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Feb 29, 2024 134 likes •473 views

r s ts str rts r rt rst r r s

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SLIDE 1

❏❛✛❛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

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SLIDE 2

❙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

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SLIDE 3

❊①❛♠♣❧❡s

❚❤❡ ✐❞❡♥t✐t② d : I → I✳ ❚❤❡ v✲♦♣❡r❛t✐♦♥ v : I → (D : (D : I))✳ ❚❤❡ t✲♦♣❡r❛t✐♦♥✿ t : I →

  • {Jv | J ⊆ I, J ✐s ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞}.

■❢ Y ⊆ ❖✈❡r(D) ❛♥❞

T∈Y T = D✱ ✇❡ ❝❛♥ ❞❡✜♥❡

∧Y : I →

  • T∈Y

IT. ■❢ ∆ ⊆ Spec(D) ❛♥❞

P∈∆ DP = D✱ ✇❡ ❝❛♥ ❞❡✜♥❡

s∆ : I →

  • P∈∆

IDP.

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 4

❚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

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SLIDE 5

❙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❛rf (D)✳

❚♦♣♦❧♦❣✐❝❛❧ str✉❝t✉r❡✿ t❤❡ t♦♣♦❧♦❣② ✐s t❤❡ ♦♥❡ ❣❡♥❡r❛t❡❞ ❜② t❤❡ s❡ts VI := {∗ ∈ ❙t❛r(D) | ✶ ∈ I ∗}.

◮ ❙t❛rf (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

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SLIDE 6

❊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) IT − → I ∗T ✐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

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SLIDE 7

❊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 ❛ ♠❛♣ λD,T : ❊①t❙t❛r(D; T) − → ❙t❛r(T) ∗ − → ∗T. λD,T ✐s ❝♦♥t✐♥✉♦✉s✳ λD,T ✐s s✉r❥❡❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐ts ✐♠❛❣❡ ❝♦♥t❛✐♥s t❤❡ v✲♦♣❡r❛t✐♦♥ ✭♦♥ T✮✳ λD,T ✐s ❛❧♠♦st ♥❡✈❡r ✐♥❥❡❝t✐✈❡✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

slide-8
SLIDE 8

❘❡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 ❛ ♠❛♣ ρT,D : ❙t❛r(T) − → ❙t❛r(D) ∗ − → ∗ ∧ v ρT,D ✐s ❝♦♥t✐♥✉♦✉s✳ ❘❡str✐❝t✐♦♥ ❞♦❡s♥✬t ♣r❡s❡r✈❡ ♣r♦♣❡rt✐❡s ✭✉♥❧❡ss v ❤❛s t❤❡♠✮✳ ρT,D ✐s ❛❧♠♦st ♥❡✈❡r s✉r❥❡❝t✐✈❡✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 9

❋❛♠✐❧✐❡s ♦❢ ♦✈❡rr✐♥❣s

■t ✐s ♠♦r❡ ✉s❡❢✉❧ t♦ ✇♦r❦ ✇✐t❤ ❢❛♠✐❧✐❡s ♦❢ ♦✈❡rr✐♥❣s✿ λΘ : ❊①t❙t❛r(D; Θ) − →

  • T∈Θ

❙t❛r(T) ∗ − → (∗T)T∈Θ ✇❤❡r❡ ❊①t❙t❛r(D; Θ) :=

T∈Θ ❊①t❙t❛r(D; T)✳

■♥ t❤❡ s❛♠❡ ✇❛②✱ ✇❡ ❝❛♥ ❞❡✜♥❡ ρΘ :

  • T∈Θ

❙t❛r(T) − → ❙t❛r(D) (∗(T))T∈Θ − → inf{ρT(∗(T)) | T ∈ Θ}.

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

slide-10
SLIDE 10

λΘ ❛♥❞ ρΘ

λΘ ✐s ❝♦♥t✐♥✉♦✉s✳ ■❢ Θ ✐s ❧♦❝❛❧❧② ✜♥✐t❡✱ ρΘ ✐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❤❡♥ λΘ ✐s ✐♥❥❡❝t✐✈❡❀ ✐❢ Θ ✐s ❛❧s♦ ❧♦❝❛❧❧② ✜♥✐t❡✱ t❤❡♥ λΘ ✐s ❛ t♦♣♦❧♦❣✐❝❛❧ ❡♠❜❡❞❞✐♥❣✳

◮ Θ ✐s ❝♦♠♣❧❡t❡ ✐❢ I =

T∈Θ IT ❢♦r ❛❧❧ I ∈ F(D)✳

Pr♦❜❧❡♠s✿

◮ ❲❤❛t ✐s ❊①t❙t❛r(D; Θ)❄ ◮ ■s λΘ 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) ≃

  • M∈Max(D)

❙t❛r(DM).

