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❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠

❜❛s❡❞ ♦♥ ❥♦✐♥t ✇♦r❦s ✇✐t❤ ❱✐❝t♦r ❇✉❝❤st❛❜❡r ❛♥❞ ◆✐❣❡❧ ❘❛② ❚❛r❛s P❛♥♦✈

▲♦♠♦♥♦s♦✈ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t②

• ❡♦♠❡tr② ❉❛②s ✐♥ ◆♦✈♦s✐❜✐rs❦✖✷✵✶✹

■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ❞❡❞✐❝❛t❡❞ t♦ t❤❡ ✽✺t❤ ❆♥♥✐✈❡rs❛r② ♦❢ ❨✉r✐ ●r✐❣♦r✐❡✈✐❝❤ ❘❡s❤❡t♥②❛❦ ✷✹✕✷✼ ❙❡♣t❡♠❜❡r ✷✵✶✹

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✶ ✴ ✶✻

✶✳ ❇♦✉♥❞❡❞ ✢❛❣ ♠❛♥✐❢♦❧❞s ❛♥❞ ❇♦tt t♦✇❡rs

❆ ❜♦✉♥❞❡❞ ✢❛❣ ✐♥ Cn+✶ ✐s U = {U✶ ⊂ U✷ ⊂ · · · ⊂ Un+✶ = Cn+✶, ❞✐♠ Ui = i} s✉❝❤ t❤❛t Uk ⊃ Ck−✶ = ❡✶, . . . , ❡k−✶✱ k = ✷, . . . , n✳ ❉❡♥♦t❡ ❜② ❇❋ n t❤❡ s❡t ♦❢ ❛❧❧ ❜♦✉♥❞❡❞ ✢❛❣s ✐♥ Cn+✶✳

❚❤❡♦r❡♠

❇❋ n ✐s ❛ s♠♦♦t❤ ❝♦♠♣❛❝t t♦r✐❝ ✈❛r✐❡t② ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ t♦r✉s (C×)n (C×)n × ❇❋ n → ❇❋ n (t✶, . . . , tn) · (w✶, . . . , wn, wn+✶) = (t✶w✶, . . . , tnwn, wn+✶), ❇❋ n ❜♦✉♥❞❡❞ ✢❛❣ ♠❛♥✐❢♦❧❞✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✷ ✴ ✶✻

ξn t❛✉t♦❧♦❣✐❝❛❧ ❧✐♥❡ ❜✉♥❞❧❡ ♦✈❡r ❇❋ n✱ ✇❤♦s❡ ✜❜r❡ ♦✈❡r U ✐s U✶ ∼ = C✳

Pr♦♣♦s✐t✐♦♥

❇❋ n = CP(C ⊕ ξn−✶)✱ ✇❤❡r❡ ξn−✶ ✐s ♦✈❡r ❇❋ n−✶✳

Pr♦♦❢✳

❈♦♥s✐❞❡r ❇❋ n → ❇❋ n−✶ U → U′ = U/C✶ ✐♥ C✷,...,n+✶ = Cn, ✇❤❡r❡ U′ = {U′

✶ ⊂ U′ ✷ ⊂ · · · ⊂ U′ n−✶}✱ U′ k = Uk+✶/C✶✳

❚♦ r❡❝♦✈❡r U ❢r♦♠ U′ ♦♥❡ ❤❛s t♦ ❝❤♦♦s❡ ❛ ❧✐♥❡ U✶ ✐♥ U✷ = C✶ ⊕ U′

✶✳

• ❡t ❛ s❡q✉❡♥❝❡ ♦❢ ✜❜r❡ ❜✉♥❞❧❡s ✇✐t❤ ✜❜r❡ CP✶

❇❋ n → ❇❋ n−✶ → · · · → ❇❋ ✶ = CP✶ → ♣t ❛ ❇♦tt t♦✇❡r str✉❝t✉r❡ ♦♥ ❇❋ n✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✸ ✴ ✶✻

