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❊♥❞♦♠♦r♣❤✐s♠s ❛♥❞ ❞❡❝♦♠♣♦s✐t✐♦♥s ♦❢ ❏❛❝♦❜✐❛♥s ❏❡r♦❡♥ ❙✐❥s❧✐♥❣ ❯♥✐✈❡rs✐tät ❯❧♠ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❊❞❣❛r ❈♦st❛✱ ❏❡r♦❡♥ ❍❛♥s❡❧♠❛♥✱ ◆✐❝♦❧❛s ▼❛s❝♦t ❛♥❞ ❏♦❤♥ ❱♦✐❣❤t ■❈❊❘▼✱ Pr♦✈✐❞❡♥❝❡ ✸ ❏✉♥❡ ✷✵✶✾

❲❡ r❡♣r❡s❡♥t ❛♥ ❡❧❡♠❡♥t ❏ ❛s ❢♦❧❧♦✇s✳ ❋✐① ❛ ❜❛s❡ ♣♦✐♥t P ✵ ❳ ✳ ❚❤✐s ❞❡t❡r♠✐♥❡s ❛ ♠❛♣ ❳ ❏ P P P ✵ ✇❤✐❝❤ ✐s ✐♥❥❡❝t✐✈❡ ✐❢ ❣ ✵✳ ❲❡ ❣❡t ❛ ❝♦♠♣♦s❡❞ ♠❛♣ ❳ ❏ ❏ ❣ P P ◗ ✐ ✐ ✶ ❚❤✐s tr❛❝❡s ♦✉t ❛ ❞✐✈✐s♦r ♦♥ ❳ ❳ ✱ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s ✳ ❙❡t✉♣ ▲❡t ❋ ❜❡ ❛ ♥✉♠❜❡r ✜❡❧❞ ✇✐t❤ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ ❋ ❛❧ ✳ ▲❡t ❳ ❜❡ ❛ ♥✐❝❡ ✭s♠♦♦t❤✱ ♣r♦❥❡❝t✐✈❡✱ ❣❡♦♠❡tr✐❝❛❧❧② ✐♥t❡❣r❛❧✮ ❝✉r✈❡ ♦✈❡r ❋ ♦❢ ❣❡♥✉s ❣ ❣✐✈❡♥ ❜② ❡q✉❛t✐♦♥s✳ ▲❡t ❏ ❜❡ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ❳ ✳ ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ ❡♥❞♦♠♦r♣❤✐s♠ r✐♥❣ End( ❏ ) ✳

❙❡t✉♣ ▲❡t ❋ ❜❡ ❛ ♥✉♠❜❡r ✜❡❧❞ ✇✐t❤ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ ❋ ❛❧ ✳ ▲❡t ❳ ❜❡ ❛ ♥✐❝❡ ✭s♠♦♦t❤✱ ♣r♦❥❡❝t✐✈❡✱ ❣❡♦♠❡tr✐❝❛❧❧② ✐♥t❡❣r❛❧✮ ❝✉r✈❡ ♦✈❡r ❋ ♦❢ ❣❡♥✉s ❣ ❣✐✈❡♥ ❜② ❡q✉❛t✐♦♥s✳ ▲❡t ❏ ❜❡ t❤❡ ❏❛❝♦❜✐❛♥ ♦❢ ❳ ✳ ❲❡ ✇❛♥t t♦ ❝♦♠♣✉t❡ t❤❡ ❡♥❞♦♠♦r♣❤✐s♠ r✐♥❣ End( ❏ ) ✳ ❲❡ r❡♣r❡s❡♥t ❛♥ ❡❧❡♠❡♥t α ∈ End( ❏ ) ❛s ❢♦❧❧♦✇s✳ ❋✐① ❛ ❜❛s❡ ♣♦✐♥t P ✵ ∈ ❳ ✳ ❚❤✐s ❞❡t❡r♠✐♥❡s ❛ ♠❛♣ ι : ❳ → ❏ P �→ [ P ] − [ P ✵ ] ✇❤✐❝❤ ✐s ✐♥❥❡❝t✐✈❡ ✐❢ ❣ > ✵✳ ❲❡ ❣❡t ❛ ❝♦♠♣♦s❡❞ ♠❛♣ α ◦ ι : ❳ → ❏ → ❏ ❣ � P �→ α ( ι ( P )) =: ι ( ◗ ✐ ) . ✐ = ✶ ❚❤✐s tr❛❝❡s ♦✉t ❛ ❞✐✈✐s♦r ♦♥ ❳ × ❳ ✱ ✇❤✐❝❤ ❞❡t❡r♠✐♥❡s α ✳