◮ ■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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SLIDE 11

❏❛✛❛r❞ ❢❛♠✐❧✐❡s

❉❡✜♥✐t✐♦♥

❆ ❏❛✛❛r❞ ❢❛♠✐❧② ♦❢ D ✐s ❛ ❢❛♠✐❧② Θ ⊆ ❖✈❡r(D) ♦❢ ✢❛t ♦✈❡rr✐♥❣s s✉❝❤ t❤❛t✿ Θ ✐s ❝♦♠♣❧❡t❡❀ Θ ✐s ❧♦❝❛❧❧② ✜♥✐t❡❀ TS = K ❢♦r ❡✈❡r② T, S ∈ Θ✱ T = S ✭Θ ✐s ✐♥❞❡♣❡♥❞❡♥t✮✳ ❚❤❡ s❡❝♦♥❞ ❛♥❞ t❤❡ t❤✐r❞ ❝♦♥❞✐t✐♦♥ ❛r❡ ❡q✉✐✈❛❧❡♥t t♦ T · Θ⊥(T) = K✱ ✇❤❡r❡ Θ⊥(T) ✐s t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ ❛❧❧ S ∈ Θ \ T✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ P ∈ Spec(D)✱ P = (✵)✱ t❤❡♥ t❤❡r❡ ✐s ❡①❛❝t❧② ♦♥❡ T ∈ Θ s✉❝❤ t❤❛t PT = T✳ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛r❡ ✐♥ ❜✐❥❡❝t✐✈❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ✇✐t❤ ♣❛rt✐❝✉❧❛r ♣❛rt✐t✐♦♥s ♦❢ Max(D) ✭▼❛t❧✐s ♣❛rt✐t✐♦♥s✮✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 12

❏❛✛❛r❞ ❢❛♠✐❧✐❡s ✭✷✮

❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❣❡♥❡r❛❧✐③❡ t❤❡ ❝♦♥❝❡♣t ♦❢ h✲❧♦❝❛❧ ❞♦♠❛✐♥✿ ✐♥❞❡❡❞✱ t❤❡ s❡t {DM | M ∈ Max(D)} ✐s ❛ ❏❛✛❛r❞ ❢❛♠✐❧② ✐❢ ❛♥❞ ♦♥❧② ✐❢ D ✐s h✲❧♦❝❛❧✳

◮ ❆ ❞♦♠❛✐♥ ✐s h✲❧♦❝❛❧ ✐❢ ✐t ✐s ❧♦❝❛❧❧② ✜♥✐t❡ ❛♥❞ ❡✈❡r② ♣r✐♠❡ ✐s ❝♦♥t❛✐♥❡❞ ✐♥

♦♥❧② ♦♥❡ ♠❛①✐♠❛❧ ✐❞❡❛❧✳

■❢ {Xα}α∈A ⊆ ❋(D) ❛♥❞

α Xα = (✵)✱ ❛♥❞ T ∈ Θ✱ t❤❡♥ α∈A

Xα

  • T =
  • α∈A

XαT. ■❢ M ✐s ❛ t♦rs✐♦♥ D✲♠♦❞✉❧❡✱ t❤❡♥ M ≃

  • T∈Θ

M ⊗D T.

◮ ■♥ ♣❛rt✐❝✉❧❛r✱ ✐❢ I = (✵) ✐s ❛♥ ✐❞❡❛❧ ♦❢ D✱ t❤❡♥

D I ≃

  • T∈Θ

T IT .

