❇♦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 ❆ ❜♦✉♥❞❡❞ ✢❛❣ ✐♥ C n + ✶ ✐s U = { U ✶ ⊂ U ✷ ⊂ · · · ⊂ U n + ✶ = C n + ✶ , ❞✐♠ U i = i } s✉❝❤ t❤❛t U k ⊃ C k − ✶ = � ❡ ✶ , . . . , ❡ k − ✶ � ✱ k = ✷ , . . . , n ✳ ❉❡♥♦t❡ ❜② ❇❋ n t❤❡ s❡t ♦❢ ❛❧❧ ❜♦✉♥❞❡❞ ✢❛❣s ✐♥ C n + ✶ ✳ ❚❤❡♦r❡♠ ❇❋ n ✐s ❛ s♠♦♦t❤ ❝♦♠♣❛❝t t♦r✐❝ ✈❛r✐❡t② ✉♥❞❡r t❤❡ ❛❝t✐♦♥ ♦❢ t❤❡ t♦r✉s ( C × ) n ( C × ) n × ❇❋ n → ❇❋ n ( t ✶ , . . . , t n ) · ( w ✶ , . . . , w n , w n + ✶ ) = ( t ✶ w ✶ , . . . , t n w n , w n + ✶ ) , ❇❋ n ❜♦✉♥❞❡❞ ✢❛❣ ♠❛♥✐❢♦❧❞✳ ❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✷ ✴ ✶✻
ξ n t❛✉t♦❧♦❣✐❝❛❧ ❧✐♥❡ ❜✉♥❞❧❡ ♦✈❡r ❇❋ n ✱ ✇❤♦s❡ ✜❜r❡ ♦✈❡r U ✐s U ✶ ∼ = C ✳ Pr♦♣♦s✐t✐♦♥ ❇❋ n = C P ( C ⊕ ξ n − ✶ ) ✱ ✇❤❡r❡ ξ n − ✶ ✐s ♦✈❡r ❇❋ n − ✶ ✳ Pr♦♦❢✳ ❈♦♥s✐❞❡r ❇❋ n → ❇❋ n − ✶ U �→ U ′ = U / C ✶ ✐♥ C ✷ ,..., n + ✶ = C n , ✇❤❡r❡ U ′ = { U ′ ✶ ⊂ U ′ ✷ ⊂ · · · ⊂ U ′ n − ✶ } ✱ U ′ k = U k + ✶ / C ✶ ✳ ❚♦ r❡❝♦✈❡r U ❢r♦♠ U ′ ♦♥❡ ❤❛s t♦ ❝❤♦♦s❡ ❛ ❧✐♥❡ U ✶ ✐♥ U ✷ = C ✶ ⊕ U ′ ✶ ✳ ●❡t ❛ s❡q✉❡♥❝❡ ♦❢ ✜❜r❡ ❜✉♥❞❧❡s ✇✐t❤ ✜❜r❡ C P ✶ ❇❋ n → ❇❋ n − ✶ → · · · → ❇❋ ✶ = C P ✶ → ♣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 B n → B n − ✶ → · · · → B ✶ = C P ✶ → ♣t ✐♥ ✇❤✐❝❤ B k = C P ( C ⊕ η k − ✶ ) ❢♦r ❛ ❧✐♥❡ ❜✉♥❞❧❡ η k − ✶ ♦✈❡r B k − ✶ ✳ ❚❤❡♦r❡♠ �� � H ∗ ( B n ) ∼ = H ∗ ( B n − ✶ )[ u n ] u ✷ n = c ✶ ( η n − ✶ ) u n , ✇❤❡r❡ u n = c ✶ ( ξ n ) ❛♥❞ ξ n ✐s t❤❡ t❛✉t♦❧♦❣✐❝❛❧ ❧✐♥❡ ❜✉♥❞❧❡ ♦✈❡r B n ✳ ❊①❛♠♣❧❡ ❲❤❡♥ η k = ξ k ❢♦r ❡❛❝❤ k ✱ ✇❡ ❣❡t B n = ❇❋ n ✱ t❤❡ ❜♦✉♥❞❡❞ ✢❛❣ ♠❛♥✐❢♦❧❞ ✇✐t❤ t❤❡ ❵✐♥tr✐♥s✐❝✬ str✉❝t✉r❡ ♦❢ ❛ ❇♦tt t♦✇❡r✳ ❲❡ ❤❛✈❡ � H ∗ ( ❇❋ n ) ∼ = H ∗ ( ❇❋ n − ✶ )[ u n ] ( u ✷ n = u n − ✶ u n ) . ❚❛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✐♥❣ Ω U = { st❛❜❧② ❝♦♠♣❧❡① ♠❛♥✐❢♦❧❞s } / ❝♦♠♣❧❡① ❜♦r❞✐s♠ r❡❧❛t✐♦♥ ❚❤❡♦r❡♠ ✭▼✐❧♥♦r✱ ◆♦✈✐❦♦✈✬✶✾✻✵✮ Ω U ∼ = Z [ a ✶ , a ✷ , . . . ] , ❞✐♠ a i = ✷ i . ❆ st❛❜❧② ❝♦♠♣❧❡① ♠❛♥✐❢♦❧❞ M ✷ n ❝❛♥ ❜❡ t❛❦❡♥ ❛s ❛ r❡♣r❡s❡♥t❛t✐✈❡ ♦❢ a n ✐✛ � n � = p k − ✶ , ± ✶ , s n [ M ✷ n ] = n = p k − ✶ . ± p , ❍❡r❡ s n ✐s t❤❡ ❝❤❛r❛❝t❡r✐st✐❝ ❝❧❛ss ❝♦rr❡s♣♦♥❞✐♥❣ t♦ t❤❡ s②♠♠❡tr✐❝ ♣♦❧②♥♦♠✐❛❧ x n ✶ + · · · + x n n ✱ ✇❤❡r❡ c n ( T M ✷ n ) = ( ✶ + x ✶ ) · · · ( ✶ + x n ) ✳ ❊✳❣✳✱ s n [ C P n ] = n + ✶✱ s♦ [ C P ✶ ] = a ✶ ✱ C P ✷ = [ a ✷ ] ✱ ❜✉t C P ✸ � = [ a ✸ ] ✳ ❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✺ ✴ ✶✻
●✐✈❡♥ i � j ✱ ❝♦♥s✐❞❡r C i + ✶ ⊂ C j + ✶ ❛♥❞ ❞❡✜♥❡ t❤❡ ▼✐❧♥♦r ❤②♣❡rs✉r❢❛❝❡ � � ℓ ✐s ❛ ❧✐♥❡ ✐♥ C i + ✶ , W ✐s ❛ ❤②♣❡r♣❧❛♥❡ ✐♥ C j + ✶ , ℓ ⊂ W H ij = ( ℓ, W ): . ■t ✐s ❣✐✈❡♥ ❜② t❤❡ ❡q✉❛t✐♦♥ z ✵ w ✵ + · · · + z i w i = ✵ ✐♥ C P i × C P j ⊂ C P ij + i + j ✳ ❊✳❣✳✱ H ✷✷ = ❋❧ ( C ✸ ) ✱ ❝♦♠♣❧❡t❡ ✢❛❣s ✐♥ C ✸ ✳ Pr♦♣♦s✐t✐♦♥ � i + j � s i + j − ✶ [ H ij ] = . i ❚❤❡r❡❢♦r❡✱ { [ H ij ] , ✵ � i � j } ❣❡♥❡r❛t❡ t❤❡ ❝♦♠♣❧❡① ❜♦r❞✐s♠ r✐♥❣ Ω U ✳ ❍♦✇❡✈❡r✱ H ij ✐s ♥♦t ❛ t♦r✐❝ ♠❛♥✐❢♦❧❞ ✇❤❡♥ i � ✷✳ ❚❛r❛s P❛♥♦✈ ✭▼❙❯✮ ❇♦tt ❚♦✇❡rs ❛♥❞ ❊q✉✐✈❛r✐❛♥t ❈♦❜♦r❞✐s♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✻ ✴ ✶✻
❚❤❡♦r❡♠ ✭❇✉❝❤st❛❜❡r✕❘❛②✮ ❚❤❡ ❝♦♠♣❧❡① ❜♦r❞✐s♠ r✐♥❣ Ω U ❤❛s ❛ ❣❡♥❡r❛t♦r s❡t ❝♦♥s✐st✐♥❣ ♦❢ t♦r✐❝ ♠❛♥✐❢♦❧❞s✳ Pr♦♦❢✳ ❈♦♥s✐❞❡r t❤❡ ♠❛♥✐❢♦❧❞s � U ✐s ❛ ❜♦✉♥❞❡❞ ✢❛❣ ✐♥ C i + ✶ , B ij = ( U , W ): � W ✐s ❛ ❤②♣❡r♣❧❛♥❡ ✐♥ C j + ✶ , U ✶ ⊂ W . B ij → H ij ( U , W ) �→ ( U ✶ , W ) ↓ ↓ ↓ ↓ C P i ❇❋ i → U �→ U ✶ B ij = C P ( 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 a i ♦❢ t❤❡ ❝♦♠♣❧❡① ❝♦❜♦r❞✐s♠ r✐♥❣ Ω U ❝❛♥ ❜❡ 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♠ ✷✹✕✷✼ ❙❡♣ ✷✵✶✹ ✽ ✴ ✶✻
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