❛ ♣❛✐r (B, S) ✇❤❡r❡ B ⊂ X ✐s ♦♣❡♥✱ S ⋐ G✱ ❛♥❞ {αg(B): g ∈ S} ✐s ❞✐s❥♦✐♥t✳ dimtow(α) ≤ d ✐✛ ∀F ⋐ G✱ ∃ t♦✇❡rs (Bi, Si) ❢♦r i ∈ I = I(0) ∪ . . . ∪ I(d)✱ s✉❝❤ t❤❛t

✶

∀l ∈ {0, . . . , d}✱ {αg(Bi): g ∈ Si, i ∈ I(l)} ✐s ❞✐s❥♦✐♥t❀

✷

∀x ∈ X✱ ∃ i ∈ I ❛♥❞ g ∈ Si s✉❝❤ t❤❛t Fg ⊂ Si ❛♥❞ x ∈ αg(Bi) ✭= ⇒ X =

g∈Si,i∈I αg(Bi)✮✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✸ ✴ ✾

❉❡✜♥✐t✐♦♥ ✭❚♦✇❡r ❞✐♠❡♥s✐♦♥✮

dimtow(α) ≤ d ✐✛ ∀F ⋐ G✱ ∃ t♦✇❡rs (Bi, Si) ❢♦r i ∈ I = I(0) ∪ . . . ∪ I(d)✱ s✉❝❤ t❤❛t

✶ ∀l ∈ {0, . . . , d}✱ {αg(Bi): g ∈ Si, i ∈ I(l)} ✐s ❞✐s❥♦✐♥t❀ ✷ ∀x ∈ X✱ ∃ i ∈ I ❛♥❞ g ∈ Si s✉❝❤ t❤❛t Fg ⊂ Si ❛♥❞ x ∈ αg(Bi)✳

❚❤❡♦r❡♠ ✭❑❡rr✮

dim+1

nuc(C(X) ⋊α G) ≤ dim+1(X) · dim+1 tow(α)✳

■❢ G ✐s ✜♥✐t❡✱ ✇❡ ♠❛② ❝❤♦♦s❡ F = G✳ ❚❤❡♥ ❲▲❖●✱ Si = G✱ ∀i✱ ❛♥❞ ❡❛❝❤ t♦✇❡r ✐s ❛ ❧♦❝❛❧ tr✐✈✐❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❝t✐♦♥✳ ❚❤❡ t♦✇❡r ❞✐♠❡♥s✐♦♥ r❡❝♦✈❡rs ❛ ♥♦t✐♦♥ ❝❛❧❧❡❞ t❤❡ G✲✐♥❞❡①✳ ❊①❛♠♣❧❡✿ ❚❤❡ ♦❞♦♠❡t❡r ❛❝t✐♦♥ α: Z Zn ❤❛s dimtow(α) = 1✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐❢ G ✐s r❡s✐❞✉❛❧❧② ✜♥✐t❡ ❛♥❞ lim − → G/Gi ✐s ✐ts ♣r♦✜♥✐t❡ ❝♦♠♣❧❡t✐♦♥ ❛ss♦❝✳ t♦ ❛ s❡♣❛r❛t✐♥❣ ❝❤❛✐♥ G1 > G2 > . . . ♦❢ ✜♥✐t❡✲✐♥❞❡① s✉❜❣r♦✉♣s✱ ✇❡ ❤❛✈❡ dimtow(G lim − → G/Gi) ≤ asdim({Gi}G)✱ t❤❡ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❜♦① s♣❛❝❡ ❛ss♦❝✳ t♦ {Gi}✳ dimtow(G βG) = asdim(G)✳ ❊✳❣✳✱ t❤✐s ✇♦r❦s ❢♦r t❤❡ ❢r❡❡ ❣r♦✉♣✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✹ ✴ ✾

❚❤❡♦r❡♠ ✭❑❡rr✮

dim+1

nuc(C(X) ⋊α G) ≤ dim+1(X) · dim+1 tow(α)✳

■♥❣r❡❞✐❡♥ts ✐♥ t❤❡ ♣r♦♦❢✿ ❆ t♦✇❡r (B, S) ❞❡t❡r♠✐♥❡s ❛ s✉❜❛❧❣❡❜r❛ ♦❢ C(X) ⋊ G✿ {ugfu∗

h : f ∈ C0(B), h, g ∈ S} ∼

= M|S|(C0(X))✳ ❚❤❡ ✏F✲♦✈❡r❧❛♣s✑ ♦❢ t❤❡ t♦✇❡rs ❤❡❧♣ ♣r♦❞✉❝❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② s✉❜❥❡❝t t♦ t❤❡ t♦✇❡rs ❝♦♥s✐st✐♥❣ ♦❢ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ❛r❡ ✏❛❧♠♦st ✢❛t ❛❧♦♥❣ ♦r❜✐ts✑✳ ❚❤✉s ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✈❡r✐❢② dimam(−) < ∞✳