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 13

❚❤❡ ♠❛✐♥ t❤❡♦r❡♠

❚❤❡♦r❡♠

▲❡t D ❜❡ ❛♥ ✐♥t❡❣r❛❧ ❞♦♠❛✐♥ ❛♥❞ Θ ❜❡ ❛ ❏❛✛❛r❞ ❢❛♠✐❧② ♦❢ D✳ ❚❤❡♥✿ ❡✈❡r② ∗ ∈ ❙t❛r(D) ✐s ❡①t❡♥❞❛❜❧❡ t♦ ❡✈❡r② T ∈ Θ❀ λΘ ❛♥❞ ρΘ ❛r❡ ❤♦♠❡♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ ❙t❛r(D) ❛♥❞

  • T∈Θ

❙t❛r(T)✳ ❚❤❡ s❛♠❡ ❤♦❧❞s ✐❢✱ ✐♥st❡❛❞ ♦❢ t❤❡ s❡t ♦❢ ❛❧❧ st❛r ♦♣❡r❛t✐♦♥s✱ ✇❡ ❝♦♥s✐❞❡r ♦♥❧② ✜♥✐t❡✲t②♣❡✱ s♣❡❝tr❛❧✱ st❛❜❧❡✱ ♦r ◆♦❡t❤❡r✐❛♥ st❛r ♦♣❡r❛t✐♦♥s✳ ❚❤❡ t❤❡♦r❡♠ ❞♦❡s ♥♦t ❤♦❧❞ ❢♦r s❡♠✐st❛r ♦♣❡r❛t✐♦♥s✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 14

❈♦♥s❡q✉❡♥❝❡s ♦❢ t❤❡ ♠❛✐♥ t❤❡♦r❡♠

■❢ dim(D) = ✶ ❛♥❞ D ✐s ❧♦❝❛❧❧② ✜♥✐t❡✱ t❤❡♥ ❙t❛r(D) ≃

  • M∈Max(D)

❙t❛r(DM). D ❤❛s ❛♥ m✲❝❛♥♦♥✐❝❛❧ ✐❞❡❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢✿

◮ D ✐s h✲❧♦❝❛❧❀ ◮ DM ❤❛s ❛♥ m✲❝❛♥♦♥✐❝❛❧ ✐❞❡❛❧ ❢♦r ❡✈❡r② M ∈ Max(D)❀ ◮ |❙t❛r(DM)| > ✶ ❢♦r ♦♥❧② ✜♥✐t❡❧② ♠❛♥② M ∈ Max(D)✳

■❢ ∗ ∈ ❙t❛r(D)✱ t❤❡♥ ❈❧∗(D) P✐❝(D) ≃

  • T∈Θ

❈❧∗T (T) P✐❝(T) .

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 15

Prü❢❡r ❞♦♠❛✐♥s

❲❤❡♥ D ✐s ❛ Prü❢❡r ❞♦♠❛✐♥✱ t❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡ ❢♦r Θ✳ ❙❛② M, N ∈ Max(D) ❛r❡ ❞❡♣❡♥❞❡♥t ✐❢ t❤❡r❡ ✐s ❛ ♣r✐♠❡ ✐❞❡❛❧ P = (✵) s✉❝❤ t❤❛t P ⊆ M ∩ N✳ ❉❡♣❡♥❞❡♥❝❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳ ❋♦r ❛❧❧ M ∈ Max(D)✱ ❞❡✜♥❡ T(M) ❛s t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ DN✱ ❛s N r❛♥❣❡s ✐♥ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss ♦❢ M✳ ❚❛❦❡ Θ := {T(M) | M ∈ Max(D)}✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 16

Prü❢❡r ❞♦♠❛✐♥s

❲❤❡♥ D ✐s ❛ Prü❢❡r ❞♦♠❛✐♥✱ t❤❡r❡ ✐s ❛ ♥❛t✉r❛❧ ❝❛♥❞✐❞❛t❡ ❢♦r Θ✳ ❙❛② M, N ∈ Max(D) ❛r❡ ❞❡♣❡♥❞❡♥t ✐❢ t❤❡r❡ ✐s ❛ ♣r✐♠❡ ✐❞❡❛❧ P = (✵) s✉❝❤ t❤❛t P ⊆ M ∩ N✳ ❉❡♣❡♥❞❡♥❝❡ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳ ❋♦r ❛❧❧ M ∈ Max(D)✱ ❞❡✜♥❡ T(M) ❛s t❤❡ ✐♥t❡rs❡❝t✐♦♥ ♦❢ DN✱ ❛s N r❛♥❣❡s ✐♥ t❤❡ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss ♦❢ M✳ ❚❛❦❡ Θ := {T(M) | M ∈ Max(D)}✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 17