❆❧t❡r♥❛t✐✈❡ r❡♣r❡s❡♥t❛t✐♦♥s α ◦ ι : ❳ → ❏ → ❏ ❣ � P �→ α ( ι ( P )) = ι ( ◗ ✐ ) ✐ = ✶ ❆❧t❡r♥❛t✐✈❡❧②✱ ✇❡ ❝❛♥ ✉s❡ ❛ ✭♣♦ss✐❜❧② s✐♥❣✉❧❛r✮ ♣❧❛♥❡ ❡q✉❛t✐♦♥ ❢ ( ① , ② ) = ✵ ❢♦r ❳ ✳ ❲❡ ❝❛♥ ❞❡s❝r✐❜❡ t❤❡ ♣♦✐♥ts ◗ ✐ ❜② ❣✐✈✐♥❣ ❛ ♣♦❧②♥♦♠✐❛❧ t❤❛t ✈❛♥✐s❤❡s ♦♥ t❤❡✐r ① ✲❝♦♦r❞✐♥❛t❡s✱ ❛❧♦♥❣ ✇✐t❤ ❛ s❡❝♦♥❞ ♣♦❧②♥♦♠✐❛❧ t❤❛t ✐♥t❡r♣♦❧❛t❡s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ② ✲✈❛❧✉❡s✳ ❚❤✐s ❧❡❛❞s t♦ ❈❛♥t♦r ❡q✉❛t✐♦♥s ① ❣ + ❛ ✶ ① ❣ − ✶ + ... + ❛ ❣ = ✵ ❜ ✶ ① ❣ − ✶ + ... + ❜ ❣ = ② ✇✐t❤ ❛ ✐ , ❜ ❥ ∈ ❋ ( ❳ ) ✳

❆❧t❡r♥❛t✐✈❡ r❡♣r❡s❡♥t❛t✐♦♥s ❚❤❡ t❛♥❣❡♥t s♣❛❝❡ ♦❢ ❏ ✐♥ ✵ ✐s ♥❛t✉r❛❧❧② ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ ❞✉❛❧ ♦❢ ❍ ✵ ( ❳ , ω ❳ ) ✱ ❛♥❞ ♦✈❡r C ✇❡ ❤❛✈❡ ❏ ( C ) = ❍ ✵ ( ❳ ( C ) , ω ❳ ) ∨ / ❍ ✶ ( ❳ ( C ) , Z ) . ■❢ ❉ ⊂ ❳ × ❳ ✐s t❤❡ ❞✐✈✐s♦r ❝♦rr❡s♣♦♥❞✐♥❣ t♦ α ✱ t❤❡♥ ❢♦r ❚ = ❚ α ✇❡ ❤❛✈❡ ❚ = (( ♣ ✶ ) ∗ ( ♣ ✷ ) ∗ ) ∨ : ❍ ✵ ( ❳ , ω ❳ ) ∨ → ❍ ✵ ( ❳ , ω ❳ ) ∨ . ❖✈❡r C ✱ ✇❡ ❛❧s♦ ❣❡t ❛ s❡❝♦♥❞✱ ❝♦♠♣❛t✐❜❧❡ ♠❛♣ ❘ : ❍ ✶ ( ❳ ( C ) , Z ) → ❍ ✶ ( ❳ ( C ) , Z ) . ■♥ ♣r❛❝t✐❝❡✱ ✇❡ ❝❤♦♦s❡ ❜❛s❡s ❛♥❞ ❝♦♥s✐❞❡r ❚ ❛s ❛♥ ❡❧❡♠❡♥t ♦❢ M ❣ ( ❋ ❛❧ ) ❛♥❞ ❘ ❛s ❛♥ ❡❧❡♠❡♥t ♦❢ M ✷ ❣ ( Z ) ✳ ❋♦r t❤❡ ♣❡r✐♦❞ ♠❛tr✐① Π ♦❢ ❳ ✇❡ t❤❡♥ ❤❛✈❡ ❚ Π = Π ❘ .

