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❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❚r✐❛♥❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ●✐✈❡♥ ❛ ❝♦♠♠✉t❛t✐✈❡ F ✲❛❧❣❡❜r❛ H ✱ ❛♥ ❛❧❣❡❜r❛ ❛✉t♦♠♦r♣❤✐s♠ θ : H → H ✱ ❛♥❞ ❡❧❡♠❡♥ts z 0 ∈ H ❛♥❞ z 1 ∈ H × ✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ tr✐❛♥❣✉❧❛r ●❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛ ✭●❲❆✮ ✐s W ( H, θ, z 0 , z 1 ) := H � d, u � / ( uh = θ ( h ) u, hd = dθ ( h ) , ud = z 0 + dz 1 ) . ●♦❛❧✿ ❙t✉❞② ❈❛t❡❣♦r② O ♦✈❡r ❛ tr✐❛♥❣✉❧❛r ●❲❆✳ ❯♥❞❡rst❛♥❞ t❤❡ str✉❝t✉r❡ ♦❢ ♣r♦❥❡❝t✐✈❡s ✐♥ ❛ ❜❧♦❝❦✱ ❛♥❞ t❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛✳ ✹ ✴ ✸✹

◗✉❛♥t✉♠ ❛❧❣❡❜r❛✿ ❏✐♥❣✲❩❤❛♥❣ st✉❞✐❡❞ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡✱ ♥♦♥✲❝♦❝♦♠♠✉t❛t✐✈❡ ❜✐❛❧❣❡❜r❛s t❤❛t ✲❞❡❢♦r♠ ✳ ❑❛❝✿ ✏❞✐s♣✐♥ ▲✐❡ s✉♣❡r❛❧❣❡❜r❛ ✑✳ ❈♦♠❜✐♥❛t♦r✐❝s✿ ❇❡♥❦❛rt✕❘♦❜② st✉❞✐❡❞ ✏❞♦✇♥✱ ✉♣ ♦♣❡r❛t♦rs✑ ♦♥ ♣♦s❡ts✿ ✭❣❡♥❡r❛❧✐③❡❞✮ ❞♦✇♥✲✉♣ ❛❧❣❡❜r❛s ✳ ❋♦r ❛❧❧ ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s✱ ❛♥❞ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❚r✐❛♥❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ✭❝♦♥t✳✮ ❚r✐❛♥❣✉❧❛r ●❲❆s ♦❝❝✉r ✐♥ ♠❛♥② s❡tt✐♥❣s✿ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✿ ❙♠✐t❤ st✉❞✐❡❞ ❞❡❢♦r♠❛t✐♦♥s ♦❢ sl 2 ✿ C � e, f, h � / ([ h, e ] = 2 e, [ h, f ] = − 2 f, [ e, f ] = z 0 ( h )) . ▼❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✿ ❲✐tt❡♥ ✐♥tr♦❞✉❝❡❞ ✼✲♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ ❞❡❢♦r♠❛t✐♦♥s ♦❢ U ( sl 2 ) ✳ ▲❡ ❇r✉②♥✿ ❈♦♥❢♦r♠❛❧ sl 2 ✲❛❧❣❡❜r❛s✳ ✺ ✴ ✸✹

❈♦♠❜✐♥❛t♦r✐❝s✿ ❇❡♥❦❛rt✕❘♦❜② st✉❞✐❡❞ ✏❞♦✇♥✱ ✉♣ ♦♣❡r❛t♦rs✑ ♦♥ ♣♦s❡ts✿ ✭❣❡♥❡r❛❧✐③❡❞✮ ❞♦✇♥✲✉♣ ❛❧❣❡❜r❛s ✳ ❋♦r ❛❧❧ ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s✱ ❛♥❞ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❚r✐❛♥❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ✭❝♦♥t✳✮ ❚r✐❛♥❣✉❧❛r ●❲❆s ♦❝❝✉r ✐♥ ♠❛♥② s❡tt✐♥❣s✿ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✿ ❙♠✐t❤ st✉❞✐❡❞ ❞❡❢♦r♠❛t✐♦♥s ♦❢ sl 2 ✿ C � e, f, h � / ([ h, e ] = 2 e, [ h, f ] = − 2 f, [ e, f ] = z 0 ( h )) . ▼❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✿ ❲✐tt❡♥ ✐♥tr♦❞✉❝❡❞ ✼✲♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ ❞❡❢♦r♠❛t✐♦♥s ♦❢ U ( sl 2 ) ✳ ▲❡ ❇r✉②♥✿ ❈♦♥❢♦r♠❛❧ sl 2 ✲❛❧❣❡❜r❛s✳ ◗✉❛♥t✉♠ ❛❧❣❡❜r❛✿ ❏✐♥❣✲❩❤❛♥❣ st✉❞✐❡❞ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡✱ ♥♦♥✲❝♦❝♦♠♠✉t❛t✐✈❡ ❜✐❛❧❣❡❜r❛s t❤❛t q ✲❞❡❢♦r♠ U ( gl 2 ) , U ( sl 2 ) ✳ ❑❛❝✿ ✏❞✐s♣✐♥ ▲✐❡ s✉♣❡r❛❧❣❡❜r❛ B [0 , 1] ✑✳ ✺ ✴ ✸✹

❋♦r ❛❧❧ ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s✱ ❛♥❞ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❚r✐❛♥❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ✭❝♦♥t✳✮ ❚r✐❛♥❣✉❧❛r ●❲❆s ♦❝❝✉r ✐♥ ♠❛♥② s❡tt✐♥❣s✿ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✿ ❙♠✐t❤ st✉❞✐❡❞ ❞❡❢♦r♠❛t✐♦♥s ♦❢ sl 2 ✿ C � e, f, h � / ([ h, e ] = 2 e, [ h, f ] = − 2 f, [ e, f ] = z 0 ( h )) . ▼❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✿ ❲✐tt❡♥ ✐♥tr♦❞✉❝❡❞ ✼✲♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ ❞❡❢♦r♠❛t✐♦♥s ♦❢ U ( sl 2 ) ✳ ▲❡ ❇r✉②♥✿ ❈♦♥❢♦r♠❛❧ sl 2 ✲❛❧❣❡❜r❛s✳ ◗✉❛♥t✉♠ ❛❧❣❡❜r❛✿ ❏✐♥❣✲❩❤❛♥❣ st✉❞✐❡❞ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡✱ ♥♦♥✲❝♦❝♦♠♠✉t❛t✐✈❡ ❜✐❛❧❣❡❜r❛s t❤❛t q ✲❞❡❢♦r♠ U ( gl 2 ) , U ( sl 2 ) ✳ ❑❛❝✿ ✏❞✐s♣✐♥ ▲✐❡ s✉♣❡r❛❧❣❡❜r❛ B [0 , 1] ✑✳ ❈♦♠❜✐♥❛t♦r✐❝s✿ ❇❡♥❦❛rt✕❘♦❜② st✉❞✐❡❞ ✏❞♦✇♥✱ ✉♣ ♦♣❡r❛t♦rs✑ ♦♥ ♣♦s❡ts✿ ✭❣❡♥❡r❛❧✐③❡❞✮ ❞♦✇♥✲✉♣ ❛❧❣❡❜r❛s ✳ ✺ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❚r✐❛♥❣✉❧❛r ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ✭❝♦♥t✳✮ ❚r✐❛♥❣✉❧❛r ●❲❆s ♦❝❝✉r ✐♥ ♠❛♥② s❡tt✐♥❣s✿ ❘❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r②✿ ❙♠✐t❤ st✉❞✐❡❞ ❞❡❢♦r♠❛t✐♦♥s ♦❢ sl 2 ✿ C � e, f, h � / ([ h, e ] = 2 e, [ h, f ] = − 2 f, [ e, f ] = z 0 ( h )) . ▼❛t❤❡♠❛t✐❝❛❧ ♣❤②s✐❝s✿ ❲✐tt❡♥ ✐♥tr♦❞✉❝❡❞ ✼✲♣❛r❛♠❡t❡r ❢❛♠✐❧② ♦❢ ❞❡❢♦r♠❛t✐♦♥s ♦❢ U ( sl 2 ) ✳ ▲❡ ❇r✉②♥✿ ❈♦♥❢♦r♠❛❧ sl 2 ✲❛❧❣❡❜r❛s✳ ◗✉❛♥t✉♠ ❛❧❣❡❜r❛✿ ❏✐♥❣✲❩❤❛♥❣ st✉❞✐❡❞ ♥♦♥✲❝♦♠♠✉t❛t✐✈❡✱ ♥♦♥✲❝♦❝♦♠♠✉t❛t✐✈❡ ❜✐❛❧❣❡❜r❛s t❤❛t q ✲❞❡❢♦r♠ U ( gl 2 ) , U ( sl 2 ) ✳ ❑❛❝✿ ✏❞✐s♣✐♥ ▲✐❡ s✉♣❡r❛❧❣❡❜r❛ B [0 , 1] ✑✳ ❈♦♠❜✐♥❛t♦r✐❝s✿ ❇❡♥❦❛rt✕❘♦❜② st✉❞✐❡❞ ✏❞♦✇♥✱ ✉♣ ♦♣❡r❛t♦rs✑ ♦♥ ♣♦s❡ts✿ ✭❣❡♥❡r❛❧✐③❡❞✮ ❞♦✇♥✲✉♣ ❛❧❣❡❜r❛s ✳ ❋♦r ❛❧❧ ♦❢ t❤❡s❡ ❛❧❣❡❜r❛s✱ H = F [ h ] ∋ z 0 ❛♥❞ z 1 ∈ F × ✳ ✺ ✴ ✸✹

❚❤❡ ✏❝❧❛ss✐❝❛❧✑ ❛❧❣❡❜r❛s ❞❡❢♦r♠ ❀ t❤❡ ✏q✉❛♥t✉♠✑ ❛❧❣❡❜r❛s ❞❡❢♦r♠ ✳ ❆r❡ t❤❡r❡ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ t❤❡s❡ ✏❝❧❛ss✐❝❛❧✑ ❛♥❞ ✏q✉❛♥t✉♠✑ ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s❄ ❨❡s✿ t❤❡ ❝❧❛ss✐❝❛❧ ❢❛♠✐❧✐❡s ❛r❡ ✏❝❧❛ss✐❝❛❧ ❧✐♠✐ts✑ ♦❢ t❤❡ q✉❛♥t✉♠ ❢❛♠✐❧✐❡s✿ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ✏◗✉❛♥t✉♠✑ ❡①❛♠♣❧❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ❋❛♠✐❧✐❡s ♦❢ ✏q✉❛♥t✉♠✑ ❡①❛♠♣❧❡s✱ ✇✐t❤ H ❛ ❣r♦✉♣ ❛❧❣❡❜r❛✿ ◗✉❛♥t✉♠ sl 2 ✿ H = F [ K ± 1 ] ✳ ❉r✐♥❢❡❧❞ ❞♦✉❜❧❡ ♦❢ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ U q ( sl 2 ) ✿ H = F [ K ± 1 , L ± 1 ] ✳ ✻ ✴ ✸✹

❨❡s✿ t❤❡ ❝❧❛ss✐❝❛❧ ❢❛♠✐❧✐❡s ❛r❡ ✏❝❧❛ss✐❝❛❧ ❧✐♠✐ts✑ ♦❢ t❤❡ q✉❛♥t✉♠ ❢❛♠✐❧✐❡s✿ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ✏◗✉❛♥t✉♠✑ ❡①❛♠♣❧❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ❋❛♠✐❧✐❡s ♦❢ ✏q✉❛♥t✉♠✑ ❡①❛♠♣❧❡s✱ ✇✐t❤ H ❛ ❣r♦✉♣ ❛❧❣❡❜r❛✿ ◗✉❛♥t✉♠ sl 2 ✿ H = F [ K ± 1 ] ✳ ❉r✐♥❢❡❧❞ ❞♦✉❜❧❡ ♦❢ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ U q ( sl 2 ) ✿ H = F [ K ± 1 , L ± 1 ] ✳ ❚❤❡ ✏❝❧❛ss✐❝❛❧✑ ❛❧❣❡❜r❛s ❞❡❢♦r♠ U ( sl 2 ) ❀ t❤❡ ✏q✉❛♥t✉♠✑ ❛❧❣❡❜r❛s ❞❡❢♦r♠ U q ( sl 2 ) ✳ ❆r❡ t❤❡r❡ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ t❤❡s❡ ✏❝❧❛ss✐❝❛❧✑ ❛♥❞ ✏q✉❛♥t✉♠✑ ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s❄ ✻ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ✏◗✉❛♥t✉♠✑ ❡①❛♠♣❧❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ❋❛♠✐❧✐❡s ♦❢ ✏q✉❛♥t✉♠✑ ❡①❛♠♣❧❡s✱ ✇✐t❤ H ❛ ❣r♦✉♣ ❛❧❣❡❜r❛✿ ◗✉❛♥t✉♠ sl 2 ✿ H = F [ K ± 1 ] ✳ ❉r✐♥❢❡❧❞ ❞♦✉❜❧❡ ♦❢ ♣♦s✐t✐✈❡ ♣❛rt ♦❢ U q ( sl 2 ) ✿ H = F [ K ± 1 , L ± 1 ] ✳ ❚❤❡ ✏❝❧❛ss✐❝❛❧✑ ❛❧❣❡❜r❛s ❞❡❢♦r♠ U ( sl 2 ) ❀ t❤❡ ✏q✉❛♥t✉♠✑ ❛❧❣❡❜r❛s ❞❡❢♦r♠ U q ( sl 2 ) ✳ ❆r❡ t❤❡r❡ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ t❤❡s❡ ✏❝❧❛ss✐❝❛❧✑ ❛♥❞ ✏q✉❛♥t✉♠✑ ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s❄ ❨❡s✿ t❤❡ ❝❧❛ss✐❝❛❧ ❢❛♠✐❧✐❡s ❛r❡ ✏❝❧❛ss✐❝❛❧ ❧✐♠✐ts✑ ♦❢ t❤❡ q✉❛♥t✉♠ ❢❛♠✐❧✐❡s✿ ✻ ✴ ✸✹

❙✉♣♣♦s❡ ✐s tr❛♥s❝❡♥❞❡♥t❛❧ ♦✈❡r ✱ ❛♥❞ ❛♥❞ ❛r❡ ✐♥t❡❣❡rs✳ ❉❡✜♥❡ t❤❡ ✏q✉❛♥t✉♠ ❛❧❣❡❜r❛✑ ✇❤❡r❡ ✳ ▲❡t ❜❡ t❤❡ ❧♦❝❛❧ s✉❜r✐♥❣ ♦❢ ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s r❡❣✉❧❛r ❛t t❤❡ ♣♦✐♥t ✳ ▲❡t ❜❡ t❤❡ ✲s✉❜❛❧❣❡❜r❛ ♦❢ ❣❡♥❡r❛t❡❞ ❜② ✳ ❚❤❡♥ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❉❡❢♦r♠❛t✐♦♥✲q✉❛♥t✐③❛t✐♦♥ ❡q✉❛❧s q✉❛♥t✐③❛t✐♦♥✲❞❡❢♦r♠❛t✐♦♥ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❋✐① s❝❛❧❛rs γ ∈ F , z 1 ∈ F × ✱ ❛♥❞ ❛ ♣♦❧②♥♦♠✐❛❧ z 0 = z 0 ( h ) ∈ F [ h ] ✳ ❈♦♥s✐❞❡r t❤❡ ❛❧❣❡❜r❛ W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ✱ ✇✐t❤ θ ( h ) := h + γ ✳ ✼ ✴ ✸✹

▲❡t ❜❡ t❤❡ ❧♦❝❛❧ s✉❜r✐♥❣ ♦❢ ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s r❡❣✉❧❛r ❛t t❤❡ ♣♦✐♥t ✳ ▲❡t ❜❡ t❤❡ ✲s✉❜❛❧❣❡❜r❛ ♦❢ ❣❡♥❡r❛t❡❞ ❜② ✳ ❚❤❡♥ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❉❡❢♦r♠❛t✐♦♥✲q✉❛♥t✐③❛t✐♦♥ ❡q✉❛❧s q✉❛♥t✐③❛t✐♦♥✲❞❡❢♦r♠❛t✐♦♥ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❋✐① s❝❛❧❛rs γ ∈ F , z 1 ∈ F × ✱ ❛♥❞ ❛ ♣♦❧②♥♦♠✐❛❧ z 0 = z 0 ( h ) ∈ F [ h ] ✳ ❈♦♥s✐❞❡r t❤❡ ❛❧❣❡❜r❛ W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ✱ ✇✐t❤ θ ( h ) := h + γ ✳ ❙✉♣♣♦s❡ q ✐s tr❛♥s❝❡♥❞❡♥t❛❧ ♦✈❡r F ✱ ❛♥❞ l � = 0 ❛♥❞ m, n ❛r❡ ✐♥t❡❣❡rs✳ ❉❡✜♥❡ t❤❡ ✏q✉❛♥t✉♠ ❛❧❣❡❜r❛✑ � � F ( q )[ K ± 1 ] , θ, q m K n z 0 ( γ (1 − K ) W q ( l, m, n ) := W l ( q − 1) ) , z 1 , ✇❤❡r❡ θ ( K ) = q − l K ✳ ✼ ✴ ✸✹