Prü❢❡r ❞♦♠❛✐♥s ✭✷✮

Pr♦❜❧❡♠✿ Θ ♠❛② ♥♦t ❜❡ ❛ ❏❛✛❛r❞ ❢❛♠✐❧②✳

◮ ❋♦r ❡①❛♠♣❧❡✱ ✐t ♠❛② ♥♦t ❜❡ ❧♦❝❛❧❧② ✜♥✐t❡ ✭❡✳❣✳✱ ❛♥ ❛❧♠♦st ❉❡❞❡❦✐♥❞

❞♦♠❛✐♥ ✇❤✐❝❤ ✐s ♥♦t ❉❡❞❡❦✐♥❞✮✳

◮ ■t ✇♦r❦s ✐❢ ✇❡ r❡str✐❝t t♦ ❧♦❝❛❧❧② ✜♥✐t❡ ❞♦♠❛✐♥s✳

Pr♦❜❧❡♠✿ ✐❢ T ✐s ♥♦t ❛ ✈❛❧✉❛t✐♦♥ ❞♦♠❛✐♥✱ ✇❡ ❞♦♥✬t ❦♥♦✇ ❙t❛r(T)✳

◮ ❙✉♣♣♦s❡ D ✐s s❡♠✐❧♦❝❛❧✱ ♦r ❧♦❝❛❧❧② ✜♥✐t❡ ❛♥❞ ✜♥✐t❡✲❞✐♠❡♥s✐♦♥❛❧✳ ◮ ❚❤❡♥✱ ❏❛❝(D) ❝♦♥t❛✐♥s ❛ ♣r✐♠❡ ✐❞❡❛❧ P✳ ◮ ❲❡ ✇❛♥t t♦ ❧✐♥❦ ❙t❛r(T) ❛♥❞ ❙t❛r(T/P)✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 18

❈✉tt✐♥❣ t❤❡ ❜r❛♥❝❤

❙✉♣♣♦s❡ t❤❡r❡ ✐s ❛ ♣r✐♠❡ ✐❞❡❛❧ P = (✵) ✐♥s✐❞❡ t❤❡ ❏❛❝♦❜s♦♥ r❛❞✐❝❛❧✳ P = PDP ✭P ✐s ❞✐✈✐❞❡❞✮✳ ❊✈❡r② D✲s✉❜♠♦❞✉❧❡ ♦❢ K ✐s ❛ ❢r❛❝t✐♦♥❛❧ ✐❞❡❛❧✳ ◆♦♥✲❞✐✈✐s♦r✐❛❧ ✐❞❡❛❧s ❝♦rr❡s♣♦♥❞ t♦ D/P✲s✉❜♠♦❞✉❧❡s ♦❢ DP/P✳ [❋♦♥t❛♥❛ ❛♥❞ P❛r❦✱ ✷✵✵✹❀ ❍♦✉st♦♥✱ ▼✐♠♦✉♥✐ ❛♥❞ P❛r❦✱ ✷✵✶✹] ❙t❛r ♦♣❡r❛t✐♦♥s ❝♦rr❡s♣♦♥❞ t♦ s❡♠✐st❛r ♦♣❡r❛t✐♦♥s s✉❝❤ t❤❛t (D/P)∗ = (D/P)✳ P ❙t❛r(D) ≃ (❙)❙t❛r(D/P)

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 19

❆♥ ✐♥❞✉❝t✐✈❡ ❛r❣✉♠❡♥t

❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❣♦ ❢r♦♠ (❙)❙t❛r(R) t♦ ❙t❛r(R)✱ ❢♦r ❡✈❡r② ✭s❡♠✐❧♦❝❛❧✮ Prü❢❡r ❞♦♠❛✐♥✳ M✶ M✷ N P

❙t❛r(D)

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 20

❆♥ ✐♥❞✉❝t✐✈❡ ❛r❣✉♠❡♥t

❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❣♦ ❢r♦♠ (❙)❙t❛r(R) t♦ ❙t❛r(R)✱ ❢♦r ❡✈❡r② ✭s❡♠✐❧♦❝❛❧✮ Prü❢❡r ❞♦♠❛✐♥✳ M✶ M✷ N P

❙t❛r(D) ❙t❛r(D✶) ❙t❛r(DN)