❖✉r ♦❜❥❡❝t✐✈❡✱ ♠♦r❡ ♣r❡❝✐s❡❧② ❋♦r ✉s✱ t♦ ❝♦♠♣✉t❡ t❤❡ ❡♥❞♦♠♦r♣❤✐s♠ r✐♥❣ ♦❢ ❏ ♠❡❛♥s t♦ ❞❡t❡r♠✐♥❡ ❛♥❞ r❡♣r❡s❡♥t t❤❡ r✐♥❣ End( ❏ ❋ ❛❧ ) ❛s ❛ Gal( ❋ ❛❧ | ❋ ) ✲♠♦❞✉❧❡✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✇❡ ✇❛♥t t♦ ❝❛❧❝✉❧❛t❡ ❛ ✜♥✐t❡ ●❛❧♦✐s ❡①t❡♥s✐♦♥ ❑ ⊇ ❋ ✇✐t❤ End( ❏ ❑ ) = End( ❏ ❋ ❛❧ ) ✱ ❛ Z ✲❜❛s✐s ❢♦r End( ❏ ❑ ) ✱ ❛♥❞ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ t❛❜❧❡ ❛s ✇❡❧❧ ❛s t❤❡ ❛❝t✐♦♥ ♦❢ Gal( ❑ | ❋ ) ✭❜♦t❤ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❛❢♦r❡♠❡♥t✐♦♥❡❞ ❜❛s✐s✮✳ ❚❤✐s ❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ❤❛s ♠❛♥② ❛♣♣❧✐❝❛t✐♦♥s✱ ❢♦r ❡①❛♠♣❧❡ ✐♥ ♠♦❞✉❧❛r✐t②✳

▼❛✐♥ ✐❞❡❛✿ ❆♥❞ ♦♥❝❡ t❤❡ t✇❛✐♥ s❤❛❧❧ ♠❡❡t ❉❛✈✐❞❡ ▲♦♠❜❛r❞♦ ❤❛s s❤♦✇♥ t❤❛t t❤❡r❡ ✐s ❛ ❞❛②✲❛♥❞✲♥✐❣❤t ❛❧❣♦r✐t❤♠ t♦ ❝♦♠♣✉t❡ t❤❡ ❣❡♦♠❡tr✐❝ ❡♥❞♦♠♦r♣❤✐s♠ r✐♥❣ ♦❢ ❏ ✳ ❇r✐❡✢②✿ ❇② ❛ t❤❡♦r❡♠ ♦❢ ❙✐❧✈❡r❜❡r❣✱ End( ❏ ❋ ❛❧ ) ✐s ❞❡✜♥❡❞ ♦✈❡r ❑ = ❋ ( ❏ [ ✸ ]) ✳ ❇② ❞❛②✱ ✇❡ ❝♦♠♣✉t❡ ❛ ❧♦✇❡r ❜♦✉♥❞ ❜② s❡❛r❝❤✐♥❣ ❢♦r ❡♥❞♦♠♦r♣❤✐s♠s ❜② ♥❛✐✈❡❧② tr②✐♥❣ ❛❧❧ ♠❛♣s ❏ ��� ❏ ✳ ❇② ♥✐❣❤t✱ ✇❡ ❝♦♠♣✉t❡ ❛♥ ✉♣♣❡r ❜♦✉♥❞ ❜② ❝r❡❡♣✐♥❣ ✉♣ ♦♥ t❤❡ ✐s♦♠♦r♣❤✐s♠ End( ❏ ❑ ) ⊗ Z ℓ ≃ End Gal( ❋ ❛❧ | ❑ ) ❚ ℓ ( ❏ ❑ ) . ❊✈❡♥t✉❛❧❧②✱ t❤❡ ❧♦✇❡r ❛♥❞ ✉♣♣❡r ❜♦✉♥❞s ✇✐❧❧ ♠❡❡t✳ ▼♦r❡ ❡✛❡❝t✐✈❡ ✈❡rs✐♦♥s ♦❢ t❤❡s❡ ✉♣♣❡r ❜♦✉♥❞s ❛r❡ t❤❡♠❡s ♦❢ ♦♥❣♦✐♥❣ ✇♦r❦ ❜② ▲♦♠❜❛r❞♦ ❡t ❛❧✳