▲❡t ❜❡ t❤❡ ✲s✉❜❛❧❣❡❜r❛ ♦❢ ❣❡♥❡r❛t❡❞ ❜② ✳ ❚❤❡♥ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❉❡❢♦r♠❛t✐♦♥✲q✉❛♥t✐③❛t✐♦♥ ❡q✉❛❧s q✉❛♥t✐③❛t✐♦♥✲❞❡❢♦r♠❛t✐♦♥ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❋✐① s❝❛❧❛rs γ ∈ F , z 1 ∈ F × ✱ ❛♥❞ ❛ ♣♦❧②♥♦♠✐❛❧ z 0 = z 0 ( h ) ∈ F [ h ] ✳ ❈♦♥s✐❞❡r t❤❡ ❛❧❣❡❜r❛ W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ✱ ✇✐t❤ θ ( h ) := h + γ ✳ ❙✉♣♣♦s❡ q ✐s tr❛♥s❝❡♥❞❡♥t❛❧ ♦✈❡r F ✱ ❛♥❞ l � = 0 ❛♥❞ m, n ❛r❡ ✐♥t❡❣❡rs✳ ❉❡✜♥❡ t❤❡ ✏q✉❛♥t✉♠ ❛❧❣❡❜r❛✑ � � F ( q )[ K ± 1 ] , θ, q m K n z 0 ( γ (1 − K ) W q ( l, m, n ) := W l ( q − 1) ) , z 1 , ✇❤❡r❡ θ ( K ) = q − l K ✳ ▲❡t R ❜❡ t❤❡ ❧♦❝❛❧ s✉❜r✐♥❣ ♦❢ F ( q ) ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s r❡❣✉❧❛r ❛t t❤❡ ♣♦✐♥t q = 1 ✳ ✼ ✴ ✸✹

❚❤❡♥ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❉❡❢♦r♠❛t✐♦♥✲q✉❛♥t✐③❛t✐♦♥ ❡q✉❛❧s q✉❛♥t✐③❛t✐♦♥✲❞❡❢♦r♠❛t✐♦♥ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❋✐① s❝❛❧❛rs γ ∈ F , z 1 ∈ F × ✱ ❛♥❞ ❛ ♣♦❧②♥♦♠✐❛❧ z 0 = z 0 ( h ) ∈ F [ h ] ✳ ❈♦♥s✐❞❡r t❤❡ ❛❧❣❡❜r❛ W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ✱ ✇✐t❤ θ ( h ) := h + γ ✳ ❙✉♣♣♦s❡ q ✐s tr❛♥s❝❡♥❞❡♥t❛❧ ♦✈❡r F ✱ ❛♥❞ l � = 0 ❛♥❞ m, n ❛r❡ ✐♥t❡❣❡rs✳ ❉❡✜♥❡ t❤❡ ✏q✉❛♥t✉♠ ❛❧❣❡❜r❛✑ � � F ( q )[ K ± 1 ] , θ, q m K n z 0 ( γ (1 − K ) W q ( l, m, n ) := W l ( q − 1) ) , z 1 , ✇❤❡r❡ θ ( K ) = q − l K ✳ ▲❡t R ❜❡ t❤❡ ❧♦❝❛❧ s✉❜r✐♥❣ ♦❢ F ( q ) ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s r❡❣✉❧❛r ❛t t❤❡ ♣♦✐♥t q = 1 ✳ ▲❡t W R q ( l, m, n ) ❜❡ t❤❡ R ✲s✉❜❛❧❣❡❜r❛ ♦❢ W q ( l, m, n ) ❣❡♥❡r❛t❡❞ ❜② u, d, K ± 1 , ( K − 1) / ( q − 1) ✳ ✼ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❉❡❢♦r♠❛t✐♦♥✲q✉❛♥t✐③❛t✐♦♥ ❡q✉❛❧s q✉❛♥t✐③❛t✐♦♥✲❞❡❢♦r♠❛t✐♦♥ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❋✐① s❝❛❧❛rs γ ∈ F , z 1 ∈ F × ✱ ❛♥❞ ❛ ♣♦❧②♥♦♠✐❛❧ z 0 = z 0 ( h ) ∈ F [ h ] ✳ ❈♦♥s✐❞❡r t❤❡ ❛❧❣❡❜r❛ W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ✱ ✇✐t❤ θ ( h ) := h + γ ✳ ❙✉♣♣♦s❡ q ✐s tr❛♥s❝❡♥❞❡♥t❛❧ ♦✈❡r F ✱ ❛♥❞ l � = 0 ❛♥❞ m, n ❛r❡ ✐♥t❡❣❡rs✳ ❉❡✜♥❡ t❤❡ ✏q✉❛♥t✉♠ ❛❧❣❡❜r❛✑ � � F ( q )[ K ± 1 ] , θ, q m K n z 0 ( γ (1 − K ) W q ( l, m, n ) := W l ( q − 1) ) , z 1 , ✇❤❡r❡ θ ( K ) = q − l K ✳ ▲❡t R ❜❡ t❤❡ ❧♦❝❛❧ s✉❜r✐♥❣ ♦❢ F ( q ) ♦❢ r❛t✐♦♥❛❧ ❢✉♥❝t✐♦♥s r❡❣✉❧❛r ❛t t❤❡ ♣♦✐♥t q = 1 ✳ ▲❡t W R q ( l, m, n ) ❜❡ t❤❡ R ✲s✉❜❛❧❣❡❜r❛ ♦❢ W q ( l, m, n ) ❣❡♥❡r❛t❡❞ ❜② u, d, K ± 1 , ( K − 1) / ( q − 1) ✳ W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ∼ = W R q ( l, m, n ) / ( q − 1) W R ❚❤❡♥ q ( l, m, n ) ✳ ✼ ✴ ✸✹

▲❡t ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s✉r❥❡❝t✐♦♥ ♦❢ ✲❛❧❣❡❜r❛s ❚♦ s❤♦✇✿ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ❊♥♦✉❣❤ t♦ s❤♦✇ t❤✐s✱ ❛❢t❡r ❝❤❛♥❣✐♥❣ s❝❛❧❛rs t♦ ✱ ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ✜❡❧❞ ❡①t❡♥s✐♦♥ ♦❢ ✳ ◆♦✇ ✜♥❞ ❛ ❱❡r♠❛ ♠♦❞✉❧❡ t❤❛t ✐s s✐♠♣❧❡ ♦✈❡r ✱ ❤❡♥❝❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ✭❘❡q✉✐r❡s ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ str✉❝t✉r❡ ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s ❛♥❞ ❜❧♦❝❦s ♦❢ ❈❛t❡❣♦r② ✳✮ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ❙❡t W R ± t♦ ❜❡ R [ u ] , R [ d ] r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ W R 0 t♦ ❜❡ t❤❡ R ✲s✉❜❛❧❣❡❜r❛ ♦❢ F ( q )[ K ± 1 ] ❣❡♥❡r❛t❡❞ ❜② K ± 1 , K − 1 q − 1 ✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♠❛♣ : W R − ⊗ R W R 0 ⊗ R W R + → W q ( l, m, n ) ✐s ❛♥ R ✲❛❧❣❡❜r❛ ✐s♦♠♦r♣❤✐s♠✳ ✽ ✴ ✸✹

❚♦ s❤♦✇✿ r❡str✐❝t❡❞ t♦ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ❊♥♦✉❣❤ t♦ s❤♦✇ t❤✐s✱ ❛❢t❡r ❝❤❛♥❣✐♥❣ s❝❛❧❛rs t♦ ✱ ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ✜❡❧❞ ❡①t❡♥s✐♦♥ ♦❢ ✳ ◆♦✇ ✜♥❞ ❛ ❱❡r♠❛ ♠♦❞✉❧❡ t❤❛t ✐s s✐♠♣❧❡ ♦✈❡r ✱ ❤❡♥❝❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ✭❘❡q✉✐r❡s ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ str✉❝t✉r❡ ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s ❛♥❞ ❜❧♦❝❦s ♦❢ ❈❛t❡❣♦r② ✳✮ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ❙❡t W R ± t♦ ❜❡ R [ u ] , R [ d ] r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ W R 0 t♦ ❜❡ t❤❡ R ✲s✉❜❛❧❣❡❜r❛ ♦❢ F ( q )[ K ± 1 ] ❣❡♥❡r❛t❡❞ ❜② K ± 1 , K − 1 q − 1 ✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♠❛♣ : W R − ⊗ R W R 0 ⊗ R W R + → W q ( l, m, n ) ✐s ❛♥ R ✲❛❧❣❡❜r❛ ✐s♦♠♦r♣❤✐s♠✳ ▲❡t W 1 := W R q ( l, m, n ) / ( q − 1) W R q ( l, m, n ) ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s✉r❥❡❝t✐♦♥ ♦❢ F ✲❛❧❣❡❜r❛s ( u �→ u, d �→ d, h �→ γ (1 − K ) π : W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ։ W 1 l ( q − 1) ) . ✽ ✴ ✸✹

◆♦✇ ✜♥❞ ❛ ❱❡r♠❛ ♠♦❞✉❧❡ t❤❛t ✐s s✐♠♣❧❡ ♦✈❡r ✱ ❤❡♥❝❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ✭❘❡q✉✐r❡s ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ str✉❝t✉r❡ ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s ❛♥❞ ❜❧♦❝❦s ♦❢ ❈❛t❡❣♦r② ✳✮ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ❙❡t W R ± t♦ ❜❡ R [ u ] , R [ d ] r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ W R 0 t♦ ❜❡ t❤❡ R ✲s✉❜❛❧❣❡❜r❛ ♦❢ F ( q )[ K ± 1 ] ❣❡♥❡r❛t❡❞ ❜② K ± 1 , K − 1 q − 1 ✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♠❛♣ : W R − ⊗ R W R 0 ⊗ R W R + → W q ( l, m, n ) ✐s ❛♥ R ✲❛❧❣❡❜r❛ ✐s♦♠♦r♣❤✐s♠✳ ▲❡t W 1 := W R q ( l, m, n ) / ( q − 1) W R q ( l, m, n ) ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s✉r❥❡❝t✐♦♥ ♦❢ F ✲❛❧❣❡❜r❛s ( u �→ u, d �→ d, h �→ γ (1 − K ) π : W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ։ W 1 l ( q − 1) ) . ❚♦ s❤♦✇✿ π r❡str✐❝t❡❞ t♦ F [ d ] ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ❊♥♦✉❣❤ t♦ s❤♦✇ t❤✐s✱ ❛❢t❡r ❝❤❛♥❣✐♥❣ s❝❛❧❛rs t♦ F u ✱ ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ✜❡❧❞ ❡①t❡♥s✐♦♥ ♦❢ F ✳ ✽ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙❦❡t❝❤ ♦❢ ♣r♦♦❢ ❙❡t W R ± t♦ ❜❡ R [ u ] , R [ d ] r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ W R 0 t♦ ❜❡ t❤❡ R ✲s✉❜❛❧❣❡❜r❛ ♦❢ F ( q )[ K ± 1 ] ❣❡♥❡r❛t❡❞ ❜② K ± 1 , K − 1 q − 1 ✳ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♠❛♣ : W R − ⊗ R W R 0 ⊗ R W R + → W q ( l, m, n ) ✐s ❛♥ R ✲❛❧❣❡❜r❛ ✐s♦♠♦r♣❤✐s♠✳ ▲❡t W 1 := W R q ( l, m, n ) / ( q − 1) W R q ( l, m, n ) ✳ ❚❤❡♥ t❤❡r❡ ✐s ❛ s✉r❥❡❝t✐♦♥ ♦❢ F ✲❛❧❣❡❜r❛s ( u �→ u, d �→ d, h �→ γ (1 − K ) π : W ( F [ h ] , θ, z 0 ( h ) , z 1 ) ։ W 1 l ( q − 1) ) . ❚♦ s❤♦✇✿ π r❡str✐❝t❡❞ t♦ F [ d ] ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠✳ ❊♥♦✉❣❤ t♦ s❤♦✇ t❤✐s✱ ❛❢t❡r ❝❤❛♥❣✐♥❣ s❝❛❧❛rs t♦ F u ✱ ❛♥ ✉♥❝♦✉♥t❛❜❧❡ ✜❡❧❞ ❡①t❡♥s✐♦♥ ♦❢ F ✳ ◆♦✇ ✜♥❞ ❛ ❱❡r♠❛ ♠♦❞✉❧❡ M F u 1 ( λ ) t❤❛t ✐s s✐♠♣❧❡ ♦✈❡r F u ⊗ F W 1 ✱ ❤❡♥❝❡ ✐♥✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ✭❘❡q✉✐r❡s ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ str✉❝t✉r❡ ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s ❛♥❞ ❜❧♦❝❦s ♦❢ ❈❛t❡❣♦r② O ✳✮ ✽ ✴ ✸✹

✷ ❈❛t❡❣♦r② ✐s t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧❡s t❤❛t ❛r❡✿ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ ✲s❡♠✐s✐♠♣❧❡✱ ✇✐t❤ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ ✲✇❡✐❣❤t s♣❛❝❡s✱ ❛♥❞ ❛❝ts ❧♦❝❛❧❧② ♥✐❧♣♦t❡♥t❧② ♦♥ ❡❛❝❤ ♠♦❞✉❧❡✳ ✸ ❲❡✐❣❤ts ❛r❡ ❝❤❛r❛❝t❡rs ✭❛❧❣❡❜r❛ ♠❛♣s✮ ✳ ✹ ●✐✈❡♥ ✱ t❤❡ ✲✇❡✐❣❤t s♣❛❝❡ ♦❢ ❛ ♠♦❞✉❧❡ ✐s ✺ ❚❤❡ ✇❡✐❣❤ts ♦❢ ❛ ♠♦❞✉❧❡ ❛r❡ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O A := W ( H, θ, z 0 , z 1 ) = tr✐❛♥❣✉❧❛r ●❲❆✳ ✶ P❇❲ ♣r♦♣❡rt②✿ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♠❛♣ : F [ d ] ⊗ H ⊗ F [ u ] → A ✐s ❛ ✈❡❝t♦r s♣❛❝❡ ✐s♦♠♦r♣❤✐s♠✳ ✾ ✴ ✸✹

✸ ❲❡✐❣❤ts ❛r❡ ❝❤❛r❛❝t❡rs ✭❛❧❣❡❜r❛ ♠❛♣s✮ ✳ ✹ ●✐✈❡♥ ✱ t❤❡ ✲✇❡✐❣❤t s♣❛❝❡ ♦❢ ❛ ♠♦❞✉❧❡ ✐s ✺ ❚❤❡ ✇❡✐❣❤ts ♦❢ ❛ ♠♦❞✉❧❡ ❛r❡ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O A := W ( H, θ, z 0 , z 1 ) = tr✐❛♥❣✉❧❛r ●❲❆✳ ✶ P❇❲ ♣r♦♣❡rt②✿ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♠❛♣ : F [ d ] ⊗ H ⊗ F [ u ] → A ✐s ❛ ✈❡❝t♦r s♣❛❝❡ ✐s♦♠♦r♣❤✐s♠✳ ✷ ❈❛t❡❣♦r② O ✐s t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧❡s t❤❛t ❛r❡✿ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ H ✲s❡♠✐s✐♠♣❧❡✱ ✇✐t❤ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ H ✲✇❡✐❣❤t s♣❛❝❡s✱ ❛♥❞ u ❛❝ts ❧♦❝❛❧❧② ♥✐❧♣♦t❡♥t❧② ♦♥ ❡❛❝❤ ♠♦❞✉❧❡✳ ✾ ✴ ✸✹

✹ ●✐✈❡♥ ✱ t❤❡ ✲✇❡✐❣❤t s♣❛❝❡ ♦❢ ❛ ♠♦❞✉❧❡ ✐s ✺ ❚❤❡ ✇❡✐❣❤ts ♦❢ ❛ ♠♦❞✉❧❡ ❛r❡ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O A := W ( H, θ, z 0 , z 1 ) = tr✐❛♥❣✉❧❛r ●❲❆✳ ✶ P❇❲ ♣r♦♣❡rt②✿ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♠❛♣ : F [ d ] ⊗ H ⊗ F [ u ] → A ✐s ❛ ✈❡❝t♦r s♣❛❝❡ ✐s♦♠♦r♣❤✐s♠✳ ✷ ❈❛t❡❣♦r② O ✐s t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧❡s t❤❛t ❛r❡✿ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ H ✲s❡♠✐s✐♠♣❧❡✱ ✇✐t❤ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ H ✲✇❡✐❣❤t s♣❛❝❡s✱ ❛♥❞ u ❛❝ts ❧♦❝❛❧❧② ♥✐❧♣♦t❡♥t❧② ♦♥ ❡❛❝❤ ♠♦❞✉❧❡✳ ✸ ❲❡✐❣❤ts ❛r❡ ❝❤❛r❛❝t❡rs ✭❛❧❣❡❜r❛ ♠❛♣s✮ � H := { λ : H → F } ✳ ✾ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O A := W ( H, θ, z 0 , z 1 ) = tr✐❛♥❣✉❧❛r ●❲❆✳ ✶ P❇❲ ♣r♦♣❡rt②✿ ❚❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ♠❛♣ : F [ d ] ⊗ H ⊗ F [ u ] → A ✐s ❛ ✈❡❝t♦r s♣❛❝❡ ✐s♦♠♦r♣❤✐s♠✳ ✷ ❈❛t❡❣♦r② O ✐s t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♠♦❞✉❧❡s t❤❛t ❛r❡✿ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ H ✲s❡♠✐s✐♠♣❧❡✱ ✇✐t❤ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧ H ✲✇❡✐❣❤t s♣❛❝❡s✱ ❛♥❞ u ❛❝ts ❧♦❝❛❧❧② ♥✐❧♣♦t❡♥t❧② ♦♥ ❡❛❝❤ ♠♦❞✉❧❡✳ ✸ ❲❡✐❣❤ts ❛r❡ ❝❤❛r❛❝t❡rs ✭❛❧❣❡❜r❛ ♠❛♣s✮ � H := { λ : H → F } ✳ ✹ ●✐✈❡♥ λ ∈ � H ✱ t❤❡ λ ✲✇❡✐❣❤t s♣❛❝❡ ♦❢ ❛ ♠♦❞✉❧❡ M ✐s M λ := { m ∈ M : h · m = λ ( h ) m, ∀ h ∈ H } . ✺ ❚❤❡ ✇❡✐❣❤ts ♦❢ ❛ ♠♦❞✉❧❡ M ❛r❡ wt M := { λ ∈ � H : M λ � = 0 } ✳ ✾ ✴ ✸✹

