❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s ❉❛r✐♦ ❙♣✐r✐t♦ ❯♥✐✈❡rs✐tà ❞✐ ❘♦♠❛ ❚r❡ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❘✐♥❣s ❛♥❞ ❋❛❝t♦r✐③❛t✐♦♥s ●r❛③✱ ❋❡❜r✉❛r② ✷✸r❞✱ ✷✵✶✽ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
❙t❛r ♦♣❡r❛t✐♦♥s ❲❡ ✇✐❧❧ ❛❧✇❛②s t❛❦❡ ❛♥ ✐♥t❡❣r❛❧ ❞♦♠❛✐♥ D ✇✐t❤ q✉♦t✐❡♥t ✜❡❧❞ K ✳ ▲❡t ❋ ( D ) ❜❡ t❤❡ s❡t ♦❢ D ✲s✉❜♠♦❞✉❧❡s ♦❢ K ✳ ▲❡t F ( D ) ❜❡ t❤❡ s❡t ♦❢ ❢r❛❝t✐♦♥❛❧ ✐❞❡❛❧s ♦❢ D ✱ ✐✳❡✳✱ ♦❢ t❤❡ I ∈ ❋ ( D ) s✉❝❤ t❤❛t xI ⊆ D ❢♦r s♦♠❡ x ∈ K ✳ ❉❡✜♥✐t✐♦♥ ❆ st❛r ♦♣❡r❛t✐♦♥ ♦♥ D ✐s ❛ ♠❛♣ ∗ : F ( D ) − → F ( D ) s✉❝❤ t❤❛t I ⊆ I ∗ ❀ ⇒ I ∗ ⊆ J ∗ ❀ I ⊆ J = ( I ∗ ) ∗ = I ∗ ✳ ( xI ) ∗ = x · I ∗ ❀ D ∗ = D ✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
❊①❛♠♣❧❡s ❚❤❡ ✐❞❡♥t✐t② d : I �→ I ✳ ❚❤❡ v ✲♦♣❡r❛t✐♦♥ v : I �→ ( D : ( D : I )) ✳ ❚❤❡ t ✲♦♣❡r❛t✐♦♥✿ { J v | J ⊆ I , J ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ } . � t : I �→ ■❢ Y ⊆ ❖✈❡r ( D ) ❛♥❞ � T ∈Y T = D ✱ ✇❡ ❝❛♥ ❞❡✜♥❡ � ∧ Y : I �→ IT . T ∈Y ■❢ ∆ ⊆ Spec( D ) ❛♥❞ � P ∈ ∆ D P = D ✱ ✇❡ ❝❛♥ ❞❡✜♥❡ � s ∆ : I �→ ID P . P ∈ ∆ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
❚②♣❡s ♦❢ ❝❧♦s✉r❡ ♦♣❡r❛t✐♦♥s ∗ ✐s ♦❢ ✜♥✐t❡ t②♣❡ ✐❢✱ ❢♦r ❡✈❡r② I ✱ I ∗ = { F ∗ | F ⊆ I , F ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ } � ∗ ✐s s♣❡❝tr❛❧ ✐❢ ✐t ✐s ✐♥ t❤❡ ❢♦r♠ s ∆ ✳ ∗ ✐s st❛❜❧❡ ✐❢ ( I ∩ J ) ∗ = I ∗ ∩ J ∗ ❢♦r ❛❧❧ I , J ✳ ∗ ✐s ◆♦❡t❤❡r✐❛♥ ✐❢ t❤❡ s❡t I ∗ ( D ) := { I ∈ F ( D ) | I ⊆ D , I = I ∗ } s❛t✐s✜❡s t❤❡ ❛s❝❡♥❞✐♥❣ ❝❤❛✐♥ ❝♦♥❞✐t✐♦♥✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