❚❤❡♦r❡♠ ✭❙③❛❜♦✲❲✲❩❛❝❤❛r✐❛s✮

▲❡t G ❜❡ ❛ ❢✳❣✳ ♥✐❧♣♦t❡♥t ❣r♦✉♣ ❛♥❞ G

α

X ❛ ❢r❡❡ ❛❝t✐♦♥ ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ dim(X) < ∞✳ ❚❤❡♥ dimtow(α) < ∞✳

❈♦r♦❧❧❛r②

❋♦r s✉❝❤ ❛ (X, G, α)✱ t❤❡ ❝r♦ss❡❞ ♣r♦❞✉❝t C(X) ⋊α G ❤❛s ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❛♥❞ ✐s t❤✉s ✐♥ t❤❡ ❝❧❛ss ♦❢ s✐♠♣❧❡ s❡♣❛r❛❜❧❡ ✉♥✐t❛❧ ♥✉❝❧❡❛r C∗✲❛❧❣❡❜r❛s t❤❛t ❛r❡ ❝❧❛ss✐✜❛❜❧❡ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✳ ❲❤❛t ✐❢ A ✐s ♥♦♥❝♦♠♠✉t❛t✐✈❡❄ ❙❡❡ t❤❡ ♥❡①t s❧✐❞❡ − →

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✺ ✴ ✾

❋♦r C∗✲❞②♥❛♠✐❝❛❧ s②st❡♠s G A✱ ♦♥❡ str❛t❡❣② ✐s t❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥✳ ❚❤✐s ✇❛s ❞❡✜♥❡❞ ✜rst ❢♦r Z ❛♥❞ ❢♦r ✜♥✐t❡ ❣r♦✉♣s ❜② ❍✐rs❤❜❡r❣✱ ❲✐♥t❡r ❛♥❞ ❩❛❝❤❛r✐❛s✳ G✿ ❝♦✉♥t❛❜❧❡ ❞✐s❝r❡t❡ r❡s✐❞✉❛❧❧② ✜♥✐t❡ ❣r♦✉♣✱ ❛♥❞ α : G A✳ {Gi}∞

i=1✿ ❛ ց s❡q✉❡♥❝❡ ♦❢ ✜♥✐t❡✲✐♥❞❡① s✉❜❣r♦✉♣s s✳t✳ ∞ i=1 Gi = {1}✳

F∞(A) ✐s ❑✐r❝❤❜❡r❣✬s ❝❡♥tr❛❧ s❡q✉❡♥❝❡ ❛❧❣❡❜r❛✳ ❲❤❡♥ A ✐s ✉♥✐t❛❧✱ F∞(A) := (l∞(N, A)/C0(N, A)) ∩ {❝♦♥st❛♥t s❡q✉❡♥❝❡s}′✳ α α∞ : G F∞(A)✳ ❆ ❝✳♣✳ ♠❛♣ φ : A → B ✐s ❝❛❧❧❡❞ ♦r❞❡r✲③❡r♦ ✐✛ aa′ = 0 = ⇒ φ(a)φ(a′) = 0✱ ∀a, a′ ∈ A ✭❡✳❣✳ ∗✲❤♦♠♦♠♦r♣❤✐s♠s ❛r❡ ♦r❞❡r✲③❡r♦✮✳

❉❡✜♥✐t✐♦♥ ✭❙③❛❜♦✲❲✲❩❛❝❤❛r✐❛s✱ ❛❢t❡r ❍✐rs❤❜❡r❣✲❲✐♥t❡r✲❩❛❝❤❛r✐❛s✮

❚❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ♦❢ α ✇✳r✳t {Gi}✱ ✇r✐tt❡♥ ❛s dimRok(α, {Gi}) = d✱ ✐s t❤❡ s♠❛❧❧❡st d ∈ N s✳t✳ ❢♦r ❡✈❡r② Gi✱ ∃ ❡q✉✐✈❛r✐❛♥t ❝✳♣✳❝✳ ♦r❞❡r✲③❡r♦ ♠❛♣s φ(l) : (C(G/Gi), G✲s❤✐❢t) → (F∞(A), α∞) ❢♦r l = 0, · · · , d✱ s✳t✳ φ(0)(1) + · · · + φ(l)(1) = 1✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✻ ✴ ✾

G r✳❢✳ ❜♦① s♣❛❝❡ G✳ ❲r✐t✐♥❣ dim+1(−) := dim(−) + 1✱ ✇❡ ❤❛✈❡✿

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮

dim+1

nuc(A ⋊α,w G) ≤ asdim+1(G) · dim+1 nuc(A) · dim+1 Rok(α)

✳

❚❤❡♦r❡♠ ✭❉❡❧❛❜✐❡✲❚♦✐♥t♦♥✱ ❛❢t❡r ❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✱ ❋✐♥♥✲❙❡❧❧✕❲✮