■♠♣♦rt❛♥t ♦❜❥❡❝ts ✐♥ ❈❛t❡❣♦r② ✿ ❱❡r♠❛ ♠♦❞✉❧❡s ✿ ❊❛❝❤ ❱❡r♠❛ ♠♦❞✉❧❡ ❤❛s ❛ ✉♥✐q✉❡ s✐♠♣❧❡ q✉♦t✐❡♥t ✳ ❆❧s♦ ❧✐❡s ✐♥ ✳ ❆❧❧ s✐♠♣❧❡ ♦❜❥❡❝ts ✐♥ ❛r❡ ♦❢ t❤❡ ❢♦r♠ ✳ ❲❤❛t ❛r❡ t❤❡ ✇❡✐❣❤ts ♦❢ ❄ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ ❄ ●❡♥❡r❛❧ ❢❛❝t✿ ❈❛t❡❣♦r② ✐s ✜♥✐t❡ ❧❡♥❣t❤✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ❱❡r♠❛ ♠♦❞✉❧❡s ❤❛✈❡ ✜♥✐t❡ ❧❡♥❣t❤✳ ❍♦✇ t♦ ❝♦♠♣✉t❡ s✉❜♠♦❞✉❧❡s ♦❢ ❄ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❱❡r♠❛ ♠♦❞✉❧❡s ❚❡❝❤♥✐❝❛❧ ❛ss✉♠♣t✐♦♥✿ ❋♦r ❛❧❧ ✇❡✐❣❤ts λ ∈ � H ✱ ✐❢ n ∈ Z ❛♥❞ λ ≡ λ ◦ θ n ♦♥ ❛❧❧ ♦❢ H ✱ t❤❡♥ n = 0 ✳ ✭❙♦ θ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✐♥✜♥✐t❡ ♦r❞❡r✳✮ ✶✵ ✴ ✸✹

❆❧s♦ ❧✐❡s ✐♥ ✳ ❆❧❧ s✐♠♣❧❡ ♦❜❥❡❝ts ✐♥ ❛r❡ ♦❢ t❤❡ ❢♦r♠ ✳ ❲❤❛t ❛r❡ t❤❡ ✇❡✐❣❤ts ♦❢ ❄ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ ❄ ●❡♥❡r❛❧ ❢❛❝t✿ ❈❛t❡❣♦r② ✐s ✜♥✐t❡ ❧❡♥❣t❤✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ❱❡r♠❛ ♠♦❞✉❧❡s ❤❛✈❡ ✜♥✐t❡ ❧❡♥❣t❤✳ ❍♦✇ t♦ ❝♦♠♣✉t❡ s✉❜♠♦❞✉❧❡s ♦❢ ❄ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❱❡r♠❛ ♠♦❞✉❧❡s ❚❡❝❤♥✐❝❛❧ ❛ss✉♠♣t✐♦♥✿ ❋♦r ❛❧❧ ✇❡✐❣❤ts λ ∈ � H ✱ ✐❢ n ∈ Z ❛♥❞ λ ≡ λ ◦ θ n ♦♥ ❛❧❧ ♦❢ H ✱ t❤❡♥ n = 0 ✳ ✭❙♦ θ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✐♥✜♥✐t❡ ♦r❞❡r✳✮ ■♠♣♦rt❛♥t ♦❜❥❡❝ts ✐♥ ❈❛t❡❣♦r② O ✿ ❱❡r♠❛ ♠♦❞✉❧❡s ✿ λ ∈ � M ( λ ) := A/ ( A · u + A · ker λ ) , H. ❊❛❝❤ ❱❡r♠❛ ♠♦❞✉❧❡ ❤❛s ❛ ✉♥✐q✉❡ s✐♠♣❧❡ q✉♦t✐❡♥t L ( λ ) ✳ ✶✵ ✴ ✸✹

❲❤❛t ❛r❡ t❤❡ ✇❡✐❣❤ts ♦❢ ❄ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ ❄ ●❡♥❡r❛❧ ❢❛❝t✿ ❈❛t❡❣♦r② ✐s ✜♥✐t❡ ❧❡♥❣t❤✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ❱❡r♠❛ ♠♦❞✉❧❡s ❤❛✈❡ ✜♥✐t❡ ❧❡♥❣t❤✳ ❍♦✇ t♦ ❝♦♠♣✉t❡ s✉❜♠♦❞✉❧❡s ♦❢ ❄ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❱❡r♠❛ ♠♦❞✉❧❡s ❚❡❝❤♥✐❝❛❧ ❛ss✉♠♣t✐♦♥✿ ❋♦r ❛❧❧ ✇❡✐❣❤ts λ ∈ � H ✱ ✐❢ n ∈ Z ❛♥❞ λ ≡ λ ◦ θ n ♦♥ ❛❧❧ ♦❢ H ✱ t❤❡♥ n = 0 ✳ ✭❙♦ θ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✐♥✜♥✐t❡ ♦r❞❡r✳✮ ■♠♣♦rt❛♥t ♦❜❥❡❝ts ✐♥ ❈❛t❡❣♦r② O ✿ ❱❡r♠❛ ♠♦❞✉❧❡s ✿ λ ∈ � M ( λ ) := A/ ( A · u + A · ker λ ) , H. ❊❛❝❤ ❱❡r♠❛ ♠♦❞✉❧❡ ❤❛s ❛ ✉♥✐q✉❡ s✐♠♣❧❡ q✉♦t✐❡♥t L ( λ ) ✳ ❆❧s♦ ❧✐❡s ✐♥ O ✳ ❆❧❧ s✐♠♣❧❡ ♦❜❥❡❝ts ✐♥ O ❛r❡ ♦❢ t❤❡ ❢♦r♠ L ( λ ) ✳ ✶✵ ✴ ✸✹

❍♦✇ t♦ ❝♦♠♣✉t❡ s✉❜♠♦❞✉❧❡s ♦❢ ❄ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❱❡r♠❛ ♠♦❞✉❧❡s ❚❡❝❤♥✐❝❛❧ ❛ss✉♠♣t✐♦♥✿ ❋♦r ❛❧❧ ✇❡✐❣❤ts λ ∈ � H ✱ ✐❢ n ∈ Z ❛♥❞ λ ≡ λ ◦ θ n ♦♥ ❛❧❧ ♦❢ H ✱ t❤❡♥ n = 0 ✳ ✭❙♦ θ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✐♥✜♥✐t❡ ♦r❞❡r✳✮ ■♠♣♦rt❛♥t ♦❜❥❡❝ts ✐♥ ❈❛t❡❣♦r② O ✿ ❱❡r♠❛ ♠♦❞✉❧❡s ✿ λ ∈ � M ( λ ) := A/ ( A · u + A · ker λ ) , H. ❊❛❝❤ ❱❡r♠❛ ♠♦❞✉❧❡ ❤❛s ❛ ✉♥✐q✉❡ s✐♠♣❧❡ q✉♦t✐❡♥t L ( λ ) ✳ ❆❧s♦ ❧✐❡s ✐♥ O ✳ ❆❧❧ s✐♠♣❧❡ ♦❜❥❡❝ts ✐♥ O ❛r❡ ♦❢ t❤❡ ❢♦r♠ L ( λ ) ✳ ❲❤❛t ❛r❡ t❤❡ ✇❡✐❣❤ts ♦❢ M ( λ ) ❄ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ M ( λ ) ❄ ●❡♥❡r❛❧ ❢❛❝t✿ ❈❛t❡❣♦r② O ✐s ✜♥✐t❡ ❧❡♥❣t❤✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ❱❡r♠❛ ♠♦❞✉❧❡s ❤❛✈❡ ✜♥✐t❡ ❧❡♥❣t❤✳ ✶✵ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❱❡r♠❛ ♠♦❞✉❧❡s ❚❡❝❤♥✐❝❛❧ ❛ss✉♠♣t✐♦♥✿ ❋♦r ❛❧❧ ✇❡✐❣❤ts λ ∈ � H ✱ ✐❢ n ∈ Z ❛♥❞ λ ≡ λ ◦ θ n ♦♥ ❛❧❧ ♦❢ H ✱ t❤❡♥ n = 0 ✳ ✭❙♦ θ ✐s ❛♥ ❛✉t♦♠♦r♣❤✐s♠ ♦❢ ✐♥✜♥✐t❡ ♦r❞❡r✳✮ ■♠♣♦rt❛♥t ♦❜❥❡❝ts ✐♥ ❈❛t❡❣♦r② O ✿ ❱❡r♠❛ ♠♦❞✉❧❡s ✿ λ ∈ � M ( λ ) := A/ ( A · u + A · ker λ ) , H. ❊❛❝❤ ❱❡r♠❛ ♠♦❞✉❧❡ ❤❛s ❛ ✉♥✐q✉❡ s✐♠♣❧❡ q✉♦t✐❡♥t L ( λ ) ✳ ❆❧s♦ ❧✐❡s ✐♥ O ✳ ❆❧❧ s✐♠♣❧❡ ♦❜❥❡❝ts ✐♥ O ❛r❡ ♦❢ t❤❡ ❢♦r♠ L ( λ ) ✳ ❲❤❛t ❛r❡ t❤❡ ✇❡✐❣❤ts ♦❢ M ( λ ) ❄ ❲❤❛t ✐s t❤❡ ❧❡♥❣t❤ ♦❢ M ( λ ) ❄ ●❡♥❡r❛❧ ❢❛❝t✿ ❈❛t❡❣♦r② O ✐s ✜♥✐t❡ ❧❡♥❣t❤✱ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ❱❡r♠❛ ♠♦❞✉❧❡s ❤❛✈❡ ✜♥✐t❡ ❧❡♥❣t❤✳ ❍♦✇ t♦ ❝♦♠♣✉t❡ s✉❜♠♦❞✉❧❡s ♦❢ M ( λ ) ❄ ✶✵ ✴ ✸✹

❛s ❢r❡❡ ✲♠♦❞✉❧❡s✳ ✸ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s A = W ( H, θ, z 0 , z 1 ) . ❋❛❝ts✿ ✶ wt M ( λ ) = { λ ◦ θ n : n ≥ 0 } ✳ ✷ ●✐✈❡♥ λ ∈ � H ❛♥❞ n ∈ Z ✱ ❞❡✜♥❡ n ∗ λ := λ ◦ θ − n ✳ ❚❤✉s✱ wt M ( λ ) = { ( − n ) ∗ λ : n ≥ 0 } ✳ ❆❧❧ ✭♥♦♥③❡r♦✮ ✇❡✐❣❤t ♠✉❧t✐♣❧✐❝✐t✐❡s ❛r❡ 1 ✳ ✶✶ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s A = W ( H, θ, z 0 , z 1 ) . ❋❛❝ts✿ ✶ wt M ( λ ) = { λ ◦ θ n : n ≥ 0 } ✳ ✷ ●✐✈❡♥ λ ∈ � H ❛♥❞ n ∈ Z ✱ ❞❡✜♥❡ n ∗ λ := λ ◦ θ − n ✳ ❚❤✉s✱ wt M ( λ ) = { ( − n ) ∗ λ : n ≥ 0 } ✳ ❆❧❧ ✭♥♦♥③❡r♦✮ ✇❡✐❣❤t ♠✉❧t✐♣❧✐❝✐t✐❡s ❛r❡ 1 ✳ ✸ M ( λ ) ∼ = F [ d ] ❛s ❢r❡❡ F [ d ] ✲♠♦❞✉❧❡s✳ ✶✶ ✴ ✸✹

Pr♦♣♦s✐t✐♦♥ ✭❑✳✱ ✷✵✶✺✮ ❋♦r ❛❧❧ ✇❡✐❣❤ts ✱ t❤❡ ❱❡r♠❛ ♠♦❞✉❧❡ ✐s ✉♥✐s❡r✐❛❧✱ ✇✐t❤ ✉♥✐q✉❡ ❝♦♠♣♦s✐t✐♦♥ s❡r✐❡s ✇❤❡r❡ ❝♦♠♣r✐s❡ t❤❡ s❡t ✳ ▼♦r❡♦✈❡r✱ ❢♦r ❛❧❧ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s ✭❝♦♥t✳✮ ❋♦r n ∈ N ✱ ❞❡✜♥❡ ❞✐st✐♥❣✉✐s❤❡❞ ❡❧❡♠❡♥ts � z n ∈ H ✿ n − 1 n − 1 � � θ i ( z 1 ) , θ j ( z 0 z ′ z ′ z ′ n := 0 := 1 , z n := � n − 1 − j ) . i =0 j =0 ❆❧s♦ ❞❡✜♥❡ � z n ❢♦r ♥♦♥✲♣♦s✐t✐✈❡ n ∈ Z ✿ z − n := θ − n ( � � z 0 := 0 , � z n ) ( n > 0) . ✶✷ ✴ ✸✹

▼♦r❡♦✈❡r✱ ❢♦r ❛❧❧ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s ✭❝♦♥t✳✮ ❋♦r n ∈ N ✱ ❞❡✜♥❡ ❞✐st✐♥❣✉✐s❤❡❞ ❡❧❡♠❡♥ts � z n ∈ H ✿ n − 1 n − 1 � � θ i ( z 1 ) , θ j ( z 0 z ′ z ′ z ′ n := 0 := 1 , z n := � n − 1 − j ) . i =0 j =0 ❆❧s♦ ❞❡✜♥❡ � z n ❢♦r ♥♦♥✲♣♦s✐t✐✈❡ n ∈ Z ✿ z − n := θ − n ( � � z 0 := 0 , � z n ) ( n > 0) . Pr♦♣♦s✐t✐♦♥ ✭❑✳✱ ✷✵✶✺✮ ❋♦r ❛❧❧ ✇❡✐❣❤ts λ ∈ � H ✱ t❤❡ ❱❡r♠❛ ♠♦❞✉❧❡ M ( λ ) ✐s ✉♥✐s❡r✐❛❧✱ ✇✐t❤ ✉♥✐q✉❡ ❝♦♠♣♦s✐t✐♦♥ s❡r✐❡s M ( λ ) ⊃ M (( − n 1 ) ∗ λ ) ⊃ M (( − n 2 ) ∗ λ ) ⊃ · · · , ✇❤❡r❡ 0 < n 1 < n 2 < · · · ❝♦♠♣r✐s❡ t❤❡ s❡t { n > 0 : λ ( � z n ) = 0 } ✳ ✶✷ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦♣❡rt✐❡s ♦❢ ❱❡r♠❛ ♠♦❞✉❧❡s ✭❝♦♥t✳✮ ❋♦r n ∈ N ✱ ❞❡✜♥❡ ❞✐st✐♥❣✉✐s❤❡❞ ❡❧❡♠❡♥ts � z n ∈ H ✿ n − 1 n − 1 � � θ i ( z 1 ) , θ j ( z 0 z ′ z ′ z ′ n := 0 := 1 , � z n := n − 1 − j ) . i =0 j =0 ❆❧s♦ ❞❡✜♥❡ � z n ❢♦r ♥♦♥✲♣♦s✐t✐✈❡ n ∈ Z ✿ z − n := θ − n ( � � z 0 := 0 , � z n ) ( n > 0) . Pr♦♣♦s✐t✐♦♥ ✭❑✳✱ ✷✵✶✺✮ ❋♦r ❛❧❧ ✇❡✐❣❤ts λ ∈ � H ✱ t❤❡ ❱❡r♠❛ ♠♦❞✉❧❡ M ( λ ) ✐s ✉♥✐s❡r✐❛❧✱ ✇✐t❤ ✉♥✐q✉❡ ❝♦♠♣♦s✐t✐♦♥ s❡r✐❡s M ( λ ) ⊃ M (( − n 1 ) ∗ λ ) ⊃ M (( − n 2 ) ∗ λ ) ⊃ · · · , ✇❤❡r❡ 0 < n 1 < n 2 < · · · ❝♦♠♣r✐s❡ t❤❡ s❡t { n > 0 : λ ( � z n ) = 0 } ✳ ▼♦r❡♦✈❡r✱ [ M ( λ ) : L ( µ )] ≤ 1 ❢♦r ❛❧❧ λ, µ ∈ � H ✳ ✶✷ ✴ ✸✹

❈❛t❡❣♦r② s❛t✐s✜❡s ♠❛♥② ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s ✐❢ ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛ ❜❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥ ✇✐t❤ ✜♥✐t❡ ❢♦r ❛❧❧ ✳ ❚❤✐s ♣❛rt✐t✐♦♥ ❞❡✜♥❡s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ ✱ ✇❤♦s❡ ❝❧❛ss❡s ❛r❡ t❤❡ ✳ ❋♦r ❛♥② s✉❝❤ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♠♦❞✉❧❡ ✭❡✳❣✳✱ ✮✱ ❛❧❧ s✐♠♣❧❡ ❢❛❝t♦rs ♦❢ ❧✐❡ ✐♥ t❤❡ s❛♠❡ ❝❧❛ss✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✏✜♥❡st✑ ♣❛rt✐t✐♦♥✳ ❊q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✿ ❋♦r ♠❛♥② ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✐s ✜♥✐t❡ ❢♦r ❛❧❧ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ♦❢ ❈❛t❡❣♦r② O ❉❡✜♥✐t✐♦♥✿ ●✐✈❡♥ T ⊂ � H ✱ ❧❡t O T ❞❡♥♦t❡ t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♦❜❥❡❝ts✱ ❛❧❧ ♦❢ ✇❤♦s❡ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs L ( µ ) s❛t✐s❢②✿ µ ∈ T ✳ ✶✸ ✴ ✸✹