❆ ❇♦tt t♦✇❡r ✐s ❛ s❡q✉❡♥❝❡ ♦❢ ✜❜r❡ ❜✉♥❞❧❡s Bn → Bn−✶ → · · · → B✶ = CP✶ → ♣t ✐♥ ✇❤✐❝❤ Bk = CP(C ⊕ ηk−✶) ❢♦r ❛ ❧✐♥❡ ❜✉♥❞❧❡ ηk−✶ ♦✈❡r Bk−✶✳

❚❤❡♦r❡♠

H∗(Bn) ∼ = H∗(Bn−✶)[un]

• u✷

n = c✶(ηn−✶)un

• ,

✇❤❡r❡ un = c✶(ξn) ❛♥❞ ξn ✐s t❤❡ t❛✉t♦❧♦❣✐❝❛❧ ❧✐♥❡ ❜✉♥❞❧❡ ♦✈❡r Bn✳

❊①❛♠♣❧❡

❲❤❡♥ ηk = ξk ❢♦r ❡❛❝❤ k✱ ✇❡ ❣❡t Bn = ❇❋ n✱ t❤❡ ❜♦✉♥❞❡❞ ✢❛❣ ♠❛♥✐❢♦❧❞ ✇✐t❤ t❤❡ ❵✐♥tr✐♥s✐❝✬ str✉❝t✉r❡ ♦❢ ❛ ❇♦tt t♦✇❡r✳ ❲❡ ❤❛✈❡ H∗(❇❋ n) ∼ = H∗(❇❋ n−✶)[un]

• (u✷

n = un−✶un).

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✹ ✴ ✶✻

✷✳ ❘❡♣r❡s❡♥t✐♥❣ ❝♦♠♣❧❡① ❜♦r❞✐s♠ ❝❧❛ss❡s

❆s ❛ ❝♦♠♣❧❡① ♠❛♥✐❢♦❧❞ ❇❋ n✱ r❡♣r❡s❡♥ts ❛ ✷n✲❞✐♠❡♥s✐♦♥❛❧ ❝❧❛ss ✐♥ t❤❡ ❝♦♠♣❧❡① ❜♦r❞✐s♠ r✐♥❣ ΩU = {st❛❜❧② ❝♦♠♣❧❡① ♠❛♥✐❢♦❧❞s}/❝♦♠♣❧❡① ❜♦r❞✐s♠ r❡❧❛t✐♦♥

❚❤❡♦r❡♠ ✭▼✐❧♥♦r✱ ◆♦✈✐❦♦✈✬✶✾✻✵✮

ΩU ∼ = Z[a✶, a✷, . . .], ❞✐♠ ai = ✷i. ❆ st❛❜❧② ❝♦♠♣❧❡① ♠❛♥✐❢♦❧❞ M✷n ❝❛♥ ❜❡ t❛❦❡♥ ❛s ❛ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ an ✐✛ sn[M✷n] =

• ±✶,

n = pk − ✶, ±p, n = pk − ✶. ❍❡r❡ sn ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❝❧❛ss ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s②♠♠❡tr✐❝ ♣♦❧②♥♦♠✐❛❧ xn

✶ + · · · + xn n✱ ✇❤❡r❡ cn(T M✷n) = (✶ + x✶) · · · (✶ + xn)✳

❊✳❣✳✱ sn[CPn] = n + ✶✱ s♦ [CP✶] = a✶✱ CP✷ = [a✷]✱ ❜✉t CP✸ = [a✸]✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✺ ✴ ✶✻

• ✐✈❡♥ i j✱ ❝♦♥s✐❞❡r Ci+✶ ⊂ Cj+✶ ❛♥❞ ❞❡✜♥❡ t❤❡ ▼✐❧♥♦r ❤②♣❡rs✉r❢❛❝❡

Hij =

• (ℓ, W ):

ℓ ✐s ❛ ❧✐♥❡ ✐♥ Ci+✶, W ✐s ❛ ❤②♣❡r♣❧❛♥❡ ✐♥ Cj+✶, ℓ ⊂ W

• .

