▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s ▼❛r❡❦ ❲ó❥t♦✇✐❝③ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ❑❛③✐♠✐❡r③ ❲✐❡❧❦✐ ❯♥✐✈❡rs✐t② ✐♥ ❇②❞❣♦s③❝③ ❚r❛♥s✜♥✐t❡ ♠❡t❤♦❞s ✐♥ ❇❛♥❛❝❤ s♣❛❝❡s ❛♥❞ ❛❧❣❡❜r❛s ♦❢ ♦♣❡r❛t♦rs ❇❡❞❧❡✇♦✱ ✶✽✲✷✷ ❏✉♥❡ ✷✵✶✺ ▼❛r❡❦ ❲ó❥t♦✇✐❝③ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s
❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❙❡t ❘ ▲❡t ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ ✳ ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r ✶ ✱ ❛♥❞ ✳ ✵ ❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲ α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ▼❛r❡❦ ❲ó❥t♦✇✐❝③ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s
❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ▲❡t ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ ✳ ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r ✶ ✱ ❛♥❞ ✳ ✵ ❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲ α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F ( X ) = { f : X → ❘ : f has property F} . ▼❛r❡❦ ❲ó❥t♦✇✐❝③ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s
❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r ✶ ✱ ❛♥❞ ✳ ✵ ❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲ α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F ( X ) = { f : X → ❘ : f has property F} . ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X ✳ ▼❛r❡❦ ❲ó❥t♦✇✐❝③ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s
❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r ✶ ✱ ❛♥❞ ✳ ✵ ❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲ α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F ( X ) = { f : X → ❘ : f has property F} . ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X ✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Ext : F ( A ) → F ( X )? ▼❛r❡❦ ❲ó❥t♦✇✐❝③ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s
❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ s✉❝❤ t❤❛t t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r ✶ ✱ ❛♥❞ ✳ ✵ ❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲ α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F ( X ) = { f : X → ❘ : f has property F} . ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X ✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Ext : F ( A ) → F ( X )? ▼❛r❡❦ ❲ó❥t♦✇✐❝③ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s
t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ s✉♠ ♦❢ t✇♦ ❢✉♥❝t✐♦♥s ✐s t❤❡ s✉♠ ♦❢ ❡①t❡♥s✐♦♥s✳ ❞❡♥♦t❡s t❤❡ s♣❛❝❡ ♦❢ ❛❧❧ ❢✉♥❝t✐♦♥s ❘ ♦❢ ❇❛✐r❡ ❝❧❛ss ✱ ✇❤❡r❡ ✐s ❛♥ ♦r❞✐♥❛❧ ♥✉♠❜❡r ✶ ✱ ❛♥❞ ✳ ✵ ❚❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♣r♦❜❧❡♠ ▲❡t X ❜❡ ❛ ❍❛✉s❞♦r✛ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡✱ ❛♥❞ ❧❡t F ❜❡ ❛ ♣r♦♣❡rt② ♦❢ ❢✉♥❝t✐♦♥s ❞❡✜♥❡❞ ♦♥ X ✭❡✳❣✳✱ ❝♦♥t✐♥✉✐t②✱ t♦ ❜❡ ❛ ❇❛✐r❡✲ α ❢✉♥❝t✐♦♥✱ ❡t❝✳✮✳ ❙❡t F ( X ) = { f : X → ❘ : f has property F} . ▲❡t A ❜❡ ❛ ♥♦♥❡♠♣t② ✭❇♦r❡❧✱ ✐♥ ❣❡♥❡r❛❧✮ s✉❜s❡t ♦❢ X ❡♥❞♦✇❡❞ ✇✐t❤ t❤❡ t♦♣♦❧♦❣② ✐♥❞✉❝❡❞ ❢r♦♠ X ✳ ❉♦❡s t❤❡r❡ ❡①✐st ❛ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦♣❡r❛t♦r Ext : F ( A ) → F ( X )? ❚❤✉s✱ ✇❡ ❛s❦ ❢♦r ❛ ♠❡t❤♦❞ ♦❢ ❡①t❡♥❞✐♥❣ ❡❧❡♠❡♥ts ♦❢ F ( A ) s✉❝❤ t❤❛t ▼❛r❡❦ ❲ó❥t♦✇✐❝③ ✭❏♦✐♥t ✇♦r❦ ✇✐t❤ ❲✳ ❙✐❡❣✮ ▲✐♥❡❛r ❡①t❡♥s✐♦♥s ♦❢ ❇❛✐r❡ ❢✉♥❝t✐♦♥s
Recommend
More recommend