❡①t❡♥s✐♦♥

D✶ :=DM✶ ∩ DM✷

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 21

❆♥ ✐♥❞✉❝t✐✈❡ ❛r❣✉♠❡♥t

❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❣♦ ❢r♦♠ (❙)❙t❛r(R) t♦ ❙t❛r(R)✱ ❢♦r ❡✈❡r② ✭s❡♠✐❧♦❝❛❧✮ Prü❢❡r ❞♦♠❛✐♥✳ M✶ M✷ N P

❙t❛r(D) ❙t❛r(D✶) ❙t❛r(DN) (❙)❙t❛r(D✶/P)

❡①t❡♥s✐♦♥ ❝✉tt✐♥❣

D✶ :=DM✶ ∩ DM✷

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 22

❆♥ ✐♥❞✉❝t✐✈❡ ❛r❣✉♠❡♥t

❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❣♦ ❢r♦♠ (❙)❙t❛r(R) t♦ ❙t❛r(R)✱ ❢♦r ❡✈❡r② ✭s❡♠✐❧♦❝❛❧✮ Prü❢❡r ❞♦♠❛✐♥✳ M✶ M✷ N P

❙t❛r(D) ❙t❛r(D✶) ❙t❛r(DN) (❙)❙t❛r(D✶/P) ❙t❛r(D✶/P)

❡①t❡♥s✐♦♥ ❝✉tt✐♥❣ ❄❄❄

D✶ :=DM✶ ∩ DM✷

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 23

❆♥ ✐♥❞✉❝t✐✈❡ ❛r❣✉♠❡♥t

❙✉♣♣♦s❡ ✇❡ ❝❛♥ ❣♦ ❢r♦♠ (❙)❙t❛r(R) t♦ ❙t❛r(R)✱ ❢♦r ❡✈❡r② ✭s❡♠✐❧♦❝❛❧✮ Prü❢❡r ❞♦♠❛✐♥✳ M✶ M✷ N P

❙t❛r(D) ❙t❛r(D✶) ❙t❛r(DN) (❙)❙t❛r(D✶/P) ❙t❛r(D✶/P) ❙t❛r(DM✶/P) ❙t❛r(DM✷/P)

❡①t❡♥s✐♦♥ ❝✉tt✐♥❣ ❄❄❄ ❡①t❡♥s✐♦♥

D✶ :=DM✶ ∩ DM✷

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 24

❇r❛♥❝❤✐♥❣ ♣♦✐♥ts

❯♣ t♦ t❤❡ ♣❛ss❛❣❡ ❙t❛r(R) → (❙)❙t❛r(R)✱ t❤❡ ❛❧❣♦r✐t❤♠s ♥❡❡❞s t✇♦ t❤✐♥❣s✿ t❤❡ ❣❡♦♠❡tr② ♦❢ Spec(D)✿ ✇❡ ♥❡❡❞ t♦ ❦♥♦✇ t❤❡ ✏❜r❛♥❝❤✐♥❣ ♣♦✐♥ts✑ ♦❢ t❤❡ s♣❡❝tr✉♠ t♦ ❦♥♦✇ t❤❡ ♣❧❛❝❡s ✐♥ ✇❤✐❝❤ ✇❡ ❝✉t ❛♥❞ ✐♥ ✇❤✐❝❤ ✇❡ ❧♦❝❛❧✐③❡❀ t❤❡ st❛r ♦♣❡r❛t✐♦♥s ♦♥ DQ/PDQ✱ ✇❤❡r❡ P Q ❛r❡ s✉❝❝❡ss✐✈❡ ❜r❛♥❝❤✐♥❣ ♣♦✐♥ts ♦❢ D✳

◮ ❊✈❡r② DQ/PDQ ✐s ❛ ✈❛❧✉❛t✐♦♥ ❞♦♠❛✐♥✳ ◮ ❙t❛r(DQ/PDQ) ❞❡♣❡♥❞s ♦♥❧② ♦♥ ✇❤❡t❤❡r Q ✐s ✐❞❡♠♣♦t❡♥t ♦r ♥♦t✳ ❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 25