■t ✐s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥ z✵w✵ + · · · + ziwi = ✵ ✐♥ CPi × CPj ⊂ CPij+i+j✳ ❊✳❣✳✱ H✷✷ = ❋❧ (C✸)✱ ❝♦♠♣❧❡t❡ ✢❛❣s ✐♥ C✸✳

Pr♦♣♦s✐t✐♦♥

si+j−✶[Hij] = i + j i

• .

❚❤❡r❡❢♦r❡✱ {[Hij], ✵ i j} ❣❡♥❡r❛t❡ t❤❡ ❝♦♠♣❧❡① ❜♦r❞✐s♠ r✐♥❣ ΩU✳ ❍♦✇❡✈❡r✱ Hij ✐s ♥♦t ❛ t♦r✐❝ ♠❛♥✐❢♦❧❞ ✇❤❡♥ i ✷✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✻ ✴ ✶✻

❚❤❡♦r❡♠ ✭❇✉❝❤st❛❜❡r✕❘❛②✮

❚❤❡ ❝♦♠♣❧❡① ❜♦r❞✐s♠ r✐♥❣ ΩU ❤❛s ❛ ❣❡♥❡r❛t♦r s❡t ❝♦♥s✐st✐♥❣ ♦❢ t♦r✐❝ ♠❛♥✐❢♦❧❞s✳

Pr♦♦❢✳

❈♦♥s✐❞❡r t❤❡ ♠❛♥✐❢♦❧❞s Bij =

• (U, W ):

U ✐s ❛ ❜♦✉♥❞❡❞ ✢❛❣ ✐♥ Ci+✶, W ✐s ❛ ❤②♣❡r♣❧❛♥❡ ✐♥ Cj+✶, U✶ ⊂ W

• .

Bij → Hij (U, W ) → (U✶, W ) ↓ ↓ ↓ ↓ ❇❋ i → CPi U → U✶ Bij = CP(s✉♠ ♦❢ ❧✐♥❡ ❜✉♥❞❧❡s)✱ s♦ ✐t ✐s t♦r✐❝✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✼ ✴ ✶✻

◗✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s ❣❡♥❡r❛❧✐s❡ t♦r✐❝ ♠❛♥✐❢♦❧❞s t♦♣♦❧♦❣✐❝❛❧❧②✳ ❆ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞ M✷n ❤❛s ❛♥ ❛❝t✐♦♥ ♦❢ ❛ t♦r✉s T n ✇✐t❤ q✉♦t✐❡♥t ❛ s✐♠♣❧❡ ♣♦❧②t♦♣❡ P✳ ◗✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s ❤❛✈❡ ❝❛♥♦♥✐❝❛❧ T n✲✐♥✈❛r✐❛♥t st❛❜❧② ❝♦♠♣❧❡① str✉❝t✉r❡s✱ ❜✉t ❛r❡ ♥♦t ❝♦♠♣❧❡① ♦r ❛❧♠♦st ❝♦♠♣❧❡① ✐♥ ❣❡♥❡r❛❧✳

❚❤❡♦r❡♠ ✭❇✉❝❤st❛❜❡r✕P✕❘❛②✮

■♥ ❞✐♠❡♥s✐♦♥s > ✷✱ ❡✈❡r② ❝♦♠♣❧❡① ❜♦r❞✐s♠ ❝❧❛ss ❝♦♥t❛✐♥s ❛ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞✳ ■t r❡♠❛✐♥s ♦♣❡♥ ✇❤❡t❤❡r r✐♥❣ ❣❡♥❡r❛t♦rs ai ♦❢ t❤❡ ❝♦♠♣❧❡① ❝♦❜♦r❞✐s♠ r✐♥❣ ΩU ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② t♦r✐❝ ♦r q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s✳ ❆ ♣❛rt✐❛❧ r❡s✉❧t ♦♥ t❤✐s ♣r♦❜❧❡♠ ❤❛s ❜❡❡♥ r❡❝❡♥t❧② ♦❜t❛✐♥❡❞ ❜② ❆✳ ❲✐❧❢♦♥❣✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✽ ✴ ✶✻