❚❤✐s ♣❛rt✐t✐♦♥ ❞❡✜♥❡s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ ✱ ✇❤♦s❡ ❝❧❛ss❡s ❛r❡ t❤❡ ✳ ❋♦r ❛♥② s✉❝❤ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♠♦❞✉❧❡ ✭❡✳❣✳✱ ✮✱ ❛❧❧ s✐♠♣❧❡ ❢❛❝t♦rs ♦❢ ❧✐❡ ✐♥ t❤❡ s❛♠❡ ❝❧❛ss✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✏✜♥❡st✑ ♣❛rt✐t✐♦♥✳ ❊q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✿ ❋♦r ♠❛♥② ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✐s ✜♥✐t❡ ❢♦r ❛❧❧ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ♦❢ ❈❛t❡❣♦r② O ❉❡✜♥✐t✐♦♥✿ ●✐✈❡♥ T ⊂ � H ✱ ❧❡t O T ❞❡♥♦t❡ t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♦❜❥❡❝ts✱ ❛❧❧ ♦❢ ✇❤♦s❡ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs L ( µ ) s❛t✐s❢②✿ µ ∈ T ✳ ❈❛t❡❣♦r② O s❛t✐s✜❡s ♠❛♥② ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s ✐❢ ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛ ❜❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥ � � � H = T i , O = O T i , i ∈ I i ∈ I ✇✐t❤ T i ✜♥✐t❡ ❢♦r ❛❧❧ i ✳ ✶✸ ✴ ✸✹

❋♦r ❛♥② s✉❝❤ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♠♦❞✉❧❡ ✭❡✳❣✳✱ ✮✱ ❛❧❧ s✐♠♣❧❡ ❢❛❝t♦rs ♦❢ ❧✐❡ ✐♥ t❤❡ s❛♠❡ ❝❧❛ss✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✏✜♥❡st✑ ♣❛rt✐t✐♦♥✳ ❊q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✿ ❋♦r ♠❛♥② ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✐s ✜♥✐t❡ ❢♦r ❛❧❧ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ♦❢ ❈❛t❡❣♦r② O ❉❡✜♥✐t✐♦♥✿ ●✐✈❡♥ T ⊂ � H ✱ ❧❡t O T ❞❡♥♦t❡ t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♦❜❥❡❝ts✱ ❛❧❧ ♦❢ ✇❤♦s❡ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs L ( µ ) s❛t✐s❢②✿ µ ∈ T ✳ ❈❛t❡❣♦r② O s❛t✐s✜❡s ♠❛♥② ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s ✐❢ ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛ ❜❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥ � � � H = T i , O = O T i , i ∈ I i ∈ I ✇✐t❤ T i ✜♥✐t❡ ❢♦r ❛❧❧ i ✳ ❚❤✐s ♣❛rt✐t✐♦♥ ❞❡✜♥❡s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ � H ✱ ✇❤♦s❡ ❝❧❛ss❡s ❛r❡ t❤❡ T i ✳ ✶✸ ✴ ✸✹

❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✏✜♥❡st✑ ♣❛rt✐t✐♦♥✳ ❊q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✿ ❋♦r ♠❛♥② ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✐s ✜♥✐t❡ ❢♦r ❛❧❧ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ♦❢ ❈❛t❡❣♦r② O ❉❡✜♥✐t✐♦♥✿ ●✐✈❡♥ T ⊂ � H ✱ ❧❡t O T ❞❡♥♦t❡ t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♦❜❥❡❝ts✱ ❛❧❧ ♦❢ ✇❤♦s❡ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs L ( µ ) s❛t✐s❢②✿ µ ∈ T ✳ ❈❛t❡❣♦r② O s❛t✐s✜❡s ♠❛♥② ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s ✐❢ ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛ ❜❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥ � � � H = T i , O = O T i , i ∈ I i ∈ I ✇✐t❤ T i ✜♥✐t❡ ❢♦r ❛❧❧ i ✳ ❚❤✐s ♣❛rt✐t✐♦♥ ❞❡✜♥❡s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ � H ✱ ✇❤♦s❡ ❝❧❛ss❡s ❛r❡ t❤❡ T i ✳ ❋♦r ❛♥② s✉❝❤ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♠♦❞✉❧❡ M ∈ O ✭❡✳❣✳✱ M = M ( λ ) ✮✱ ❛❧❧ s✐♠♣❧❡ ❢❛❝t♦rs ♦❢ M ❧✐❡ ✐♥ t❤❡ s❛♠❡ ❝❧❛ss✳ ✶✸ ✴ ✸✹

❋♦r ♠❛♥② ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ✐s ✜♥✐t❡ ❢♦r ❛❧❧ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ♦❢ ❈❛t❡❣♦r② O ❉❡✜♥✐t✐♦♥✿ ●✐✈❡♥ T ⊂ � H ✱ ❧❡t O T ❞❡♥♦t❡ t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♦❜❥❡❝ts✱ ❛❧❧ ♦❢ ✇❤♦s❡ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs L ( µ ) s❛t✐s❢②✿ µ ∈ T ✳ ❈❛t❡❣♦r② O s❛t✐s✜❡s ♠❛♥② ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s ✐❢ ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛ ❜❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥ � � � H = T i , O = O T i , i ∈ I i ∈ I ✇✐t❤ T i ✜♥✐t❡ ❢♦r ❛❧❧ i ✳ ❚❤✐s ♣❛rt✐t✐♦♥ ❞❡✜♥❡s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ � H ✱ ✇❤♦s❡ ❝❧❛ss❡s ❛r❡ t❤❡ T i ✳ ❋♦r ❛♥② s✉❝❤ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♠♦❞✉❧❡ M ∈ O ✭❡✳❣✳✱ M = M ( λ ) ✮✱ ❛❧❧ s✐♠♣❧❡ ❢❛❝t♦rs ♦❢ M ❧✐❡ ✐♥ t❤❡ s❛♠❡ ❝❧❛ss✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✏✜♥❡st✑ ♣❛rt✐t✐♦♥✳ ❊q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✿ z n ) = 0 } ⊂ � λ � [ λ ] := { ( − n ) ∗ λ : n ∈ Z , λ ( � H. ✶✸ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ♦❢ ❈❛t❡❣♦r② O ❉❡✜♥✐t✐♦♥✿ ●✐✈❡♥ T ⊂ � H ✱ ❧❡t O T ❞❡♥♦t❡ t❤❡ ❢✉❧❧ s✉❜❝❛t❡❣♦r② ♦❢ ♦❜❥❡❝ts✱ ❛❧❧ ♦❢ ✇❤♦s❡ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs L ( µ ) s❛t✐s❢②✿ µ ∈ T ✳ ❈❛t❡❣♦r② O s❛t✐s✜❡s ♠❛♥② ❞❡s✐r❛❜❧❡ ♣r♦♣❡rt✐❡s ✐❢ ✇❡ ❝❛♥ ♦❜t❛✐♥ ❛ ❜❧♦❝❦ ❞❡❝♦♠♣♦s✐t✐♦♥ � � � H = T i , O = O T i , i ∈ I i ∈ I ✇✐t❤ T i ✜♥✐t❡ ❢♦r ❛❧❧ i ✳ ❚❤✐s ♣❛rt✐t✐♦♥ ❞❡✜♥❡s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ♦♥ � H ✱ ✇❤♦s❡ ❝❧❛ss❡s ❛r❡ t❤❡ T i ✳ ❋♦r ❛♥② s✉❝❤ ♣❛rt✐t✐♦♥✱ ❛♥❞ ❛♥② ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♠♦❞✉❧❡ M ∈ O ✭❡✳❣✳✱ M = M ( λ ) ✮✱ ❛❧❧ s✐♠♣❧❡ ❢❛❝t♦rs ♦❢ M ❧✐❡ ✐♥ t❤❡ s❛♠❡ ❝❧❛ss✳ ❚❤❡r❡ ❡①✐sts ❛ ✉♥✐q✉❡ ✏✜♥❡st✑ ♣❛rt✐t✐♦♥✳ ❊q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s✿ z n ) = 0 } ⊂ � λ � [ λ ] := { ( − n ) ∗ λ : n ∈ Z , λ ( � H. ❋♦r ♠❛♥② ❢❛♠✐❧✐❡s ♦❢ tr✐❛♥❣✉❧❛r ●❲❆s ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ [ λ ] ✐s ✜♥✐t❡ ❢♦r ❛❧❧ λ ∈ � H ✳ ✶✸ ✴ ✸✹

❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❯♥❞❡r t❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s✱ ✐s ❛ ❞✐r❡❝t s✉♠ ♦❢ ❜❧♦❝❦s✳ ◆♦✇ ✜① ❛♥❞ s✉♣♣♦s❡ ✳ ❚❤❡♥✱ ✐s ❛ ✜♥✐t❡ ❧❡♥❣t❤✱ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② ✇✐t❤ ❡♥♦✉❣❤ ♣r♦❥❡❝t✐✈❡s ❛♥❞ ✐♥❥❡❝t✐✈❡s✳ ❚❤❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♣r♦❥❡❝t✐✈❡s ✐♥ ❛r❡ t❤❡ ♣r♦❥❡❝t✐✈❡ ❝♦✈❡rs ♦❢ t❤❡ s✐♠♣❧❡ ♠♦❞✉❧❡s ✳ ❚❤❡r❡ ✐s ❛ ❡①❛❝t✱ ❝♦♥tr❛✈❛r✐❛♥t ❞✉❛❧✐t② ❡♥❞♦❢✉♥❝t♦r ♦❢ t❤❛t ✏✜①❡s✑ ✱ ❛♥❞ s❡♥❞s t♦ t❤❡ ✐♥❥❡❝t✐✈❡ ❤✉❧❧ ♦❢ ✳ ✲ ❢♦r ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✱ q✉❛s✐✲❤❡r❡❞✐t❛r② ❛❧❣❡❜r❛ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② s✐♠♣❧❡s ❍❡♥❝❡❢♦rt❤✱ ❛ss✉♠❡✿ ✶ ❚❤❡r❡ ❞♦ ♥♦t ❡①✐st λ ∈ � H ❛♥❞ n > 0 s✉❝❤ t❤❛t λ ≡ λ ◦ θ n ✳ ✷ ❚❤❡ s❡t [ λ ] ✐s ✜♥✐t❡ ❢♦r ❛❧❧ λ ∈ � H ✳ ✶✹ ✴ ✸✹

✐s ❛ ✜♥✐t❡ ❧❡♥❣t❤✱ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② ✇✐t❤ ❡♥♦✉❣❤ ♣r♦❥❡❝t✐✈❡s ❛♥❞ ✐♥❥❡❝t✐✈❡s✳ ❚❤❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♣r♦❥❡❝t✐✈❡s ✐♥ ❛r❡ t❤❡ ♣r♦❥❡❝t✐✈❡ ❝♦✈❡rs ♦❢ t❤❡ s✐♠♣❧❡ ♠♦❞✉❧❡s ✳ ❚❤❡r❡ ✐s ❛ ❡①❛❝t✱ ❝♦♥tr❛✈❛r✐❛♥t ❞✉❛❧✐t② ❡♥❞♦❢✉♥❝t♦r ♦❢ t❤❛t ✏✜①❡s✑ ✱ ❛♥❞ s❡♥❞s t♦ t❤❡ ✐♥❥❡❝t✐✈❡ ❤✉❧❧ ♦❢ ✳ ✲ ❢♦r ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✱ q✉❛s✐✲❤❡r❡❞✐t❛r② ❛❧❣❡❜r❛ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② s✐♠♣❧❡s ❍❡♥❝❡❢♦rt❤✱ ❛ss✉♠❡✿ ✶ ❚❤❡r❡ ❞♦ ♥♦t ❡①✐st λ ∈ � H ❛♥❞ n > 0 s✉❝❤ t❤❛t λ ≡ λ ◦ θ n ✳ ✷ ❚❤❡ s❡t [ λ ] ✐s ✜♥✐t❡ ❢♦r ❛❧❧ λ ∈ � H ✳ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❯♥❞❡r t❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s✱ O = � H O [ λ ] ✐s ❛ ❞✐r❡❝t s✉♠ ♦❢ [ λ ] ⊂ � ❜❧♦❝❦s✳ ◆♦✇ ✜① λ ∈ � H ❛♥❞ s✉♣♣♦s❡ [ λ ] = { λ 1 , . . . , λ n } ✳ ❚❤❡♥✱ ✶✹ ✴ ✸✹

❚❤❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♣r♦❥❡❝t✐✈❡s ✐♥ ❛r❡ t❤❡ ♣r♦❥❡❝t✐✈❡ ❝♦✈❡rs ♦❢ t❤❡ s✐♠♣❧❡ ♠♦❞✉❧❡s ✳ ❚❤❡r❡ ✐s ❛ ❡①❛❝t✱ ❝♦♥tr❛✈❛r✐❛♥t ❞✉❛❧✐t② ❡♥❞♦❢✉♥❝t♦r ♦❢ t❤❛t ✏✜①❡s✑ ✱ ❛♥❞ s❡♥❞s t♦ t❤❡ ✐♥❥❡❝t✐✈❡ ❤✉❧❧ ♦❢ ✳ ✲ ❢♦r ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✱ q✉❛s✐✲❤❡r❡❞✐t❛r② ❛❧❣❡❜r❛ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② s✐♠♣❧❡s ❍❡♥❝❡❢♦rt❤✱ ❛ss✉♠❡✿ ✶ ❚❤❡r❡ ❞♦ ♥♦t ❡①✐st λ ∈ � H ❛♥❞ n > 0 s✉❝❤ t❤❛t λ ≡ λ ◦ θ n ✳ ✷ ❚❤❡ s❡t [ λ ] ✐s ✜♥✐t❡ ❢♦r ❛❧❧ λ ∈ � H ✳ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❯♥❞❡r t❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s✱ O = � H O [ λ ] ✐s ❛ ❞✐r❡❝t s✉♠ ♦❢ [ λ ] ⊂ � ❜❧♦❝❦s✳ ◆♦✇ ✜① λ ∈ � H ❛♥❞ s✉♣♣♦s❡ [ λ ] = { λ 1 , . . . , λ n } ✳ ❚❤❡♥✱ O [ λ ] ✐s ❛ ✜♥✐t❡ ❧❡♥❣t❤✱ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② ✇✐t❤ ❡♥♦✉❣❤ ♣r♦❥❡❝t✐✈❡s ❛♥❞ ✐♥❥❡❝t✐✈❡s✳ ✶✹ ✴ ✸✹

❚❤❡r❡ ✐s ❛ ❡①❛❝t✱ ❝♦♥tr❛✈❛r✐❛♥t ❞✉❛❧✐t② ❡♥❞♦❢✉♥❝t♦r ♦❢ t❤❛t ✏✜①❡s✑ ✱ ❛♥❞ s❡♥❞s t♦ t❤❡ ✐♥❥❡❝t✐✈❡ ❤✉❧❧ ♦❢ ✳ ✲ ❢♦r ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✱ q✉❛s✐✲❤❡r❡❞✐t❛r② ❛❧❣❡❜r❛ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② s✐♠♣❧❡s ❍❡♥❝❡❢♦rt❤✱ ❛ss✉♠❡✿ ✶ ❚❤❡r❡ ❞♦ ♥♦t ❡①✐st λ ∈ � H ❛♥❞ n > 0 s✉❝❤ t❤❛t λ ≡ λ ◦ θ n ✳ ✷ ❚❤❡ s❡t [ λ ] ✐s ✜♥✐t❡ ❢♦r ❛❧❧ λ ∈ � H ✳ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❯♥❞❡r t❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s✱ O = � H O [ λ ] ✐s ❛ ❞✐r❡❝t s✉♠ ♦❢ [ λ ] ⊂ � ❜❧♦❝❦s✳ ◆♦✇ ✜① λ ∈ � H ❛♥❞ s✉♣♣♦s❡ [ λ ] = { λ 1 , . . . , λ n } ✳ ❚❤❡♥✱ O [ λ ] ✐s ❛ ✜♥✐t❡ ❧❡♥❣t❤✱ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② ✇✐t❤ ❡♥♦✉❣❤ ♣r♦❥❡❝t✐✈❡s ❛♥❞ ✐♥❥❡❝t✐✈❡s✳ ❚❤❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♣r♦❥❡❝t✐✈❡s ✐♥ O [ λ ] ❛r❡ t❤❡ ♣r♦❥❡❝t✐✈❡ ❝♦✈❡rs P ( µ ) ♦❢ t❤❡ s✐♠♣❧❡ ♠♦❞✉❧❡s { L ( µ ) : µ ∈ [ λ ] } ✳ ✶✹ ✴ ✸✹