❋r❛❝t✐♦♥❛❧ st❛r ♦♣❡r❛t✐♦♥s

❙t❛r ❛♥❞ s❡♠✐st❛r ♦♣❡r❛t✐♦♥s ❛r❡ ❞❡✜♥❡❞ s✐♠✐❧❛r❧②✿ ❝❧♦s✉r❡ ♦♣❡r❛t✐♦♥s ♦♥ ❛ s❡t ♦❢ ♠♦❞✉❧❡s s❛t✐s❢②✐♥❣ (xI)∗ = x · I ∗✳ st❛r ♦♣❡r❛t✐♦♥s s❡♠✐st❛r ♦♣❡r❛t✐♦♥s D = D∗ D∗ ❛r❜✐tr❛r② F(D) ❋(D) ❋r❛❝t✐♦♥❛❧ st❛r ♦♣❡r❛t✐♦♥s s❛t✐s❢② t❤❡ ♠❛✐♥ t❤❡♦r❡♠✳ ❲❡ ❝❛♥ ❝♦♥tr♦❧ t❤❡ ♣❛ss❛❣❡ ❢r♦♠ ❋❙t❛r t♦ ❙❙t❛r ✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 26

❋r❛❝t✐♦♥❛❧ st❛r ♦♣❡r❛t✐♦♥s

❙t❛r ❛♥❞ s❡♠✐st❛r ♦♣❡r❛t✐♦♥s ❛r❡ ❞❡✜♥❡❞ s✐♠✐❧❛r❧②✿ ❝❧♦s✉r❡ ♦♣❡r❛t✐♦♥s ♦♥ ❛ s❡t ♦❢ ♠♦❞✉❧❡s s❛t✐s❢②✐♥❣ (xI)∗ = x · I ∗✳ st❛r ♦♣❡r❛t✐♦♥s s❡♠✐st❛r ♦♣❡r❛t✐♦♥s ✭s❡♠✐✮st❛r ♦♣❡r❛t✐♦♥s D = D∗ D∗ ❛r❜✐tr❛r② F(D) ❋(D) ❋r❛❝t✐♦♥❛❧ st❛r ♦♣❡r❛t✐♦♥s s❛t✐s❢② t❤❡ ♠❛✐♥ t❤❡♦r❡♠✳ ❲❡ ❝❛♥ ❝♦♥tr♦❧ t❤❡ ♣❛ss❛❣❡ ❢r♦♠ ❋❙t❛r t♦ ❙❙t❛r ✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 27

❋r❛❝t✐♦♥❛❧ st❛r ♦♣❡r❛t✐♦♥s

❙t❛r ❛♥❞ s❡♠✐st❛r ♦♣❡r❛t✐♦♥s ❛r❡ ❞❡✜♥❡❞ s✐♠✐❧❛r❧②✿ ❝❧♦s✉r❡ ♦♣❡r❛t✐♦♥s ♦♥ ❛ s❡t ♦❢ ♠♦❞✉❧❡s s❛t✐s❢②✐♥❣ (xI)∗ = x · I ∗✳ st❛r ♦♣❡r❛t✐♦♥s ❢r❛❝t✐♦♥❛❧ st❛r ♦♣❡r❛t✐♦♥s s❡♠✐st❛r ♦♣❡r❛t✐♦♥s ✭s❡♠✐✮st❛r ♦♣❡r❛t✐♦♥s D = D∗ D∗ ❛r❜✐tr❛r② F(D) ❋(D) ❋r❛❝t✐♦♥❛❧ st❛r ♦♣❡r❛t✐♦♥s s❛t✐s❢② t❤❡ ♠❛✐♥ t❤❡♦r❡♠✳ ❲❡ ❝❛♥ ❝♦♥tr♦❧ t❤❡ ♣❛ss❛❣❡ ❢r♦♠ ❋❙t❛r(D) t♦ ❙❙t❛r(D/P)✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 28
  • ❧✉✐♥❣ ❢r❛❝t✐♦♥❛❧ st❛r ♦♣❡r❛t✐♦♥s

▲❡t D ❜❡ ❛ s❡♠✐❧♦❝❛❧ Prü❢❡r ❞♦♠❛✐♥ ❛♥❞ ❧❡t Θ ❜❡ ✐ts ❏❛✛❛r❞ ❢❛♠✐❧②✳ ❚❤❡r❡ ✐s ❛ ✭❡①♣❧✐❝✐t✱ ✉♥✐q✉❡✮ ❢❛♠✐❧② ❙❦❖✈❡r(D) ♦❢ ♦✈❡rr✐♥❣s s✉❝❤ t❤❛t ❡✈❡r② D✲s✉❜♠♦❞✉❧❡ ♦❢ K ✐s ❛ ❢r❛❝t✐♦♥❛❧ ✐❞❡❛❧ ♦✈❡r ❡①❛❝t❧② ♦♥❡ U ∈ ❙❦❖✈❡r(D)✳ ∗ ∈ ❙❙t❛r(D) ✐s ❞❡t❡r♠✐♥❡❞ ❜②✿