✸✳ ❚❤❡ ✉♥✐✈❡rs❛❧ t♦r✐❝ ❣❡♥✉s

• ✐✈❡♥ ❛ T k✲♠❛♥✐❢♦❧❞✱ ♦♥❡ ❤❛s ❛ ✉♥✐✈❡rs❛❧ tr❛♥s❢♦r♠❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡

t❤r❡❡ ✈❡rs✐♦♥ ♦❢ ❡q✉✐✈❛r✐❛♥t ❝♦❜♦r❞✐s♠✿ U∗

T k(X)

→ MU∗

T k(X)

→ U∗(ET k ×T k X) ❣❡♦♠❡tr✐❝ ❤♦♠♦t♦♣✐❝ ❇♦r❡❧ ❋♦r X = ♣t ♦♥❡ ❣❡ts ❛ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ ΩU✲♠♦❞✉❧❡s Φ: ΩU:T k → U∗(BT k) = ΩU[[u✶, . . . , uk]] ❝❛❧❧❡❞ t❤❡ ✉♥✐✈❡rs❛❧ t♦r✐❝ ❣❡♥✉s✳ ■t ❛ss✐❣♥s t♦ t❤❡ ❡q✉✐✈❛r✐❛♥t ❝♦❜♦r❞✐s♠ ❝❧❛ss ♦❢ ❛ T k✲♠❛♥✐❢♦❧❞ M t❤❡ ❵❝♦❜♦r❞✐s♠ ❝❧❛ss✬ ♦❢ t❤❡ ♠❛♣ ET k ×T k M → BT k✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✾ ✴ ✶✻

❲❡ ❤❛✈❡ Φ(M) = [M] +

• ω : |ω|>✵

gω(M) uω, ✐♥ U∗(BT k) = ΩU[[u✶, . . . , uk]]✱ ✇❤❡r❡ [M] ∈ ΩU✱ uω = uω✶

✶ · · · uωk k ✳

❲❤❛t ❛r❡ t❤❡ ❝♦❡✣❝✐❡♥ts gω(M)❄

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✶✵ ✴ ✶✻

❚❤❡ ❜♦✉♥❞❡❞ ✢❛❣ ♠❛♥✐❢♦❧❞ ❇❋ n ✐s t❤❡ q✉♦t✐❡♥t ♦❢ (S✸)n = {(z✶, . . . , z✷n) ∈ C✷n : |zk|✷ + |zk+n|✷ = ✶, ✶ k n} ❜② t❤❡ T n✲❛❝t✐♦♥ ❣✐✈❡♥ ❜② (z✶, . . . , z✷n) → (t✶z✶, t−✶

✶ t✷z✷, . . . , t−✶ n−✶tnzn, t✶zn+✶, t✷zn+✷, . . . , tnz✷n)

❚❤✐s ❣✐✈❡s t❤❡ st❛❜❧❡ s♣❧✐tt✐♥❣ T (❇❋ n) ⊕ Cn ∼ = ¯ ξ✶ ⊕ ξ✶¯ ξ✷ ⊕ · · · ⊕ ξn−✶¯ ξn ⊕ ¯ ξ✶ ⊕ ¯ ξ✷ ⊕ · · · ⊕ ¯ ξn ✇❤❡r❡ ξk ✐s t❤❡ t❛✉t♦❧♦❣✐❝❛❧ ❧✐♥❡ ❜✉♥❞❧❡ ♦✈❡r ❇❋ k ♣✉❧❧❡❞ ❜❛❝❦ t♦ ❇❋ n✳ ❊✳❣✳✱ ❢♦r n = ✶ ✇❡ ♦❜t❛✐♥ t❤❡ st❛♥❞❛r❞ ✐s♦♠♦r♣❤✐s♠ T CP✶ ⊕ C ∼ = ¯ ξ ⊕ ¯ ξ✱ ✇❤❡r❡ ξ = ξ✶ ✐s t❤❡ t❛✉t♦❧♦❣✐❝❛❧ ❧✐♥❡ ❜✉♥❞❧❡✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✶✶ ✴ ✶✻

◆♦✇ ✇❡ t✇✐st t❤❡ t♦r✉s ❛❝t✐♦♥ ♦♥ (S✸)n ❛s ❢♦❧❧♦✇s✿ (z✶, . . . , z✷n) → (t✶z✶, t−✶

✶ t✷z✷, . . . , t−✶ n−✶tnzn, t−✶ ✶ zn+✶, t−✶ ✷ zn+✷, . . . , t−✶ n z✷n).