✲ ❢♦r ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✱ q✉❛s✐✲❤❡r❡❞✐t❛r② ❛❧❣❡❜r❛ ✳ ❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② s✐♠♣❧❡s ❍❡♥❝❡❢♦rt❤✱ ❛ss✉♠❡✿ ✶ ❚❤❡r❡ ❞♦ ♥♦t ❡①✐st λ ∈ � H ❛♥❞ n > 0 s✉❝❤ t❤❛t λ ≡ λ ◦ θ n ✳ ✷ ❚❤❡ s❡t [ λ ] ✐s ✜♥✐t❡ ❢♦r ❛❧❧ λ ∈ � H ✳ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❯♥❞❡r t❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s✱ O = � H O [ λ ] ✐s ❛ ❞✐r❡❝t s✉♠ ♦❢ [ λ ] ⊂ � ❜❧♦❝❦s✳ ◆♦✇ ✜① λ ∈ � H ❛♥❞ s✉♣♣♦s❡ [ λ ] = { λ 1 , . . . , λ n } ✳ ❚❤❡♥✱ O [ λ ] ✐s ❛ ✜♥✐t❡ ❧❡♥❣t❤✱ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② ✇✐t❤ ❡♥♦✉❣❤ ♣r♦❥❡❝t✐✈❡s ❛♥❞ ✐♥❥❡❝t✐✈❡s✳ ❚❤❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♣r♦❥❡❝t✐✈❡s ✐♥ O [ λ ] ❛r❡ t❤❡ ♣r♦❥❡❝t✐✈❡ ❝♦✈❡rs P ( µ ) ♦❢ t❤❡ s✐♠♣❧❡ ♠♦❞✉❧❡s { L ( µ ) : µ ∈ [ λ ] } ✳ ❚❤❡r❡ ✐s ❛ ❡①❛❝t✱ ❝♦♥tr❛✈❛r✐❛♥t ❞✉❛❧✐t② ❡♥❞♦❢✉♥❝t♦r F ♦❢ O [ λ ] t❤❛t ✏✜①❡s✑ L ( µ ) ✱ ❛♥❞ s❡♥❞s P ( µ ) t♦ t❤❡ ✐♥❥❡❝t✐✈❡ ❤✉❧❧ ♦❢ L ( µ ) ✳ ✶✹ ✴ ✸✹

❋✐rst r❡s✉❧ts ❈❧❛ss✐❝❛❧ ❛♥❞ q✉❛♥t✉♠ ❣❡♥❡r❛❧✐③❡❞ ❲❡②❧ ❛❧❣❡❜r❛s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❇❛s✐❝ ♣r♦♣❡rt✐❡s ♦❢ ❈❛t❡❣♦r② O ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❇❧♦❝❦s ✇✐t❤ ✜♥✐t❡❧② ♠❛♥② s✐♠♣❧❡s ❍❡♥❝❡❢♦rt❤✱ ❛ss✉♠❡✿ ✶ ❚❤❡r❡ ❞♦ ♥♦t ❡①✐st λ ∈ � H ❛♥❞ n > 0 s✉❝❤ t❤❛t λ ≡ λ ◦ θ n ✳ ✷ ❚❤❡ s❡t [ λ ] ✐s ✜♥✐t❡ ❢♦r ❛❧❧ λ ∈ � H ✳ ❚❤❡♦r❡♠ ✭❑✳✱ ✷✵✶✺✮ ❯♥❞❡r t❤❡ ❛❜♦✈❡ ❛ss✉♠♣t✐♦♥s✱ O = � H O [ λ ] ✐s ❛ ❞✐r❡❝t s✉♠ ♦❢ [ λ ] ⊂ � ❜❧♦❝❦s✳ ◆♦✇ ✜① λ ∈ � H ❛♥❞ s✉♣♣♦s❡ [ λ ] = { λ 1 , . . . , λ n } ✳ ❚❤❡♥✱ O [ λ ] ✐s ❛ ✜♥✐t❡ ❧❡♥❣t❤✱ ❛❜❡❧✐❛♥ ❝❛t❡❣♦r② ✇✐t❤ ❡♥♦✉❣❤ ♣r♦❥❡❝t✐✈❡s ❛♥❞ ✐♥❥❡❝t✐✈❡s✳ ❚❤❡ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ♣r♦❥❡❝t✐✈❡s ✐♥ O [ λ ] ❛r❡ t❤❡ ♣r♦❥❡❝t✐✈❡ ❝♦✈❡rs P ( µ ) ♦❢ t❤❡ s✐♠♣❧❡ ♠♦❞✉❧❡s { L ( µ ) : µ ∈ [ λ ] } ✳ ❚❤❡r❡ ✐s ❛ ❡①❛❝t✱ ❝♦♥tr❛✈❛r✐❛♥t ❞✉❛❧✐t② ❡♥❞♦❢✉♥❝t♦r F ♦❢ O [ λ ] t❤❛t ✏✜①❡s✑ L ( µ ) ✱ ❛♥❞ s❡♥❞s P ( µ ) t♦ t❤❡ ✐♥❥❡❝t✐✈❡ ❤✉❧❧ ♦❢ L ( µ ) ✳ O [ λ ] ∼ = A [ λ ] ✲ Mod ❢♦r ❛ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✱ q✉❛s✐✲❤❡r❡❞✐t❛r② ❛❧❣❡❜r❛ A [ λ ] := End O ( ⊕ n j =1 P ( λ j )) op ✳ ✶✹ ✴ ✸✹

✷ ❯♥❞❡rst❛♥❞ t✐❧t✐♥❣ ♦❜❥❡❝ts ✐♥ t❤❡ ❜❧♦❝❦✳ ✸ ❯♥❞❡rst❛♥❞ ❛❧❧ ❊①t✬s ❜❡t✇❡❡♥ s✐♠♣❧❡s✱ ❱❡r♠❛s✱ ❛♥❞ ♣r♦❥❡❝t✐✈❡s✳ ✹ ◗✉❛❞r❛t✐❝ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛ ✳ ✺ ◗✉❛❞r❛t✐❝ ❞✉❛❧ ♦❢ ✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❋✉t✉r❡ ❣♦❛❧s ❘❡st ♦❢ t❤❡ t❛❧❦✿ ❲♦r❦ ✐♥ ❛ ✏✜♥✐t❡✑ ❜❧♦❝❦ O [ λ ] ✳ ✶ ❯♥❞❡rst❛♥❞ t❤❡ ❞❡t❛✐❧❡❞ str✉❝t✉r❡ ♦❢ ♣r♦❥❡❝t✐✈❡ ♦❜❥❡❝ts ✭❡✳❣✳✱ ❝❧❛ss✐❢② ❛❧❧ s✉❜♠♦❞✉❧❡s✮✱ ❛♥❞ ♠❛♣s ❜❡t✇❡❡♥ t❤❡♠✳ ✶✺ ✴ ✸✹

✸ ❯♥❞❡rst❛♥❞ ❛❧❧ ❊①t✬s ❜❡t✇❡❡♥ s✐♠♣❧❡s✱ ❱❡r♠❛s✱ ❛♥❞ ♣r♦❥❡❝t✐✈❡s✳ ✹ ◗✉❛❞r❛t✐❝ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛ ✳ ✺ ◗✉❛❞r❛t✐❝ ❞✉❛❧ ♦❢ ✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❋✉t✉r❡ ❣♦❛❧s ❘❡st ♦❢ t❤❡ t❛❧❦✿ ❲♦r❦ ✐♥ ❛ ✏✜♥✐t❡✑ ❜❧♦❝❦ O [ λ ] ✳ ✶ ❯♥❞❡rst❛♥❞ t❤❡ ❞❡t❛✐❧❡❞ str✉❝t✉r❡ ♦❢ ♣r♦❥❡❝t✐✈❡ ♦❜❥❡❝ts ✭❡✳❣✳✱ ❝❧❛ss✐❢② ❛❧❧ s✉❜♠♦❞✉❧❡s✮✱ ❛♥❞ ♠❛♣s ❜❡t✇❡❡♥ t❤❡♠✳ ✷ ❯♥❞❡rst❛♥❞ t✐❧t✐♥❣ ♦❜❥❡❝ts ✐♥ t❤❡ ❜❧♦❝❦✳ ✶✺ ✴ ✸✹

✹ ◗✉❛❞r❛t✐❝ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛ ✳ ✺ ◗✉❛❞r❛t✐❝ ❞✉❛❧ ♦❢ ✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❋✉t✉r❡ ❣♦❛❧s ❘❡st ♦❢ t❤❡ t❛❧❦✿ ❲♦r❦ ✐♥ ❛ ✏✜♥✐t❡✑ ❜❧♦❝❦ O [ λ ] ✳ ✶ ❯♥❞❡rst❛♥❞ t❤❡ ❞❡t❛✐❧❡❞ str✉❝t✉r❡ ♦❢ ♣r♦❥❡❝t✐✈❡ ♦❜❥❡❝ts ✭❡✳❣✳✱ ❝❧❛ss✐❢② ❛❧❧ s✉❜♠♦❞✉❧❡s✮✱ ❛♥❞ ♠❛♣s ❜❡t✇❡❡♥ t❤❡♠✳ ✷ ❯♥❞❡rst❛♥❞ t✐❧t✐♥❣ ♦❜❥❡❝ts ✐♥ t❤❡ ❜❧♦❝❦✳ ✸ ❯♥❞❡rst❛♥❞ ❛❧❧ ❊①t✬s ❜❡t✇❡❡♥ s✐♠♣❧❡s✱ ❱❡r♠❛s✱ ❛♥❞ ♣r♦❥❡❝t✐✈❡s✳ ✶✺ ✴ ✸✹

❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❋✉t✉r❡ ❣♦❛❧s ❘❡st ♦❢ t❤❡ t❛❧❦✿ ❲♦r❦ ✐♥ ❛ ✏✜♥✐t❡✑ ❜❧♦❝❦ O [ λ ] ✳ ✶ ❯♥❞❡rst❛♥❞ t❤❡ ❞❡t❛✐❧❡❞ str✉❝t✉r❡ ♦❢ ♣r♦❥❡❝t✐✈❡ ♦❜❥❡❝ts ✭❡✳❣✳✱ ❝❧❛ss✐❢② ❛❧❧ s✉❜♠♦❞✉❧❡s✮✱ ❛♥❞ ♠❛♣s ❜❡t✇❡❡♥ t❤❡♠✳ ✷ ❯♥❞❡rst❛♥❞ t✐❧t✐♥❣ ♦❜❥❡❝ts ✐♥ t❤❡ ❜❧♦❝❦✳ ✸ ❯♥❞❡rst❛♥❞ ❛❧❧ ❊①t✬s ❜❡t✇❡❡♥ s✐♠♣❧❡s✱ ❱❡r♠❛s✱ ❛♥❞ ♣r♦❥❡❝t✐✈❡s✳ ✹ ◗✉❛❞r❛t✐❝ ♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❛❧❣❡❜r❛ A [ λ ] ✳ ✺ ◗✉❛❞r❛t✐❝ ❞✉❛❧ ♦❢ A [ λ ] ✳ ✶✺ ✴ ✸✹

❉✉❛❧❧②✱ ❡✈❡r② ❤❛s ❛ ✜♥✐t❡ ✜❧tr❛t✐♦♥ ✇✐t❤ s✉❝❝❡ss✐✈❡ s✉❜q✉♦t✐❡♥ts ❢♦r ✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❱❡r♠❛ ✢❛❣ ♦❢ ♣r♦❥❡❝t✐✈❡s ◆♦t❛t✐♦♥✿ ❱❡r♠❛ ♠♦❞✉❧❡s ❛r❡ ✉♥✐s❡r✐❛❧✱ s♦ s✉♣♣♦s❡ M ( λ n ) ⊃ M ( λ n − 1 ) ⊃ · · · ⊃ M ( λ 1 ) ⊃ 0 , ✇✐t❤ s✉❜q✉♦t✐❡♥ts L ( λ n ) , . . . , L ( λ 1 ) r❡s♣❡❝t✐✈❡❧②✳ ❚❤✉s✱ λ n > λ n − 1 > · · · > λ 1 ✳ ◆♦✇ ❞❡✜♥❡ M j := M ( λ j ) , L j := L ( λ j ) , P j := P ( λ j ) . Pr♦♣♦s✐t✐♦♥ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❋♦r ❛❧❧ 1 ≤ j ≤ n ✱ M j ❤❛s ❛ ✜♥✐t❡ ✜❧tr❛t✐♦♥ M j ⊃ M j − 1 ⊃ · · · ⊃ M 1 ⊃ 0 , ✇✐t❤ s✉❝❝❡ss✐✈❡ s✉❜q✉♦t✐❡♥ts L k ❢♦r 1 ≤ k ≤ j ✳ ✶✻ ✴ ✸✹

❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❱❡r♠❛ ✢❛❣ ♦❢ ♣r♦❥❡❝t✐✈❡s ◆♦t❛t✐♦♥✿ ❱❡r♠❛ ♠♦❞✉❧❡s ❛r❡ ✉♥✐s❡r✐❛❧✱ s♦ s✉♣♣♦s❡ M ( λ n ) ⊃ M ( λ n − 1 ) ⊃ · · · ⊃ M ( λ 1 ) ⊃ 0 , ✇✐t❤ s✉❜q✉♦t✐❡♥ts L ( λ n ) , . . . , L ( λ 1 ) r❡s♣❡❝t✐✈❡❧②✳ ❚❤✉s✱ λ n > λ n − 1 > · · · > λ 1 ✳ ◆♦✇ ❞❡✜♥❡ M j := M ( λ j ) , L j := L ( λ j ) , P j := P ( λ j ) . Pr♦♣♦s✐t✐♦♥ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❋♦r ❛❧❧ 1 ≤ j ≤ n ✱ M j ❤❛s ❛ ✜♥✐t❡ ✜❧tr❛t✐♦♥ M j ⊃ M j − 1 ⊃ · · · ⊃ M 1 ⊃ 0 , ✇✐t❤ s✉❝❝❡ss✐✈❡ s✉❜q✉♦t✐❡♥ts L k ❢♦r 1 ≤ k ≤ j ✳ ❉✉❛❧❧②✱ ❡✈❡r② P j ❤❛s ❛ ✜♥✐t❡ ✜❧tr❛t✐♦♥ P j ⊃ P j +1 ⊃ · · · ⊃ P n ⊃ 0 , ✇✐t❤ s✉❝❝❡ss✐✈❡ s✉❜q✉♦t✐❡♥ts M k ❢♦r j ≤ k ≤ n ✳ ✶✻ ✴ ✸✹

■❢ ✱ t❤❡♥ t❤❡ ❱❡r♠❛ ♠♦❞✉❧❡ ❤❛s ❛ ♣r♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥✿ ❚❤❡ ♣r♦♦❢s ✉s❡ t❤❡ ❡①♣❧✐❝✐t ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐✈❡ ♠♦❞✉❧❡ ✱ ❛s t❤❡ ✲❞✐r❡❝t s✉♠♠❛♥❞ ♦❢ t❤❡ ✲♠♦❞✉❧❡ ✳ ✭❍❛s ✳✮ ❆❧s♦ ✉s❡ st❛♥❞❛r❞ ❢❛❝ts ✐♥ t❤❡ ❤✐❣❤❡st ✇❡✐❣❤t ❝❛t❡❣♦r② ✿ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❘❡s♦❧✉t✐♦♥ ♦❢ ❤✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡s ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❙✉♣♣♦s❡ 0 < j < k ≤ n ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♣r♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡ ❤✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡ M k /M j ✐♥ O ✿ 0 → P j +1 → P j ⊕ P k +1 → P k → M k /M j → 0 , ✇✐t❤ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ t❤❛t P n +1 = 0 ✳ ✶✼ ✴ ✸✹

❚❤❡ ♣r♦♦❢s ✉s❡ t❤❡ ❡①♣❧✐❝✐t ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐✈❡ ♠♦❞✉❧❡ ✱ ❛s t❤❡ ✲❞✐r❡❝t s✉♠♠❛♥❞ ♦❢ t❤❡ ✲♠♦❞✉❧❡ ✳ ✭❍❛s ✳✮ ❆❧s♦ ✉s❡ st❛♥❞❛r❞ ❢❛❝ts ✐♥ t❤❡ ❤✐❣❤❡st ✇❡✐❣❤t ❝❛t❡❣♦r② ✿ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❘❡s♦❧✉t✐♦♥ ♦❢ ❤✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡s ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❙✉♣♣♦s❡ 0 < j < k ≤ n ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♣r♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡ ❤✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡ M k /M j ✐♥ O ✿ 0 → P j +1 → P j ⊕ P k +1 → P k → M k /M j → 0 , ✇✐t❤ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ t❤❛t P n +1 = 0 ✳ ■❢ 0 = j < k ≤ n ✱ t❤❡♥ t❤❡ ❱❡r♠❛ ♠♦❞✉❧❡ M k ❤❛s ❛ ♣r♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥✿ 0 → P k +1 → P k → M k → 0 , ∀ 1 ≤ k ≤ n. ✶✼ ✴ ✸✹

❆❧s♦ ✉s❡ st❛♥❞❛r❞ ❢❛❝ts ✐♥ t❤❡ ❤✐❣❤❡st ✇❡✐❣❤t ❝❛t❡❣♦r② ✿ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❘❡s♦❧✉t✐♦♥ ♦❢ ❤✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡s ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❙✉♣♣♦s❡ 0 < j < k ≤ n ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♣r♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡ ❤✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡ M k /M j ✐♥ O ✿ 0 → P j +1 → P j ⊕ P k +1 → P k → M k /M j → 0 , ✇✐t❤ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ t❤❛t P n +1 = 0 ✳ ■❢ 0 = j < k ≤ n ✱ t❤❡♥ t❤❡ ❱❡r♠❛ ♠♦❞✉❧❡ M k ❤❛s ❛ ♣r♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥✿ 0 → P k +1 → P k → M k → 0 , ∀ 1 ≤ k ≤ n. ❚❤❡ ♣r♦♦❢s ✉s❡ t❤❡ ❡①♣❧✐❝✐t ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐✈❡ ♠♦❞✉❧❡ P j ✱ ❛s t❤❡ [ λ ] ✲❞✐r❡❝t s✉♠♠❛♥❞ ♦❢ t❤❡ A ✲♠♦❞✉❧❡ A/ ( Au λ n − λ j +1 + A · ker( λ j )) ∈ O ✳ ✭❍❛s 1 P j ∈ ( P j ) λ j ✳✮ ✶✼ ✴ ✸✹

❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❘❡s♦❧✉t✐♦♥ ♦❢ ❤✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡s ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❙✉♣♣♦s❡ 0 < j < k ≤ n ✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s ❛ ♣r♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥ ♦❢ t❤❡ ❤✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡ M k /M j ✐♥ O ✿ 0 → P j +1 → P j ⊕ P k +1 → P k → M k /M j → 0 , ✇✐t❤ t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ t❤❛t P n +1 = 0 ✳ ■❢ 0 = j < k ≤ n ✱ t❤❡♥ t❤❡ ❱❡r♠❛ ♠♦❞✉❧❡ M k ❤❛s ❛ ♣r♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥✿ 0 → P k +1 → P k → M k → 0 , ∀ 1 ≤ k ≤ n. ❚❤❡ ♣r♦♦❢s ✉s❡ t❤❡ ❡①♣❧✐❝✐t ❝♦♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣r♦❥❡❝t✐✈❡ ♠♦❞✉❧❡ P j ✱ ❛s t❤❡ [ λ ] ✲❞✐r❡❝t s✉♠♠❛♥❞ ♦❢ t❤❡ A ✲♠♦❞✉❧❡ A/ ( Au λ n − λ j +1 + A · ker( λ j )) ∈ O ✳ ✭❍❛s 1 P j ∈ ( P j ) λ j ✳✮ ❆❧s♦ ✉s❡ st❛♥❞❛r❞ ❢❛❝ts ✐♥ t❤❡ ❤✐❣❤❡st ✇❡✐❣❤t ❝❛t❡❣♦r② O [ λ ] ✿ dim Hom O ( P j , − ) = [ − : L j ] , dim Hom O ( P j , L k ) = δ j,k . ✶✼ ✴ ✸✹

✷ ✳ ✸ ❋♦r ❛❧❧ ❛♥❞ ✱ ✐❢ ❀ ✐❢ ❀ ✐❢ ❛♥❞ ❀ ♦t❤❡r✇✐s❡✳ ❯s❡s ❝♦♥str✉❝t✐♦♥ ❛♥❞ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs ♦❢ ✱ ❛♥❞ ❤♦♠♦❧♦❣✐❝❛❧ ❛r❣✉♠❡♥ts✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❊①t✲❢♦r♠✉❧❛s ❈❛♥ ❝♦♠♣✉t❡ ❧♦t ♦❢ ❤♦♠♦❧♦❣✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❜❧♦❝❦✿ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❋✐① 1 ≤ j < k ≤ n + 1 ❛♥❞ 0 ≤ s < r ≤ n ✳ ❚❤❡♥✱ dim Ext l O ( M r , P j /P k ) = δ l, 0 1 ( r < k ) + δ l, 1 1 ( r < j ) ✳ ✶ ✶✽ ✴ ✸✹

✸ ❋♦r ❛❧❧ ❛♥❞ ✱ ✐❢ ❀ ✐❢ ❀ ✐❢ ❛♥❞ ❀ ♦t❤❡r✇✐s❡✳ ❯s❡s ❝♦♥str✉❝t✐♦♥ ❛♥❞ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs ♦❢ ✱ ❛♥❞ ❤♦♠♦❧♦❣✐❝❛❧ ❛r❣✉♠❡♥ts✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❊①t✲❢♦r♠✉❧❛s ❈❛♥ ❝♦♠♣✉t❡ ❧♦t ♦❢ ❤♦♠♦❧♦❣✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❜❧♦❝❦✿ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❋✐① 1 ≤ j < k ≤ n + 1 ❛♥❞ 0 ≤ s < r ≤ n ✳ ❚❤❡♥✱ dim Ext l O ( M r , P j /P k ) = δ l, 0 1 ( r < k ) + δ l, 1 1 ( r < j ) ✳ ✶ dim Ext l O ( P j /P k , M r /M s ) ✷ = δ l, 0 1 ( s < j ≤ r ) + δ l, 1 1 ( s < k ≤ r ) ✳ ✶✽ ✴ ✸✹

❯s❡s ❝♦♥str✉❝t✐♦♥ ❛♥❞ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs ♦❢ ✱ ❛♥❞ ❤♦♠♦❧♦❣✐❝❛❧ ❛r❣✉♠❡♥ts✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❊①t✲❢♦r♠✉❧❛s ❈❛♥ ❝♦♠♣✉t❡ ❧♦t ♦❢ ❤♦♠♦❧♦❣✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❜❧♦❝❦✿ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❋✐① 1 ≤ j < k ≤ n + 1 ❛♥❞ 0 ≤ s < r ≤ n ✳ ❚❤❡♥✱ dim Ext l O ( M r , P j /P k ) = δ l, 0 1 ( r < k ) + δ l, 1 1 ( r < j ) ✳ ✶ dim Ext l O ( P j /P k , M r /M s ) ✷ = δ l, 0 1 ( s < j ≤ r ) + δ l, 1 1 ( s < k ≤ r ) ✳ ✸ ❋♦r ❛❧❧ 1 ≤ j, k ≤ n ❛♥❞ l > 0 ✱   F , ✐❢ | j − k | = l = 0 ❀     F , ✐❢ | j − k | = l = 1 ❀ Ext l O ( L j , L k ) =  F , ✐❢ j = k � = 1 ❛♥❞ l = 2 ❀     0 , ♦t❤❡r✇✐s❡✳ ✶✽ ✴ ✸✹

❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❊①t✲❢♦r♠✉❧❛s ❈❛♥ ❝♦♠♣✉t❡ ❧♦t ♦❢ ❤♦♠♦❧♦❣✐❝❛❧ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t t❤❡ ❜❧♦❝❦✿ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❋✐① 1 ≤ j < k ≤ n + 1 ❛♥❞ 0 ≤ s < r ≤ n ✳ ❚❤❡♥✱ dim Ext l O ( M r , P j /P k ) = δ l, 0 1 ( r < k ) + δ l, 1 1 ( r < j ) ✳ ✶ dim Ext l O ( P j /P k , M r /M s ) ✷ = δ l, 0 1 ( s < j ≤ r ) + δ l, 1 1 ( s < k ≤ r ) ✳ ✸ ❋♦r ❛❧❧ 1 ≤ j, k ≤ n ❛♥❞ l > 0 ✱   F , ✐❢ | j − k | = l = 0 ❀     F , ✐❢ | j − k | = l = 1 ❀ Ext l O ( L j , L k ) =  F , ✐❢ j = k � = 1 ❛♥❞ l = 2 ❀     0 , ♦t❤❡r✇✐s❡✳ ❯s❡s ❝♦♥str✉❝t✐♦♥ ❛♥❞ ❏♦r❞❛♥✲❍♦❧❞❡r ❢❛❝t♦rs ♦❢ P j ✱ ❛♥❞ ❤♦♠♦❧♦❣✐❝❛❧ ❛r❣✉♠❡♥ts✳ ✶✽ ✴ ✸✹

❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr♦❥❡❝t✐✈❡s✱ ❱❡r♠❛s✱ ❛♥❞ ❨♦✉♥❣ ❞✐❛❣r❛♠s ❍✐❣❤❡st ✇❡✐❣❤t ♠♦❞✉❧❡s ✭♦r t❤❡✐r ❝♦♠♣♦s✐t✐♦♥ s❡r✐❡s✮ ❝❛♥ ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❨♦✉♥❣ ❞✐❛❣r❛♠s✿ P j /P k = k k − 1 k − 2 ··· j k − 1 k − 2 ··· ··· j − 1 ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ M k /M j = j +1 4 3 2 1 3 2 1 2 1 1 k k − 1 ··· j +1 F ( M k /M j ) = ✶✾ ✴ ✸✹

❋✐rst st✉❞② t❤❡ ❧❛r❣❡r ❛❧❣❡❜r❛ ✇❤❡r❡ Pr♦♣♦s✐t✐♦♥ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ●✐✈❡♥ ✐♥t❡❣❡rs ✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ ✐♥ t❤❡ ❜❧♦❝❦ ✿ ■♥ ✏♣✐❝t✉r❡s✑✱ ❛❞❞s ❛ ✭t♦♣♠♦st✮ r♦✇ t♦ t❤❡ ❞✐❛❣r❛♠✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ❲❛♥t t♦ st✉❞② t❤❡ ❛❧❣❡❜r❛ � A [ λ ] = End O ( P [ λ ] ) op , ✇❤❡r❡ P [ λ ] = P j . 1 ≤ j ≤ n ✷✵ ✴ ✸✹

Pr♦♣♦s✐t✐♦♥ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ●✐✈❡♥ ✐♥t❡❣❡rs ✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ ✐♥ t❤❡ ❜❧♦❝❦ ✿ ■♥ ✏♣✐❝t✉r❡s✑✱ ❛❞❞s ❛ ✭t♦♣♠♦st✮ r♦✇ t♦ t❤❡ ❞✐❛❣r❛♠✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ❲❛♥t t♦ st✉❞② t❤❡ ❛❧❣❡❜r❛ � A [ λ ] = End O ( P [ λ ] ) op , ✇❤❡r❡ P [ λ ] = P j . 1 ≤ j ≤ n ❋✐rst st✉❞② t❤❡ ❧❛r❣❡r ❛❧❣❡❜r❛ � A [ λ ] = End O ( � � P [ λ ] ) op , � ✇❤❡r❡ P [ λ ] = P j /P k . 1 ≤ j<k ≤ n +1 ✷✵ ✴ ✸✹

■♥ ✏♣✐❝t✉r❡s✑✱ ❛❞❞s ❛ ✭t♦♣♠♦st✮ r♦✇ t♦ t❤❡ ❞✐❛❣r❛♠✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ❲❛♥t t♦ st✉❞② t❤❡ ❛❧❣❡❜r❛ � A [ λ ] = End O ( P [ λ ] ) op , ✇❤❡r❡ P [ λ ] = P j . 1 ≤ j ≤ n ❋✐rst st✉❞② t❤❡ ❧❛r❣❡r ❛❧❣❡❜r❛ � A [ λ ] = End O ( � � P [ λ ] ) op , � ✇❤❡r❡ P [ λ ] = P j /P k . 1 ≤ j<k ≤ n +1 Pr♦♣♦s✐t✐♦♥ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ●✐✈❡♥ ✐♥t❡❣❡rs 1 ≤ j ≤ k ≤ n ✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ ✐♥ t❤❡ ❜❧♦❝❦ O [ λ ] ✿ f ++ j,k 0 → P j /P k − → P j +1 /P k +1 → F ( M k /M j ) → 0 . ✷✵ ✴ ✸✹

❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ❲❛♥t t♦ st✉❞② t❤❡ ❛❧❣❡❜r❛ � A [ λ ] = End O ( P [ λ ] ) op , ✇❤❡r❡ P [ λ ] = P j . 1 ≤ j ≤ n ❋✐rst st✉❞② t❤❡ ❧❛r❣❡r ❛❧❣❡❜r❛ � A [ λ ] = End O ( � � P [ λ ] ) op , � ✇❤❡r❡ P [ λ ] = P j /P k . 1 ≤ j<k ≤ n +1 Pr♦♣♦s✐t✐♦♥ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ●✐✈❡♥ ✐♥t❡❣❡rs 1 ≤ j ≤ k ≤ n ✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ s❤♦rt ❡①❛❝t s❡q✉❡♥❝❡ ✐♥ t❤❡ ❜❧♦❝❦ O [ λ ] ✿ f ++ j,k 0 → P j /P k − → P j +1 /P k +1 → F ( M k /M j ) → 0 . ■♥ ✏♣✐❝t✉r❡s✑✱ f ++ ❛❞❞s ❛ ✭t♦♣♠♦st✮ r♦✇ t♦ t❤❡ ❞✐❛❣r❛♠✳ jk ✷✵ ✴ ✸✹

❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❋✐① ✐♥t❡❣❡rs ✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ✈❡❝t♦r ❣❡♥❡r❛t❡s t❤❡ s✉❜♠♦❞✉❧❡ ♦❢ ✳ ✷ ❚❤❡ ♠❛♣s ❣❡♥❡r❛t❡ t❤❡ ✲❛❧❣❡❜r❛ ✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ ▼♦r❡ ❡①❛♠♣❧❡s ♦❢ ♠❛♣s ✐♥ � A [ λ ] ✿ f −• f •− jk : P j /P k ֒ → P j − 1 /P k , jk : P j /P k ։ P j /P k − 1 . ❆❞❞ t❤❡ r✐❣❤t♠♦st ❝♦❧✉♠♥✱ ❛♥❞ r❡♠♦✈❡ t❤❡ ❧❡❢t♠♦st ❝♦❧✉♠♥✱ r❡s♣❡❝t✐✈❡❧②✳ ✷✶ ✴ ✸✹

✷ ❚❤❡ ♠❛♣s ❣❡♥❡r❛t❡ t❤❡ ✲❛❧❣❡❜r❛ ✳ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ ▼♦r❡ ❡①❛♠♣❧❡s ♦❢ ♠❛♣s ✐♥ � A [ λ ] ✿ f −• f •− jk : P j /P k ֒ → P j − 1 /P k , jk : P j /P k ։ P j /P k − 1 . ❆❞❞ t❤❡ r✐❣❤t♠♦st ❝♦❧✉♠♥✱ ❛♥❞ r❡♠♦✈❡ t❤❡ ❧❡❢t♠♦st ❝♦❧✉♠♥✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❋✐① ✐♥t❡❣❡rs 1 ≤ { r, s } ≤ j ≤ k ≤ n + 1 ✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ✈❡❝t♦r d λ j − λ s u λ j − λ r 1 P r /P k ∈ P r /P k ❣❡♥❡r❛t❡s t❤❡ s✉❜♠♦❞✉❧❡ P s /P s + k − j ♦❢ P j /P k ֒ → P r /P k ✳ ✷✶ ✴ ✸✹

❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ ▼♦r❡ ❡①❛♠♣❧❡s ♦❢ ♠❛♣s ✐♥ � A [ λ ] ✿ f −• f •− jk : P j /P k ֒ → P j − 1 /P k , jk : P j /P k ։ P j /P k − 1 . ❆❞❞ t❤❡ r✐❣❤t♠♦st ❝♦❧✉♠♥✱ ❛♥❞ r❡♠♦✈❡ t❤❡ ❧❡❢t♠♦st ❝♦❧✉♠♥✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❋✐① ✐♥t❡❣❡rs 1 ≤ { r, s } ≤ j ≤ k ≤ n + 1 ✳ ❚❤❡♥ t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ✈❡❝t♦r d λ j − λ s u λ j − λ r 1 P r /P k ∈ P r /P k ❣❡♥❡r❛t❡s t❤❡ s✉❜♠♦❞✉❧❡ P s /P s + k − j ♦❢ P j /P k ֒ → P r /P k ✳ ✷ ❚❤❡ ♠❛♣s f ++ jk , f −• jk , f •− jk ❣❡♥❡r❛t❡ t❤❡ F ✲❛❧❣❡❜r❛ A [ λ ] = End O ( � � P [ λ ] ) op ✳ ✷✶ ✴ ✸✹

❉❡✜♥❡ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ✐s ❛ ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ ✳ ✷ ❯♥❞❡r t❤✐s ❣r❛❞✐♥❣ ♦♥ ✱ ✱ ❛♥❞ ✸ ■❢ ✱ t❤❡♥ ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ s✉✐t❛❜❧❡ ✱ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ Pr♦❞✉❝❡ ❛ Z + ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ � A [ λ ] = End O ( � P [ λ ] ) op ❄ ✷✷ ✴ ✸✹

❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ✐s ❛ ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ ✳ ✷ ❯♥❞❡r t❤✐s ❣r❛❞✐♥❣ ♦♥ ✱ ✱ ❛♥❞ ✸ ■❢ ✱ t❤❡♥ ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ s✉✐t❛❜❧❡ ✱ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ Pr♦❞✉❝❡ ❛ Z + ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ � A [ λ ] = End O ( � P [ λ ] ) op ❄ ❉❡✜♥❡ ϕ ( t ) ( r,s ) , ( j,k ) := ◦ f ++ k − t − 1 ,k − 1 ◦ · · · ◦ f ++ f −• j +1 ,k ◦ · · · ◦ f −• ◦ f •− r,r + t +1 ◦ · · · ◦ f •− . r,r + t r,s k − t,k � �� � � �� � � �� � s − r − t k − j − t k − r − t ✷✷ ✴ ✸✹

✷ ❯♥❞❡r t❤✐s ❣r❛❞✐♥❣ ♦♥ ✱ ✱ ❛♥❞ ✸ ■❢ ✱ t❤❡♥ ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ s✉✐t❛❜❧❡ ✱ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ Pr♦❞✉❝❡ ❛ Z + ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ � A [ λ ] = End O ( � P [ λ ] ) op ❄ ❉❡✜♥❡ ϕ ( t ) ( r,s ) , ( j,k ) := ◦ f ++ k − t − 1 ,k − 1 ◦ · · · ◦ f ++ f −• j +1 ,k ◦ · · · ◦ f −• ◦ f •− r,r + t +1 ◦ · · · ◦ f •− . r,r + t r,s k − t,k � �� � � �� � � �� � s − r − t k − j − t k − r − t ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ { ϕ ( t ) ( r,s ) , ( j,k ) : r < s, j < k, t ≤ min( s − r, k − r, k − j ) } ✐s ❛ Z + ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ � A [ λ ] ✳ ✷✷ ✴ ✸✹