◮ t❤❡ s❡t ∆∗ ♦❢ U ∈ ❙❦❖✈❡r(D) s✉❝❤ t❤❛t U∗ ∈ F(U)❀ ◮ ∗|F(U) ∈ ❋❙t❛r(U)✱ ❢♦r U ∈ ∆∗✳

❚❤❡ s❡t ♦❢ ∗ s✉❝❤ t❤❛t ∆∗ = ∆ ✐s ❡♠♣t② ♦r ✐s♦♠♦r♣❤✐❝ t♦

  • T∈Θ

hom(∆(T), ❋❙t❛r(T)) ✇❤❡r❡ ∆(T) := {U ∈ ❙❦❖✈❡r(D) | U ⊆ T} ❛♥❞ hom ❛r❡ t❤❡ ♦r❞❡r✲♣r❡s❡r✈✐♥❣ ♠❛♣s✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 29

❆♥ ✐♥❞✉❝t✐✈❡ ❛r❣✉♠❡♥t ✭✷✮

M✶ M✷ N P

❙❙t❛r(D) ❋❙t❛r(D) ❋❙t❛r(D✶) ❋❙t❛r(DN) ❙❙t❛r(D✶/P) ❋❙t❛r(DP) ❋❙t❛r(D✶/P) ❋❙t❛r(DM✶/P) ❋❙t❛r(DM✷/P)

❣❧✉✐♥❣ ❡①t❡♥s✐♦♥ ❝✉tt✐♥❣ ❣❧✉✐♥❣ ❡①t❡♥s✐♦♥

D✶ :=DM✶ ∩ DM✷

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 30

Prü❢❡r ❞♦♠❛✐♥s ✇✐t❤ t❤❡ s❛♠❡ st❛r ♦♣❡r❛t✐♦♥s

❋♦r (❙)❙t❛r(D) ❛♥❞ ❙t❛r(D) t❤❡ r❡❛s♦♥✐♥❣ ✐s s✐♠✐❧❛r✳

◮ ❊✈❡♥ ❢♦r t❤❡♠✱ ②♦✉ st✐❧❧ ❤❛✈❡ t♦ ✉s❡ ❋❙t❛r(D)✳

▲❡t D, D′ ❜❡ s❡♠✐❧♦❝❛❧ Prü❢❡r ❞♦♠❛✐♥s s✉❝❤ t❤❛t✿

◮ t❤❡r❡ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ φ ❜❡t✇❡❡♥ t❤❡ s❡t ♦❢ t❤❡ ❜r❛♥❝❤✐♥❣ ♣♦✐♥ts ♦❢ D

❛♥❞ D′❀

◮ ❙❙t❛r(DP) ≃ ❙❙t❛r(D′

φ(P)) ❢♦r ❛❧❧ ❜r❛♥❝❤✐♥❣ ♣♦✐♥ts P✳

❚❤❡♥✱ ❙❙t❛r(D) ≃ ❙❙t❛r(D′)✳

◮ ❙✉♣♣♦s❡ M ✐s ✐❞❡♠♣♦t❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ φ(M) ✐s ✐❞❡♠♣♦t❡♥t✱ ❢♦r ❡✈❡r②

♠❛①✐♠❛❧ ✐❞❡❛❧ M✳ ❚❤❡♥✱ ❙t❛r(D) ≃ ❙t❛r(D′)✳

▲❡t D, D′ ❜❡ s❡♠✐❧♦❝❛❧ Prü❢❡r ❞♦♠❛✐♥s s✉❝❤ t❤❛t✿

◮ t❤❡r❡ ✐s ❛ ❤♦♠❡♦♠♦r♣❤✐s♠ φ : Spec(D) −

→ Spec(D′)❀

◮ P ✐s ✐❞❡♠♣♦t❡♥t ✐❢ ❛♥❞ ♦♥❧② ✐❢ φ(P) ✐s ✐❞❡♠♣♦t❡♥t✳

❚❤❡♥✱ ❙❙t❛r(D) ≃ ❙❙t❛r(D′) ❛♥❞ ❙t❛r(D) ≃ ❙t❛r(D′)✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 31