❚❤✐s ❣✐✈❡s t❤❡ s♣❧✐tt✐♥❣ T (❇❋ n) ⊕ R✷n ∼ = ¯ ξ✶ ⊕ ξ✶¯ ξ✷ ⊕ · · · ⊕ ξn−✶¯ ξn ⊕ ξ✶ ⊕ ξ✷ ⊕ · · · ⊕ ξn, ❛♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝♦♠♣❧❡① ❜♦r❞✐s♠ ❝❧❛ss ✐s ③❡r♦ ✐♥ ΩU

✷n✱ ❛s ❛♥ ✐t❡r❛t❡❞

s♣❤❡r❡ ❜✉♥❞❧❡✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✶✷ ✴ ✶✻

❲❡ ❞❡♥♦t❡ ❜② βn ∈ Ω✷n(CP∞) t❤❡ ❜♦r❞✐s♠ ❝❧❛ss ♦❢ ❇❋ n

ξn

− → CP∞✳

❚❤❡♦r❡♠ ✭❘❛②✮

❚❤❡ ❜♦r❞✐s♠ ❝❧❛ss❡s {βn : n ✵} ❢♦r♠ ❛ ❜❛s✐s ♦❢ t❤❡ ❢r❡❡ ΩU✲♠♦❞✉❧❡ U∗(CP∞) ✇❤✐❝❤ ✐s ❞✉❛❧ t♦ t❤❡ ❜❛s✐s {uk : k ✵} ♦❢ t❤❡ ΩU✲♠♦❞✉❧❡ U∗(CP∞) = ΩU[[u]]✳ ❙✐♠✐❧❛r❧②✱ ❞❡✜♥❡ βω ∈ Ω✷|ω|(BT k) t❤❡ ❜♦r❞✐s♠ ❝❧❛ss ♦❢

• ❇❋ ω =

❇❋ ω✶ × · · · × ❇❋ ωn → BT k✳

• ✐✈❡♥ ❛ T k✲♠❛♥✐❢♦❧❞ M✱ ❞❡✜♥❡ t❤❡ ❜✉♥❞❧❡

Gω(M) = (S✸)ω ×T ω M − → ❇❋ ω = (S✸)ω/T ω.

❚❤❡♦r❡♠ ✭❇✉❝❤st❛❜❡r✲P✲❘❛②✮

❚❤❡ ♠❛♥✐❢♦❧❞ Gω(M) r❡♣r❡s❡♥ts t❤❡ ❝♦❡✣❝✐❡♥t gω(M) ✐♥ t❤❡ ❡①♣❛♥s✐♦♥ ♦❢ t❤❡ ✉♥✐✈❡rs❛❧ t♦r✐❝ ❣❡♥✉s✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✶✸ ✴ ✶✻

✹✳ ❘✐❣✐❞✐t② ❛♥❞ ✜❜r❡ ♠✉❧t✐♣❧✐❝❛t✐✈✐t②

❆ ❣❡♥✉s ✐s ❛ ❤♦♠♦♠♦r♣❤✐s♠ ϕ: ΩU → R ✇❤❡r❡ R ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ r✐♥❣ ✇✐t❤ ✉♥✐t ✭✉s✉❛❧❧② Z✮✳ ❇② t❤❡ ❍✐r③❡❜r✉❝❤ ❝♦rr❡s♣♦♥❞❡♥❝❡✱ ❛ ❣❡♥✉s ϕ ✐s ❞❡t❡r♠✐♥❡❞ ❜② ❛ s❡r✐❡s f (x) = x + · · · ∈ R ⊗ Q[[x]]. ◆❛♠❡❧②✱ ϕ(M) = n