❙tr✉❝t✉r❡s ♦♥ ❙t❛r ( D ) ❖r❞❡r str✉❝t✉r❡✿ ∗ ✶ ≤ ∗ ✷ ✐❢ I ∗ ✶ ⊆ I ∗ ✷ ❢♦r ❡✈❡r② I ∈ F ( D ) ✳ ◮ ❙t❛r ( D ) ✐s ❛ ❝♦♠♣❧❡t❡ ❧❛tt✐❝❡✳ ◮ v ✐s t❤❡ ♠❛①✐♠✉♠ ♦❢ ❙t❛r ( D ) ✳ ◮ t ✐s t❤❡ ♠❛①✐♠✉♠ ♦❢ ❙t❛r f ( D ) ✳ ❚♦♣♦❧♦❣✐❝❛❧ str✉❝t✉r❡✿ t❤❡ t♦♣♦❧♦❣② ✐s t❤❡ ♦♥❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡ts V I := {∗ ∈ ❙t❛r ( D ) | ✶ ∈ I ∗ } . ◮ ❙t❛r f ( D ) ✐s ❜❡tt❡r ❜❡❤❛✈❡❞ t❤❛♥ ❙t❛r ( D ) ✳ ❙❡t str✉❝t✉r❡✿ st✉❞② ♦❢ t❤❡ ❝❛r❞✐♥❛❧✐t②✳ ◮ ❋♦r ❡①❛♠♣❧❡✱ ✇❤❡♥ ✐s ❙t❛r ( D ) ✜♥✐t❡❄ ❲❤❡♥ | ❙t❛r ( D ) | = ✶❄ ◮ ❚❤❡r❡ ❛r❡ ♥♦ ❣❡♥❡r❛❧ r❡s✉❧ts✱ ❜✉t s♦♠❡ ❝❛♥ ❜❡ s❛✐❞ ✇❤❡♥ D ✐s ◆♦❡t❤❡r✐❛♥ ♦r ✇❤❡♥ ✐t ✐s ✐♥t❡❣r❛❧❧② ❝❧♦s❡❞✳ ◮ ❋♦r ❡①❛♠♣❧❡✱ ✐❢ D ✐s ◆♦❡t❤❡r✐❛♥ t❤❡♥ | ❙t❛r ( D ) | = ✶ ✐❢ ❛♥❞ ♦♥❧② ✐❢ D ✐s ●♦r❡♥st❡✐♥ ♦❢ ❞✐♠❡♥s✐♦♥ ✶✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
❊①t❡♥s✐♦♥s ♦❢ st❛r ♦♣❡r❛t✐♦♥s ▲❡t D ❜❡ ❛♥ ✐♥t❡❣r❛❧ ❞♦♠❛✐♥ ❛♥❞ T ❛ ✢❛t ♦✈❡rr✐♥❣ ♦❢ D ✳ ❉❡✜♥✐t✐♦♥ ❆ st❛r ♦♣❡r❛t✐♦♥ ∗ ♦♥ D ✐s ❡①t❡♥❞❛❜❧❡ t♦ T ✐❢ t❤❡ ♠❛♣ ∗ T : F ( T ) − → F ( T ) → I ∗ T IT �− ✐s ✇❡❧❧✲❞❡✜♥❡❞✳ ❊q✉✐✈❛❧❡♥t❧②✱ ∗ ✐s ❡①t❡♥❞❛❜❧❡ ✐❢ IT = JT ✐♠♣❧✐❡s I ∗ T = J ∗ T ✳ ❙✐♥❝❡ T ✐s ✢❛t✱ ❡✈❡r② ✐❞❡❛❧ ♦❢ T ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ ❛♥ ✐❞❡❛❧ ♦❢ D ✳ ◆♦t ❡✈❡r② st❛r ♦♣❡r❛t✐♦♥ ✐s ❡①t❡♥❞❛❜❧❡✳ ❋✐♥✐t❡✲t②♣❡ ♦♣❡r❛t✐♦♥s ❛r❡ ❡①t❡♥❞❛❜❧❡✳ ■❢ ∗ ✐s ♦❢ ✜♥✐t❡ t②♣❡ ✭r❡s♣❡❝t✐✈❡❧②✱ s♣❡❝tr❛❧✱ ◆♦❡t❤❡r✐❛♥✮ t❤❡♥ s♦ ✐s ∗ T ✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
❊①t❡♥s✐♦♥ ❛s ❛ ♠❛♣ ▲❡t ❊①t❙t❛r ( D ; T ) ❜❡ t❤❡ s❡t ♦❢ st❛r ♦♣❡r❛t✐♦♥s ♦❢ D t❤❛t ❛r❡ ❡①t❡♥❞❛❜❧❡ t♦ T ✳ ❊①t❡♥s✐♦♥ ❞❡✜♥❡s ❛ ♠❛♣ λ D , T : ❊①t❙t❛r ( D ; T ) − → ❙t❛r ( T ) ∗ �− → ∗ T . λ D , T ✐s ❝♦♥t✐♥✉♦✉s✳ λ D , T ✐s s✉r❥❡❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ✐♠❛❣❡ ❝♦♥t❛✐♥s t❤❡ v ✲♦♣❡r❛t✐♦♥ ✭♦♥ T ✮✳ λ D , T ✐s ❛❧♠♦st ♥❡✈❡r ✐♥❥❡❝t✐✈❡✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