✸ ■❢ ✱ t❤❡♥ ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ s✉✐t❛❜❧❡ ✱ ❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ Pr♦❞✉❝❡ ❛ Z + ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ � A [ λ ] = End O ( � P [ λ ] ) op ❄ ❉❡✜♥❡ ϕ ( t ) ( r,s ) , ( j,k ) := ◦ f ++ k − t − 1 ,k − 1 ◦ · · · ◦ f ++ f −• j +1 ,k ◦ · · · ◦ f −• ◦ f •− r,r + t +1 ◦ · · · ◦ f •− . r,r + t r,s k − t,k � �� � � �� � � �� � s − r − t k − j − t k − r − t ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ { ϕ ( t ) ( r,s ) , ( j,k ) : r < s, j < k, t ≤ min( s − r, k − r, k − j ) } ✐s ❛ Z + ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ � A [ λ ] ✳ ✷ ❯♥❞❡r t❤✐s ❣r❛❞✐♥❣ ♦♥ � A [ λ ] ✱ deg f ++ = deg f −• jk = 1 ✱ ❛♥❞ jk deg ϕ ( t ) deg f •− jk = 0 , ( r,s ) , ( j,k ) = 2( k − t ) − r − j. ✷✷ ✴ ✸✹

❋✐rst r❡s✉❧ts Pr♦❥❡❝t✐✈❡ r❡s♦❧✉t✐♦♥s ❛♥❞ ❊①t✬s ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ❨♦✉♥❣ ❞✐❛❣r❛♠s ❛♥❞ ♣r♦❥❡❝t✐✈❡s ✐♥ t❤❡ ❜❧♦❝❦ ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ●r❛❞❡❞ ♠❛♣s ❜❡t✇❡❡♥ q✉♦t✐❡♥ts ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ Pr♦❞✉❝❡ ❛ Z + ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ � A [ λ ] = End O ( � P [ λ ] ) op ❄ ❉❡✜♥❡ ϕ ( t ) ( r,s ) , ( j,k ) := ◦ f ++ k − t − 1 ,k − 1 ◦ · · · ◦ f ++ f −• j +1 ,k ◦ · · · ◦ f −• ◦ f •− r,r + t +1 ◦ · · · ◦ f •− . r,r + t r,s k − t,k � �� � � �� � � �� � s − r − t k − j − t k − r − t ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ { ϕ ( t ) ( r,s ) , ( j,k ) : r < s, j < k, t ≤ min( s − r, k − r, k − j ) } ✐s ❛ Z + ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ � A [ λ ] ✳ ✷ ❯♥❞❡r t❤✐s ❣r❛❞✐♥❣ ♦♥ � A [ λ ] ✱ deg f ++ = deg f −• jk = 1 ✱ ❛♥❞ jk deg ϕ ( t ) deg f •− jk = 0 , ( r,s ) , ( j,k ) = 2( k − t ) − r − j. ✸ ■❢ 1 ≤ a < b ≤ n + 1 ✱ t❤❡♥ ❢♦r ❛❧❧ ❝❤♦✐❝❡s ♦❢ s✉✐t❛❜❧❡ u, t ✱ ϕ ( u ) ( j,k ) , ( a,b ) ◦ ϕ ( t ) ( r,s ) , ( j,k ) = 1 ( u + t + j − k > 0) ϕ ( u + t + j − k ) ( r,s ) , ( a,b ) . ✷✷ ✴ ✸✹

❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❚❤❡ ♠❛♣s ❢♦r♠ ❛ ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ ✳ ✭❉✐♠❡♥s✐♦♥ ✳✮ ✷ ❚❤❡ ✲q✉✐✈❡r ♦❢ ✐s t❤❡ ❞♦✉❜❧❡ ♦❢ t❤❡ ✲q✉✐✈❡r ✳ ✸ ▲❛❜❡❧ t❤❡ ❛rr♦✇s ❛s ❛♥❞ ✳ ❚❤❡♥ ✱ ✱ ❛♥❞ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♣❛t❤ ❛❧❣❡❜r❛ ♦❢ t❤❡ q✉✐✈❡r ✇✐t❤ r❡❧❛t✐♦♥s ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr❡s❡♥t❛t✐♦♥ ♦❢ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ Pr♦✈✐❞❡s ❝♦♠♣❧❡t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛❧❣❡❜r❛ End O ( ⊕ j<k P j /P k ) ✳ ❲❤❛t ✐s ❛ ❜❛s✐s ♦❢ t❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( ⊕ j P j ) op ❄ ✭❘❡❝❛❧❧ t❤❛t O [ λ ] ∼ = A [ λ ] ✲ Mod ✳✮ ✷✸ ✴ ✸✹

✷ ❚❤❡ ✲q✉✐✈❡r ♦❢ ✐s t❤❡ ❞♦✉❜❧❡ ♦❢ t❤❡ ✲q✉✐✈❡r ✳ ✸ ▲❛❜❡❧ t❤❡ ❛rr♦✇s ❛s ❛♥❞ ✳ ❚❤❡♥ ✱ ✱ ❛♥❞ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♣❛t❤ ❛❧❣❡❜r❛ ♦❢ t❤❡ q✉✐✈❡r ✇✐t❤ r❡❧❛t✐♦♥s ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr❡s❡♥t❛t✐♦♥ ♦❢ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ Pr♦✈✐❞❡s ❝♦♠♣❧❡t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛❧❣❡❜r❛ End O ( ⊕ j<k P j /P k ) ✳ ❲❤❛t ✐s ❛ ❜❛s✐s ♦❢ t❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( ⊕ j P j ) op ❄ ✭❘❡❝❛❧❧ t❤❛t O [ λ ] ∼ = A [ λ ] ✲ Mod ✳✮ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❚❤❡ ♠❛♣s { ϕ ( t ) ( r,n +1) , ( j,n +1) : t ≤ n + 1 − max( r, j ) } ❢♦r♠ ❛ Z ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ A [ λ ] ✳ ✭❉✐♠❡♥s✐♦♥ = 1 2 + · · · + n 2 ✳✮ ✷✸ ✴ ✸✹

❚❤❡♥ ✱ ✱ ❛♥❞ ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♣❛t❤ ❛❧❣❡❜r❛ ♦❢ t❤❡ q✉✐✈❡r ✇✐t❤ r❡❧❛t✐♦♥s ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr❡s❡♥t❛t✐♦♥ ♦❢ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ Pr♦✈✐❞❡s ❝♦♠♣❧❡t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛❧❣❡❜r❛ End O ( ⊕ j<k P j /P k ) ✳ ❲❤❛t ✐s ❛ ❜❛s✐s ♦❢ t❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( ⊕ j P j ) op ❄ ✭❘❡❝❛❧❧ t❤❛t O [ λ ] ∼ = A [ λ ] ✲ Mod ✳✮ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❚❤❡ ♠❛♣s { ϕ ( t ) ( r,n +1) , ( j,n +1) : t ≤ n + 1 − max( r, j ) } ❢♦r♠ ❛ Z ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ A [ λ ] ✳ ✭❉✐♠❡♥s✐♦♥ = 1 2 + · · · + n 2 ✳✮ ✷ ❚❤❡ Ext ✲q✉✐✈❡r ♦❢ A [ λ ] ✐s t❤❡ ❞♦✉❜❧❡ A n ♦❢ t❤❡ A n ✲q✉✐✈❡r [1] → [2] → · · · → [ n ] ✳ ✸ ▲❛❜❡❧ t❤❡ ❛rr♦✇s ❛s γ i : [ i + 1] → [ i ] ❛♥❞ δ i : [ i ] → [ i + 1] ✳ ✷✸ ✴ ✸✹

❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ Pr❡s❡♥t❛t✐♦♥ ♦❢ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ Pr♦✈✐❞❡s ❝♦♠♣❧❡t❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❛❧❣❡❜r❛ End O ( ⊕ j<k P j /P k ) ✳ ❲❤❛t ✐s ❛ ❜❛s✐s ♦❢ t❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( ⊕ j P j ) op ❄ ✭❘❡❝❛❧❧ t❤❛t O [ λ ] ∼ = A [ λ ] ✲ Mod ✳✮ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❚❤❡ ♠❛♣s { ϕ ( t ) ( r,n +1) , ( j,n +1) : t ≤ n + 1 − max( r, j ) } ❢♦r♠ ❛ Z ✲❣r❛❞❡❞ ❜❛s✐s ♦❢ A [ λ ] ✳ ✭❉✐♠❡♥s✐♦♥ = 1 2 + · · · + n 2 ✳✮ ✷ ❚❤❡ Ext ✲q✉✐✈❡r ♦❢ A [ λ ] ✐s t❤❡ ❞♦✉❜❧❡ A n ♦❢ t❤❡ A n ✲q✉✐✈❡r [1] → [2] → · · · → [ n ] ✳ ✸ ▲❛❜❡❧ t❤❡ ❛rr♦✇s ❛s γ i : [ i + 1] → [ i ] ❛♥❞ δ i : [ i ] → [ i + 1] ✳ i +1 ,n − 1 ✱ δ i = f ++ i,n +1 ✱ ❛♥❞ A op ❚❤❡♥ γ i = f −• [ λ ] ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ♣❛t❤ ❛❧❣❡❜r❛ ♦❢ t❤❡ q✉✐✈❡r A n ✇✐t❤ r❡❧❛t✐♦♥s δ i ◦ γ i = γ i +1 ◦ δ i +1 ∀ 0 < i < n − 1 , δ n − 1 ◦ γ n − 1 = 0 . ✷✸ ✴ ✸✹

❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ●❲❆✱ ❜❧♦❝❦s ✇✐t❤ s❛♠❡ ♥✉♠❜❡r ♦❢ s✐♠♣❧❡s ❛r❡ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢✿ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ ✿ ✳ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ ✿ ✐s ❣r❛❞❡❞✱ q✉❛❞r❛t✐❝❀ ✳ ◆♦✇ ✉s❡ ♥✉♠❡r✐❝❛❧ ❝r✐t❡r✐♦♥ ❢♦r ❑♦s③✉❧✐t② ❬❇●❙❪✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❑♦s③✉❧✐t② ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❚❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( P [ λ ] ) op ✐s ❑♦s③✉❧✱ ❛♥❞ ❞❡♣❡♥❞s ♦♥❧② ♦♥ n = | [ λ ] | ✳ ✷✹ ✴ ✸✹

Pr♦♦❢✿ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ ✿ ✳ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ ✿ ✐s ❣r❛❞❡❞✱ q✉❛❞r❛t✐❝❀ ✳ ◆♦✇ ✉s❡ ♥✉♠❡r✐❝❛❧ ❝r✐t❡r✐♦♥ ❢♦r ❑♦s③✉❧✐t② ❬❇●❙❪✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❑♦s③✉❧✐t② ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❚❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( P [ λ ] ) op ✐s ❑♦s③✉❧✱ ❛♥❞ ❞❡♣❡♥❞s ♦♥❧② ♦♥ n = | [ λ ] | ✳ ❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ●❲❆✱ ❜❧♦❝❦s ✇✐t❤ s❛♠❡ ♥✉♠❜❡r ♦❢ s✐♠♣❧❡s ❛r❡ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t✳ ✷✹ ✴ ✸✹

❍✐❧❜❡rt ♠❛tr✐① ♦❢ ✿ ✐s ❣r❛❞❡❞✱ q✉❛❞r❛t✐❝❀ ✳ ◆♦✇ ✉s❡ ♥✉♠❡r✐❝❛❧ ❝r✐t❡r✐♦♥ ❢♦r ❑♦s③✉❧✐t② ❬❇●❙❪✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❑♦s③✉❧✐t② ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❚❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( P [ λ ] ) op ✐s ❑♦s③✉❧✱ ❛♥❞ ❞❡♣❡♥❞s ♦♥❧② ♦♥ n = | [ λ ] | ✳ ❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ●❲❆✱ ❜❧♦❝❦s ✇✐t❤ s❛♠❡ ♥✉♠❜❡r ♦❢ s✐♠♣❧❡s ❛r❡ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢✿ n � t 2 u − j − k ✳ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ A [ λ ] ✿ H ( A [ λ ] , t ) j,k = u =max( j,k ) ✷✹ ✴ ✸✹

✐s ❣r❛❞❡❞✱ q✉❛❞r❛t✐❝❀ ✳ ◆♦✇ ✉s❡ ♥✉♠❡r✐❝❛❧ ❝r✐t❡r✐♦♥ ❢♦r ❑♦s③✉❧✐t② ❬❇●❙❪✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❑♦s③✉❧✐t② ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❚❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( P [ λ ] ) op ✐s ❑♦s③✉❧✱ ❛♥❞ ❞❡♣❡♥❞s ♦♥❧② ♦♥ n = | [ λ ] | ✳ ❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ●❲❆✱ ❜❧♦❝❦s ✇✐t❤ s❛♠❡ ♥✉♠❜❡r ♦❢ s✐♠♣❧❡s ❛r❡ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢✿ n � t 2 u − j − k ✳ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ A [ λ ] ✿ H ( A [ λ ] , t ) j,k = u =max( j,k ) ❍✐❧❜❡rt ♠❛tr✐① ♦❢ E ( A [ λ ] ) = Ext • O ( P [ λ ] , P [ λ ] ) ✿ H ( E ( A [ λ ] ) , t ) = Toeplitz(1+ t 2 , t, 0 , . . . , 0) − t 2 E 11 ✷✹ ✴ ✸✹

✐s ❣r❛❞❡❞✱ q✉❛❞r❛t✐❝❀ ✳ ◆♦✇ ✉s❡ ♥✉♠❡r✐❝❛❧ ❝r✐t❡r✐♦♥ ❢♦r ❑♦s③✉❧✐t② ❬❇●❙❪✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❑♦s③✉❧✐t② ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❚❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( P [ λ ] ) op ✐s ❑♦s③✉❧✱ ❛♥❞ ❞❡♣❡♥❞s ♦♥❧② ♦♥ n = | [ λ ] | ✳ ❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ●❲❆✱ ❜❧♦❝❦s ✇✐t❤ s❛♠❡ ♥✉♠❜❡r ♦❢ s✐♠♣❧❡s ❛r❡ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢✿ n � t 2 u − j − k ✳ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ A [ λ ] ✿ H ( A [ λ ] , t ) j,k = u =max( j,k ) ❍✐❧❜❡rt ♠❛tr✐① ♦❢ E ( A [ λ ] ) = Ext • O ( P [ λ ] , P [ λ ] ) ✿ H ( E ( A [ λ ] ) , t ) = Toeplitz(1+ t 2 , t, 0 , . . . , 0) − t 2 E 11 = H ( A [ λ ] , t ) − 1 . ✷✹ ✴ ✸✹

◆♦✇ ✉s❡ ♥✉♠❡r✐❝❛❧ ❝r✐t❡r✐♦♥ ❢♦r ❑♦s③✉❧✐t② ❬❇●❙❪✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❑♦s③✉❧✐t② ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❚❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( P [ λ ] ) op ✐s ❑♦s③✉❧✱ ❛♥❞ ❞❡♣❡♥❞s ♦♥❧② ♦♥ n = | [ λ ] | ✳ ❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ●❲❆✱ ❜❧♦❝❦s ✇✐t❤ s❛♠❡ ♥✉♠❜❡r ♦❢ s✐♠♣❧❡s ❛r❡ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢✿ n � t 2 u − j − k ✳ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ A [ λ ] ✿ H ( A [ λ ] , t ) j,k = u =max( j,k ) ❍✐❧❜❡rt ♠❛tr✐① ♦❢ E ( A [ λ ] ) = Ext • O ( P [ λ ] , P [ λ ] ) ✿ H ( E ( A [ λ ] ) , t ) = Toeplitz(1+ t 2 , t, 0 , . . . , 0) − t 2 E 11 = H ( A [ λ ] , t ) − 1 . A [ λ ] ✐s ❣r❛❞❡❞✱ q✉❛❞r❛t✐❝❀ A [ λ ] [0] = span F { id P j : 1 ≤ j ≤ n } ✳ ✷✹ ✴ ✸✹

❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❑♦s③✉❧✐t② ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❚❤❡ ❛❧❣❡❜r❛ A [ λ ] = End O ( P [ λ ] ) op ✐s ❑♦s③✉❧✱ ❛♥❞ ❞❡♣❡♥❞s ♦♥❧② ♦♥ n = | [ λ ] | ✳ ❘❡❣❛r❞❧❡ss ♦❢ t❤❡ ●❲❆✱ ❜❧♦❝❦s ✇✐t❤ s❛♠❡ ♥✉♠❜❡r ♦❢ s✐♠♣❧❡s ❛r❡ ▼♦r✐t❛ ❡q✉✐✈❛❧❡♥t✳ Pr♦♦❢✿ n � t 2 u − j − k ✳ ❍✐❧❜❡rt ♠❛tr✐① ♦❢ A [ λ ] ✿ H ( A [ λ ] , t ) j,k = u =max( j,k ) ❍✐❧❜❡rt ♠❛tr✐① ♦❢ E ( A [ λ ] ) = Ext • O ( P [ λ ] , P [ λ ] ) ✿ H ( E ( A [ λ ] ) , t ) = Toeplitz(1+ t 2 , t, 0 , . . . , 0) − t 2 E 11 = H ( A [ λ ] , t ) − 1 . A [ λ ] ✐s ❣r❛❞❡❞✱ q✉❛❞r❛t✐❝❀ A [ λ ] [0] = span F { id P j : 1 ≤ j ≤ n } ✳ ◆♦✇ ✉s❡ ♥✉♠❡r✐❝❛❧ ❝r✐t❡r✐♦♥ ❢♦r ❑♦s③✉❧✐t② ❬❇●❙❪✳ ✷✹ ✴ ✸✹

❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❋✐① ✳ ✶ ❚❤❡r❡ ❡①✐sts ❛ ❜✐❥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s✉❜♠♦❞✉❧❡s ♦❢ ✱ ❛♥❞ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s ♦❢ ✐♥t❡❣❡rs ✱ ❢♦r s♦♠❡ ✳ ✷ ❊✈❡r② s✉❝❤ s✉❜♠♦❞✉❧❡ ✐s ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❛♥❞ ❤❛s ❛ ❱❡r♠❛ ✢❛❣✱ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡s❡ s✉❜♠♦❞✉❧❡s ✐s ✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙✉❜♠♦❞✉❧❡s ♦❢ ♣r♦❥❡❝t✐✈❡s ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ A [ λ ] ❢♦❧❧♦✇❡❞ ❢r♦♠ ❞❡t❛✐❧❡❞ ❛♥❛❧②s✐s ♦❢ ♠❛♣s ❜❡t✇❡❡♥ ♠♦❞✉❧❡s P j /P k ✳ ❈❛♥ ❛❧s♦ ❝❧❛ss✐❢② ❛❧❧ s✉❜♠♦❞✉❧❡s ♦❢ t❤❡s❡ ♠♦❞✉❧❡s✿ ✷✺ ✴ ✸✹

❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙✉❜♠♦❞✉❧❡s ♦❢ ♣r♦❥❡❝t✐✈❡s ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ♦❢ A [ λ ] ❢♦❧❧♦✇❡❞ ❢r♦♠ ❞❡t❛✐❧❡❞ ❛♥❛❧②s✐s ♦❢ ♠❛♣s ❜❡t✇❡❡♥ ♠♦❞✉❧❡s P j /P k ✳ ❈❛♥ ❛❧s♦ ❝❧❛ss✐❢② ❛❧❧ s✉❜♠♦❞✉❧❡s ♦❢ t❤❡s❡ ♠♦❞✉❧❡s✿ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❋✐① 1 ≤ j < k ≤ n + 1 ✳ ✶ ❚❤❡r❡ ❡①✐sts ❛ ❜✐❥❡❝t✐♦♥ ❜❡t✇❡❡♥ t❤❡ s✉❜♠♦❞✉❧❡s ♦❢ P j /P k ✱ ❛♥❞ str✐❝t❧② ❞❡❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s ♦❢ ✐♥t❡❣❡rs k − 1 ≥ m l > m l − 1 > · · · > m 1 ≥ 1 ✱ ❢♦r s♦♠❡ 0 ≤ l ≤ k − j ✳ ✷ ❊✈❡r② s✉❝❤ s✉❜♠♦❞✉❧❡ ✐s ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❛♥❞ ❤❛s ❛ ❱❡r♠❛ � k − 1 � k − j � ✢❛❣✱ ❛♥❞ t❤❡ ♥✉♠❜❡r ♦❢ t❤❡s❡ s✉❜♠♦❞✉❧❡s ✐s ✳ l l =0 ✷✺ ✴ ✸✹

❚❤❡♥ ❡❛❝❤ s✉❜q✉♦t✐❡♥t ✐s ❛ s✉❜♠♦❞✉❧❡ ❢♦r s♦♠❡ ✳ ▲❡❛❞s t♦ tr❛♥s❢❡r ♠❛♣ ✳ ❆❧s♦ ❧❡❛❞s t♦ ❞✐❛❣r❛♠ ♠❛♣ ❢r♦♠ t♦ ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ ✳ ❊✳❣✳✱ ❞✐❛❣r❛♠ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✱ ✇✐t❤ ✿ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙✉❜♠♦❞✉❧❡s ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ ❚❤❡ ❜✐❥❡❝t✐♦♥✿ ●✐✈❡♥ N ⊂ P j /P k ✱ ❝♦♥s✐❞❡r t❤❡ ✜❧tr❛t✐♦♥✿ 0 ⊂ N ∩ ( P k − 1 /P k ) ⊂ N ∩ ( P k − 2 /P k ) ⊂ · · · ⊂ N ∩ ( P j /P k ) . ✷✻ ✴ ✸✹

▲❡❛❞s t♦ tr❛♥s❢❡r ♠❛♣ ✳ ❆❧s♦ ❧❡❛❞s t♦ ❞✐❛❣r❛♠ ♠❛♣ ❢r♦♠ t♦ ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ ✳ ❊✳❣✳✱ ❞✐❛❣r❛♠ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✱ ✇✐t❤ ✿ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙✉❜♠♦❞✉❧❡s ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ ❚❤❡ ❜✐❥❡❝t✐♦♥✿ ●✐✈❡♥ N ⊂ P j /P k ✱ ❝♦♥s✐❞❡r t❤❡ ✜❧tr❛t✐♦♥✿ 0 ⊂ N ∩ ( P k − 1 /P k ) ⊂ N ∩ ( P k − 2 /P k ) ⊂ · · · ⊂ N ∩ ( P j /P k ) . ❚❤❡♥ ❡❛❝❤ s✉❜q✉♦t✐❡♥t ✐s ❛ s✉❜♠♦❞✉❧❡ M m r ⊂ M k − r ❢♦r s♦♠❡ r ✳ ✷✻ ✴ ✸✹

❆❧s♦ ❧❡❛❞s t♦ ❞✐❛❣r❛♠ ♠❛♣ ❢r♦♠ t♦ ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ ✳ ❊✳❣✳✱ ❞✐❛❣r❛♠ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ✱ ✇✐t❤ ✿ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙✉❜♠♦❞✉❧❡s ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ ❚❤❡ ❜✐❥❡❝t✐♦♥✿ ●✐✈❡♥ N ⊂ P j /P k ✱ ❝♦♥s✐❞❡r t❤❡ ✜❧tr❛t✐♦♥✿ 0 ⊂ N ∩ ( P k − 1 /P k ) ⊂ N ∩ ( P k − 2 /P k ) ⊂ · · · ⊂ N ∩ ( P j /P k ) . ❚❤❡♥ ❡❛❝❤ s✉❜q✉♦t✐❡♥t ✐s ❛ s✉❜♠♦❞✉❧❡ M m r ⊂ M k − r ❢♦r s♦♠❡ r ✳ ▲❡❛❞s t♦ tr❛♥s❢❡r ♠❛♣ N � Ψ( N ) = ( m l , . . . , m 1 ) ✳ ✷✻ ✴ ✸✹

❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❙✉❜♠♦❞✉❧❡s ♦❢ ♣r♦❥❡❝t✐✈❡s ✭❝♦♥t✳✮ ❚❤❡ ❜✐❥❡❝t✐♦♥✿ ●✐✈❡♥ N ⊂ P j /P k ✱ ❝♦♥s✐❞❡r t❤❡ ✜❧tr❛t✐♦♥✿ 0 ⊂ N ∩ ( P k − 1 /P k ) ⊂ N ∩ ( P k − 2 /P k ) ⊂ · · · ⊂ N ∩ ( P j /P k ) . ❚❤❡♥ ❡❛❝❤ s✉❜q✉♦t✐❡♥t ✐s ❛ s✉❜♠♦❞✉❧❡ M m r ⊂ M k − r ❢♦r s♦♠❡ r ✳ ▲❡❛❞s t♦ tr❛♥s❢❡r ♠❛♣ N � Ψ( N ) = ( m l , . . . , m 1 ) ✳ ❆❧s♦ ❧❡❛❞s t♦ ❞✐❛❣r❛♠ ♠❛♣ YT ❢r♦♠ N t♦ 5 ❛ ❨♦✉♥❣ ❞✐❛❣r❛♠ YT ( N ) ✳ 4 3 2 ❊✳❣✳✱ ❞✐❛❣r❛♠ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ (5 , 3 , 2) ✱ 3 2 1 ✇✐t❤ Ψ − 1 ((5 , 3 , 2)) ⊂ P 3 /P 6 ✿ 2 1 1 ✷✻ ✴ ✸✹

❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❙✉♣♣♦s❡ ✳ ✶ ❋♦r ❡❛❝❤ ✱ t❤❡ ♥✉♠❜❡r ♦❢ ❝❡❧❧s ✐♥ ♥✉♠❜❡r❡❞ ✱ ♣r❡❝✐s❡❧② ❡q✉❛❧s ✳ ✷ ◗✉♦t✐❡♥t✐♥❣ ❡q✉❛❧s ❡①❝✐s✐♦♥✿ ✱ ❛♥❞ ❞✉❛❧✐t② ❡q✉❛❧s tr❛♥s♣♦s❡✿ ✳ ✸ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♠❛♣✿ ♣r♦♣❡rt✐❡s ❲❤❛t ♣r♦♣❡rt✐❡s ❞♦❡s t❤❡ ♠❛♣ N �→ YT ( N ) s❛t✐s❢②❄ ❘❡❝❛❧❧✿ ❛❧❧ ♠♦❞✉❧❡s P r /P s ❡♠❜❡❞ ✐♥t♦ P 1 ✭❧❛r❣❡st ♣r♦❥❡❝t✐✈❡✮✳ ✷✼ ✴ ✸✹

✷ ◗✉♦t✐❡♥t✐♥❣ ❡q✉❛❧s ❡①❝✐s✐♦♥✿ ✱ ❛♥❞ ❞✉❛❧✐t② ❡q✉❛❧s tr❛♥s♣♦s❡✿ ✳ ✸ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♠❛♣✿ ♣r♦♣❡rt✐❡s ❲❤❛t ♣r♦♣❡rt✐❡s ❞♦❡s t❤❡ ♠❛♣ N �→ YT ( N ) s❛t✐s❢②❄ ❘❡❝❛❧❧✿ ❛❧❧ ♠♦❞✉❧❡s P r /P s ❡♠❜❡❞ ✐♥t♦ P 1 ✭❧❛r❣❡st ♣r♦❥❡❝t✐✈❡✮✳ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❙✉♣♣♦s❡ N ′ ⊂ N ⊂ P 1 ✳ ✶ ❋♦r ❡❛❝❤ 1 ≤ j ≤ n ✱ t❤❡ ♥✉♠❜❡r ♦❢ ❝❡❧❧s ✐♥ YT ( N ) ♥✉♠❜❡r❡❞ j ✱ ♣r❡❝✐s❡❧② ❡q✉❛❧s [ N : L j ] ✳ ✷✼ ✴ ✸✹

✸ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♠❛♣✿ ♣r♦♣❡rt✐❡s ❲❤❛t ♣r♦♣❡rt✐❡s ❞♦❡s t❤❡ ♠❛♣ N �→ YT ( N ) s❛t✐s❢②❄ ❘❡❝❛❧❧✿ ❛❧❧ ♠♦❞✉❧❡s P r /P s ❡♠❜❡❞ ✐♥t♦ P 1 ✭❧❛r❣❡st ♣r♦❥❡❝t✐✈❡✮✳ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❙✉♣♣♦s❡ N ′ ⊂ N ⊂ P 1 ✳ ✶ ❋♦r ❡❛❝❤ 1 ≤ j ≤ n ✱ t❤❡ ♥✉♠❜❡r ♦❢ ❝❡❧❧s ✐♥ YT ( N ) ♥✉♠❜❡r❡❞ j ✱ ♣r❡❝✐s❡❧② ❡q✉❛❧s [ N : L j ] ✳ ✷ ◗✉♦t✐❡♥t✐♥❣ ❡q✉❛❧s ❡①❝✐s✐♦♥✿ YT ( N/N ′ ) = YT ( N ) \ YT ( N ′ ) ✱ ❛♥❞ ❞✉❛❧✐t② ❡q✉❛❧s tr❛♥s♣♦s❡✿ YT ( F ( N )) = YT ( N ) T ✳ ✷✼ ✴ ✸✹

❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❨♦✉♥❣ ❞✐❛❣r❛♠ ♠❛♣✿ ♣r♦♣❡rt✐❡s ❲❤❛t ♣r♦♣❡rt✐❡s ❞♦❡s t❤❡ ♠❛♣ N �→ YT ( N ) s❛t✐s❢②❄ ❘❡❝❛❧❧✿ ❛❧❧ ♠♦❞✉❧❡s P r /P s ❡♠❜❡❞ ✐♥t♦ P 1 ✭❧❛r❣❡st ♣r♦❥❡❝t✐✈❡✮✳ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ❙✉♣♣♦s❡ N ′ ⊂ N ⊂ P 1 ✳ ✶ ❋♦r ❡❛❝❤ 1 ≤ j ≤ n ✱ t❤❡ ♥✉♠❜❡r ♦❢ ❝❡❧❧s ✐♥ YT ( N ) ♥✉♠❜❡r❡❞ j ✱ ♣r❡❝✐s❡❧② ❡q✉❛❧s [ N : L j ] ✳ ✷ ◗✉♦t✐❡♥t✐♥❣ ❡q✉❛❧s ❡①❝✐s✐♦♥✿ YT ( N/N ′ ) = YT ( N ) \ YT ( N ′ ) ✱ ❛♥❞ ❞✉❛❧✐t② ❡q✉❛❧s tr❛♥s♣♦s❡✿ YT ( F ( N )) = YT ( N ) T ✳ ✸ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ YT ( F ( N/N ′ )) = YT ( N ) T \ YT ( N ′ ) T ✳ ✷✼ ✴ ✸✹

✳ ✳ ✳✳✳✳✳✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ❉❡✜♥❡ t♦ ❜❡ t❤❡ ✳ ❧❛❜❡❧❧❡❞ tr✐❛♥❣✉❧❛r ❞✐❛❣r❛♠✿ ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ ✳ ❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❚❤❡ ♣❛rt✐❛❧✴✐♥❞❡❝♦♠♣♦s❛❜❧❡ t✐❧t✐♥❣ ♠♦❞✉❧❡s ✐♥ t❤❡ ❜❧♦❝❦ ❛r❡ ❢♦r ✳ ✷ ❊❛❝❤ ✐s s❡❧❢✲❞✉❛❧✳ ✸ ❚❤❡ ✐♥❥❡❝t✐✈❡ ❤✉❧❧ ✐♥ t❤❡ ❜❧♦❝❦ ♦❢ t❤❡ s✐♠♣❧❡ ♠♦❞✉❧❡ ✐s ❡q✉❛❧ t♦ ✱ ✇❤❡r❡ ✇❡ s❡t ✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❚✐❧t✐♥❣ ♠♦❞✉❧❡s ❚✐❧t✐♥❣ ♠♦❞✉❧❡s T s❛t✐s❢②✿ ❜♦t❤ T, F ( T ) ❤❛✈❡ ❛ ❱❡r♠❛ ✢❛❣✳ ❲❤✐❝❤ ❞✐❛❣r❛♠s ✇♦✉❧❞ ♦♥❡ ❣❡t❄ ✷✽ ✴ ✸✹

❚❤❡♦r❡♠ ✭❑❤❛r❡✲❚✐❦❛r❛❞③❡✱ ✷✵✶✺✮ ✶ ❚❤❡ ♣❛rt✐❛❧✴✐♥❞❡❝♦♠♣♦s❛❜❧❡ t✐❧t✐♥❣ ♠♦❞✉❧❡s ✐♥ t❤❡ ❜❧♦❝❦ ❛r❡ ❢♦r ✳ ✷ ❊❛❝❤ ✐s s❡❧❢✲❞✉❛❧✳ ✸ ❚❤❡ ✐♥❥❡❝t✐✈❡ ❤✉❧❧ ✐♥ t❤❡ ❜❧♦❝❦ ♦❢ t❤❡ s✐♠♣❧❡ ♠♦❞✉❧❡ ✐s ❡q✉❛❧ t♦ ✱ ✇❤❡r❡ ✇❡ s❡t ✳ ❋✐rst r❡s✉❧ts ❑♦s③✉❧✐t② ❛♥❞ ♣r❡s❡♥t❛t✐♦♥ ❚❤❡ ❡♥❞♦♠♦r♣❤✐s♠ ❛❧❣❡❜r❛ ♦❢ ♣r♦❥❡❝t✐✈❡s ●❲❆s ❝❛t❡❣♦r✐❢② ❨♦✉♥❣ ❞✐❛❣r❛♠s ❑♦s③✉❧✐t② ❛♥❞ ❝❛t❡❣♦r✐✜❝❛t✐♦♥ ❚✐❧t✐♥❣ ♠♦❞✉❧❡s k − 1 k − 2 · · · 2 1 ❚✐❧t✐♥❣ ♠♦❞✉❧❡s T s❛t✐s❢②✿ k k − 1 k − 2 · · ·· · · 1 ❜♦t❤ T, F ( T ) ❤❛✈❡ ❛ ❱❡r♠❛ ✢❛❣✳ k − 2 ✳ ✳ ✳✳✳✳✳✳ ✳ ❲❤✐❝❤ ❞✐❛❣r❛♠s ✇♦✉❧❞ ♦♥❡ ❣❡t❄ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ❉❡✜♥❡ YT k t♦ ❜❡ t❤❡ ✳ ❧❛❜❡❧❧❡❞ tr✐❛♥❣✉❧❛r ❞✐❛❣r❛♠✿ 2 1 1 ❚❤✐s ❝♦rr❡s♣♦♥❞s t♦ P 1 /P k +1 ✳ ✷✽ ✴ ✸✹