• i=✶

xi f (xi), [M]

• ,

✇❤❡r❡ c(T M) = (✶ + x✶) · · · (✶ + xn)✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✶✹ ✴ ✶✻

• ✐✈❡♥ ❛ ❣❡♥✉s ϕ: ΩU → R✱ ❞❡✜♥❡ ✐ts ❡q✉✐✈❛r✐❛♥t ❡①t❡♥s✐♦♥

ϕT : ΩU:T k

Φ

− → ΩU[[u✶, . . . , uk]]

hϕ

− → R ⊗ Q[[x✶, . . . , xk]] ♠❛♣♣✐♥❣ [M] → ϕ(M) ❛♥❞ ui → f (xi)✳

❉❡✜♥✐t✐♦♥

❆ ❣❡♥✉s ϕ ✐s r✐❣✐❞ ♦♥ M ✐❢ ϕT = ϕ ✭❛ ❝♦♥st❛♥t✮✳ ❆ ❣❡♥✉s ϕ ✐s ✜❜r❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ✇✐t❤ r❡s♣❡❝t t♦ M ✐❢ ϕ(N) = ϕ(M)ϕ(B) ❢♦r ❛♥② ✜❜r❡ ❜✉♥❞❧❡ N → M → B ✇✐t❤ str✉❝t✉r❡ ❣r♦✉♣ G ♦❢ ♣♦s✐t✐✈❡ r❛♥❦✳

❚❤❡♦r❡♠ ✭❇✉❝❤st❛❜❡r✲P✲❘❛②✮

❆ ❣❡♥✉s ϕ ✐s r✐❣✐❞ ♦♥ M ✐✛ ✐t ✐s ✜❜r❡ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ✇✐t❤ r❡s♣❡❝t t♦ M✳

Pr♦♦❢✳

❯s❡ t❤❡ ❡①♣❛♥s✐♦♥ Φ(M) = [M] + · · · ✇✐t❤ ❝♦❡✣❝✐❡♥ts r❡♣r❡s❡♥t❡❞ ❜② Gω(M)✱ ❛ ❜✉♥❞❧❡ ♦✈❡r ♥✉❧❧✲❜♦r❞❛♥t ❜❛s❡ ❇❋ ω✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✶✺ ✴ ✶✻

❘❡❢❡r❡♥❝❡s

❱✐❝t♦r ▼✳ ❇✉❝❤st❛❜❡r ❛♥❞ ❚❛r❛s ❊✳ P❛♥♦✈✳ ❚♦r✐❝ ❚♦♣♦❧♦❣②✳ ❆ ❜♦♦❦ ♣r♦❥❡❝t❀ ❛r❳✐✈✿✶✷✶✵✳✷✸✻✽✳ ❱✐❝t♦r ▼✳ ❇✉❝❤st❛❜❡r✱ ❚❛r❛s ❊✳ P❛♥♦✈ ❛♥❞ ◆✐❣❡❧ ❘❛②✳ ❚♦r✐❝ ❣❡♥❡r❛✳ ■♥t❡r♥❛t✳ ▼❛t❤✳ ❘❡s✳ ◆♦t✐❝❡s ✷✵✶✵✱ ♥♦✳ ✶✻✱ ✸✷✵✼✕✸✷✻✷✳ ❱✐❝t♦r ▼✳ ❇✉❝❤st❛❜❡r✱ ❚❛r❛s ❊✳ P❛♥♦✈ ❛♥❞ ◆✐❣❡❧ ❘❛②✳ ❙♣❛❝❡s ♦❢ ♣♦❧②t♦♣❡s ❛♥❞ ❝♦❜♦r❞✐s♠ ♦❢ q✉❛s✐t♦r✐❝ ♠❛♥✐❢♦❧❞s✳ ▼♦s❝♦✇ ▼❛t❤✳ ❏✳ ✼ ✭✷✵✵✼✮✱ ♥♦✳ ✷✱ ✷✶✾✕✷✹✷✳

❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✶✻ ✴ ✶✻