❘❡str✐❝t✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s ❚❤❡ ❝♦♥❝❡♣t ❞✉❛❧ t♦ ❡①t❡♥s✐♦♥ ✐s r❡str✐❝t✐♦♥✿ ✐❢ ∗ ∈ ❙t❛r ( T ) ✱ ✐ts r❡str✐❝t✐♦♥ t♦ D ✐s ∗ ∧ v : I �→ ( IT ) ∗ ∩ I v . ❲❡ ❝❛♥ s❡❡ r❡str✐❝t✐♦♥ ❛s ❛ ♠❛♣ ρ T , D : ❙t❛r ( T ) − → ❙t❛r ( D ) ∗ �− → ∗ ∧ v ρ T , D ✐s ❝♦♥t✐♥✉♦✉s✳ ❘❡str✐❝t✐♦♥ ❞♦❡s♥✬t ♣r❡s❡r✈❡ ♣r♦♣❡rt✐❡s ✭✉♥❧❡ss v ❤❛s t❤❡♠✮✳ ρ T , D ✐s ❛❧♠♦st ♥❡✈❡r s✉r❥❡❝t✐✈❡✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
❋❛♠✐❧✐❡s ♦❢ ♦✈❡rr✐♥❣s ■t ✐s ♠♦r❡ ✉s❡❢✉❧ t♦ ✇♦r❦ ✇✐t❤ ❢❛♠✐❧✐❡s ♦❢ ♦✈❡rr✐♥❣s✿ � λ Θ : ❊①t❙t❛r ( D ; Θ) − → ❙t❛r ( T ) T ∈ Θ ∗ �− → ( ∗ T ) T ∈ Θ ✇❤❡r❡ ❊①t❙t❛r ( D ; Θ) := � T ∈ Θ ❊①t❙t❛r ( D ; T ) ✳ ■♥ t❤❡ s❛♠❡ ✇❛②✱ ✇❡ ❝❛♥ ❞❡✜♥❡ � ρ Θ : ❙t❛r ( T ) − → ❙t❛r ( D ) T ∈ Θ ( ∗ ( T ) ) T ∈ Θ �− → inf { ρ T ( ∗ ( T ) ) | T ∈ Θ } . ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
λ Θ ❛♥❞ ρ Θ λ Θ ✐s ❝♦♥t✐♥✉♦✉s✳ ■❢ Θ ✐s ❧♦❝❛❧❧② ✜♥✐t❡✱ ρ Θ ✐s ❝♦♥t✐♥✉♦✉s✳ ◮ Θ ✐s ❧♦❝❛❧❧② ✜♥✐t❡ ✭♦r ♦❢ ✜♥✐t❡ ❝❤❛r❛❝t❡r ✮ ✐❢✱ ❢♦r ❡✈❡r② x ∈ K ✱ t❤❡r❡ ❛r❡ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② T ∈ Θ s✉❝❤ t❤❛t xT � T ✳ ■❢ Θ ✐s ❝♦♠♣❧❡t❡✱ t❤❡♥ λ Θ ✐s ✐♥❥❡❝t✐✈❡❀ ✐❢ Θ ✐s ❛❧s♦ ❧♦❝❛❧❧② ✜♥✐t❡✱ t❤❡♥ λ Θ ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ ❡♠❜❡❞❞✐♥❣✳ ◮ Θ ✐s ❝♦♠♣❧❡t❡ ✐❢ I = � T ∈ Θ IT ❢♦r ❛❧❧ I ∈ F ( D ) ✳ Pr♦❜❧❡♠s✿ ◮ ❲❤❛t ✐s ❊①t❙t❛r ( D ; Θ) ❄ ◮ ■s λ Θ s✉r❥❡❝t✐✈❡❄ ❲✐t❤ s♦♠❡ ❤②♣♦t❤❡s✐s✱ ✇❡ ❝❛♥ s♦❧✈❡ t❤❡s❡ ♣r♦❜❧❡♠s✿ ◮ ❙✉♣♣♦s❡ D ✐s ◆♦❡t❤❡r✐❛♥✱ ✐♥t❡❣r❛❧❧② ❝❧♦s❡❞✱ ❧♦❝❛❧❧② ✜♥✐t❡✱ ❛♥❞ t❤❛t dim( D ) = ✷✿ t❤❡♥✱ � ❙t❛r ( D ) ≃ ❙t❛r ( D M ) . M ∈ Max( D ) ◮ ■t ✐s ♣♦ss✐❜❧❡ t♦ ✇❡❛❦❡♥ ✏✐♥t❡❣r❛❧❧② ❝❧♦s❡❞✑✱ ❜✉t ♥♦t t❤❡ ♦t❤❡r ❤②♣♦t❤❡s✐s✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s
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