❙t❛r ♦♣❡r❛t✐♦♥s ♦♥ Prü❢❡r ❞♦♠❛✐♥s

❙✉♣♣♦s❡ D ✐s ❛ s❡♠✐❧♦❝❛❧ Prü❢❡r ❞♦♠❛✐♥✳ ■❢ dim(D) ✐s ✜♥✐t❡✱ s♦ ❛r❡ ❙t❛r(D) ❛♥❞ ❙❙t❛r(D)✳

◮ ❨♦✉ ❝❛♥ ❛❝t✉❛❧❧② ❝❛❧❝✉❧❛t❡ t❤❡ ❝❛r❞✐♥❛❧✐t②✳

❚❤❡ s❡t ♦❢ st❛❜❧❡ st❛r ♦♣❡r❛t✐♦♥s ✐s ✐s♦♠♦r♣❤✐❝ t♦

  • M∈Max(D)

❙t❛r(DM)✳

◮ ❊q✉✐✈❛❧❡♥t❧②✱ t♦ t❤❡ ♣♦✇❡r s❡t ♦❢ {M ∈ Max(D) | M = Mv}✳ ◮ D ✐s h✲❧♦❝❛❧ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡✈❡r② st❛r ♦♣❡r❛t✐♦♥ ✐s st❛❜❧❡✳

■❢ I = I ∗ ❛♥❞ J = J∗ ❛r❡ ∗✲✐♥✈❡rt✐❜❧❡✱ t❤❡♥ s♦ ✐s I + J✳

◮ L ✐s ∗✲✐♥✈❡rt✐❜❧❡ ✐❢ (L(D : L))∗ = D✳

❋♦r ❡✈❡r② ∗ ∈ ❙t❛r(D)✱ ❈❧∗(D) ≃

  • M∈Max(D)

M=M∗

❈❧v(DM) ❛♥❞ ✇❡ ❦♥♦✇ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡✿ ✐t ✐s (✵) ♦r R/H ❢♦r s♦♠❡ s✉❜❣r♦✉♣ H ❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ✈❛❧✉❡ ❣r♦✉♣ ♦❢ DM✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s

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SLIDE 32

❇✐❜❧✐♦❣r❛♣❤②

▼❛r❝♦ ❋♦♥t❛♥❛ ❛♥❞ ▼✐ ❍❡❡ P❛r❦✳ ❙t❛r ♦♣❡r❛t✐♦♥s ❛♥❞ ♣✉❧❧❜❛❝❦s✳ ❏✳ ❆❧❣❡❜r❛✱ ✷✼✹✭✶✮✿✸✽✼✕✹✷✶✱ ✷✵✵✹✳ ❊✈❛♥ ●✳ ❍♦✉st♦♥✱ ❆❜❞❡s❧❛♠ ▼✐♠♦✉♥✐✱ ❛♥❞ ▼✐ ❍❡❡ P❛r❦✳ ■♥t❡❣r❛❧❧② ❝❧♦s❡❞ ❞♦♠❛✐♥s ✇✐t❤ ♦♥❧② ✜♥✐t❡❧② ♠❛♥② st❛r ♦♣❡r❛t✐♦♥s✳ ❈♦♠♠✳ ❆❧❣❡❜r❛✱ ✹✷✭✶✷✮✿✺✷✻✹✕✺✷✽✻✱ ✷✵✶✹✳ ❉❛r✐♦ ❙♣✐r✐t♦✳ ❚❤❡ s❡ts ♦❢ st❛r ❛♥❞ s❡♠✐st❛r ♦♣❡r❛t✐♦♥s ♦♥ Prü❢❡r ❞♦♠❛✐♥s✳ s✉❜♠✐tt❡❞✳ ❉❛r✐♦ ❙♣✐r✐t♦✳ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❧♦❝❛❧✐③❛t✐♦♥s ♦❢ st❛r ♦♣❡r❛t✐♦♥s✳ ❏✳ ❈♦♠♠✉t✳ ❆❧❣❡❜r❛✱ t♦ ❛♣♣❡❛r✳

❉❛r✐♦ ❙♣✐r✐t♦ ❏❛✛❛r❞ ❢❛♠✐❧✐❡s ❛♥❞ ❡①t❡♥s✐♦♥ ♦❢ st❛r ♦♣❡r❛t✐♦♥s