❙t❛t❡ ♦❢ t❤❡ ❛rt ♦♥ ✉♣♣❡r ❜♦✉♥❞s t � ❆ ♥ ✐ ✐ , dim ▲ ✐ ❇ ✐ = ❡ ✷ ❆ ∼ ✐ . ✐ = ✶ ❚❤❡♦r❡♠ ■❢ t❤❡ ▼✉♠❢♦r❞✕❚❛t❡ ❝♦♥❥❡❝t✉r❡ ❤♦❧❞s ❢♦r ❆✱ t❤❡♥ ✇❡ ❝❛♥ ❝♦♠♣✉t❡ ❚❤❡ ♥✉♠❜❡r ♦❢ ❢❛❝t♦rs t❀ ❚❤❡ q✉❛♥t✐t② � ✐ ❡ ✐ ♥ ✷ ✐ dim ❆ ✐ ❀ ❚❤❡ s❡t ♦❢ t✉♣❧❡s { ( ❡ ✐ ♥ ✐ , ♥ ✐ dim ❆ ✐ ) } ✐ ✳ ❚❤❡ ❝❡♥t❡rs ▲ ✐ ✳

❨❡ ♦❧❞❡ ❤❡✉r✐st✐❝ ❛♣♣r♦❛❝❤❡ ❚♦ ✜♥❞ ❛ ❧♦✇❡r ❜♦✉♥❞✱ ✇❡ ✜rst ❛♣♣r♦①✐♠❛t❡ t❤❡ ♥✉♠❡r✐❝❛❧ ❡♥❞♦♠♦r♣❤✐s♠ r✐♥❣ ♦❢ ❏ C = C ❣ / Λ ✳ ❚❤❡s❡ ♠❡t❤♦❞s ✇❡r❡ ❛❧s♦ ✉s❡❞ ✐♥ ❣❡♥✉s ❣ = ✷ ❜② ❱❛♥ ❲❛♠❡❧❡♥ ✭❈▼✮ ❛♥❞ ❑✉♠❛r✕▼✉❦❛♠❡❧ ✭❘▼✮✱ ✉s✐♥❣ t❤❡ ❢♦r♠❡r✬s ▼❛❣♠❛ ❛❧❣♦r✐t❤♠s✳ ❊♠❜❡❞ ❋ ❛❧ ֒ → C ✱ ❛♥❞ ❝♦♠♣✉t❡ ✭✈✐❛ ▼♦❧✐♥✕◆❡✉r♦❤r ♦r ❇r✉✐♥✮ ❛ ♣❡r✐♦❞ ♠❛tr✐① Π ❢♦r ❏ t♦ s♦♠❡ ♣r❡❝✐s✐♦♥✱ ✇✐t❤ ♣❡r✐♦❞ ❧❛tt✐❝❡ Λ ✳ ❯s❡ ▲▲▲ t♦ ❞❡t❡r♠✐♥❡ ❛ ❜❛s✐s ♦❢ t❤❡ Z ✲♠♦❞✉❧❡ ♦❢ ♠❛tr✐❝❡s ❘ ∈ M ✷ ❣ ( Z ) s✉❝❤ t❤❛t ❚ Π = Π ❘ ❢♦r s♦♠❡ ❚ ✳ ❉❡t❡r♠✐♥❡ t❤❡ ♠❛tr✐❝❡s ❚ ✐♥ t❤❡ ❡q✉❛❧✐t② ❚ Π = Π ❘ t♦ ♦❜t❛✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ End( ❏ ❑ ) ♦♥ t❤❡ t❛♥❣❡♥t s♣❛❝❡ ❛t ✵✱ ❛♥❞ r❡❝♦❣♥✐③❡ ❚ ❛s ❛♥ ❡❧❡♠❡♥t ♦❢ M ❣ ( ❑ ) ✉s✐♥❣ ▲▲▲✳ ✭✦✦✦✮ ❇② ❡①❛❝t ❝♦♠♣✉t❛t✐♦♥✱ ❝❡rt✐❢② t❤❡ ❡♥❞♦♠♦r♣❤✐s♠s ✐♥ t❤❡ ♣r❡✈✐♦✉s st❡♣✳ ❘❡❝♦✈❡r t❤❡ ●❛❧♦✐s ❛❝t✐♦♥ Gal( ❑ | ❋ ) ❜② t❤❡ ❛❝t✐♦♥ ♦♥ t❤❡ ♠❛tr✐❝❡s ❚ ✳