■❢ G ✐s ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t✱ t❤❡♥ asdim(G) = asdim(G) (= Hirsch length(G))✳ ❋♦r r❡s✐❞✉❛❧❧② ✜♥✐t❡ G✱ t❤❡r❡ ❛r❡ ❝❧♦s❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ t✇♦ ♥♦t✐♦♥s✿

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮

dim+1

Rok(α) ≤ dimtow(α) ≤ asdim+1(G) · dim+1 Rok(α)

✳ ❉r❛✇❜❛❝❦✿ ❚❤❡ r❡q✉✐r❡♠❡♥t t❤❛t asdim(G) < ∞ ❧✐♠✐ts ✉s t♦ ❝❡rt❛✐♥ r❡s✐❞✉❛❧❧② ✜♥✐t❡ ❛♠❡♥❛❜❧❡ ❣r♦✉♣s✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✼ ✴ ✾

❚❤❡ t❤❡♦r❡♠ ♦❢ ❑❡rr ❣❡♥❡r❛❧✐③❡s t♦ G✲X✲C∗✲❛❧❣❡❜r❛s✳ ❊✳❣✳✱ ✇❤❡♥ B = A ⊗ C(X) ✇✐t❤ G ❛❝t✐♥❣ ❞✐❛❣♦♥❛❧❧② ❜② α ⊗ β✱ ✇❡ ❤❛✈❡ dim+1

nuc((A ⊗ C(X)) ⋊α⊗β G) ≤ dim+1 nuc(A) · dim+1(X) · dim+1 tow(β) .

❋♦r ❣❡♥❡r❛❧ ♥♦♥❝♦♠♠✉t❛t✐✈❡ A✱ ✇❡ ♠❛② ❞❡✜♥❡ ❛ r❡❧❛t✐✈❡ ♥♦t✐♦♥✿

❉❡✜♥✐t✐♦♥ ✭❍❛♠❜❧✐♥✲❲✲❩❛❝❤❛r✐❛s✱ ❛❢t❡r ❍✐rs❤❜❡r❣✲❲✐♥t❡r✲❩❛❝❤❛r✐❛s✮✿ dimRok(α|β)✱ t❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ♦❢ α: G A ✇✳r✳t β : G X

dimRok(α|β) ≤ d ✐✛ ∃ ❡q✉✐✈❛r✐❛♥t ❝✳♣✳ ❝♦♥tr❛❝t✐✈❡ ♦r❞❡r✲③❡r♦ ♠❛♣s φ(0), . . . , φ(d) : (C(X), β) → (F∞(A), α)✱ s✳t✳ d

l=0 φ(l)(1) = 1✳

F∞(A) ✐s ❑✐r❝❤❜❡r❣✬s ❝❡♥tr❛❧ s❡q✉❡♥❝❡ ❛❧❣❡❜r❛✳ ❲❤❡♥ A ✐s ✉♥✐t❛❧✱ F∞(A) := (l∞(N, A)/C0(N, A)) ∩ {❝♦♥st❛♥t s❡q✉❡♥❝❡s}′✳ ❖r❞❡r✲③❡r♦ ❂ ✏♦rt❤♦❣♦♥❛❧✐t② ♣r❡s❡r✈✐♥❣✑✳ ❘❡♠❛r❦✿ C(X) ♠❛② ❜❡ r❡♣❧❛❝❡❞ ❜② ♦t❤❡r ✉♥✐t❛❧ C∗✲❛❧❣❡❜r❛s✳

❚❤❡♦r❡♠ ✭❍❛♠❜❧✐♥✲❲✲❩❛❝❤❛r✐❛s✮

dim+1

nuc(A ⋊α,w G) ≤ dim+1 nuc(A) · dim+1 tow(β) · dim+1 Rok(α|β)

✳ ❚❤✐s ❣❡♥❡r❛❧✐③❡s t❤❡ t❤❡♦r② ❢♦r r❡s✐❞✉❛❧❧② ✜♥✐t❡ ❣r♦✉♣s G ✭❡✳❣✳✱ Z✮✿ ❣✐✈❡♥ ❛ s❡♣❛r❛t✐♥❣ ❝❤❛✐♥ ♦❢ ✜♥✐t❡✲✐♥❞❡① s✉❜❣r♦✉♣s {Gi}✱ ✇❡ t❛❦❡ X = lim ← −G/Gi✱ t❤❡ ♣r♦✜♥✐t❡ ❝♦♠♣❧❡t✐♦♥ = ⇒ dimam(β) ≤ asdim({Gi}G)✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✽ ✴ ✾

❋♦r r✳❢✳ ❣r♦✉♣s✱ t❛❦❡ X = lim ← −G/Gi = ⇒ dimam(β) ≤ asdim({Gi}G)✳ ❲❡ r❡❝♦✈❡r✿

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮

dim+1

nuc(A ⋊α,w G) ≤ dim+1 nuc(A) · dim+1 Rok(α) · asdim+1(G)

✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✾ ✴ ✾