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▼❛❝❦❡② ❢✉♥❝t♦rs ❛♥❞ ●r❡❡♥ ❢✉♥❝t♦rs ❊❧❛♥❣♦ P❛♥❝❤❛❞❝❤❛r❛♠ ❏♦✐♥t ✇♦r❦ ✇✐t❤ Pr♦❢❡ss♦r ❘♦ss ❙tr❡❡t ❈❡♥tr❡ ♦❢ ❆✉str❛❧✐❛♥ ❈❛t❡❣♦r② ❚❤❡♦r② ▼❛❝q✉❛r✐❡ ❯♥✐✈❡rs✐t② ❙②❞♥❡② ❆✉str❛❧✐❛ ✶

▼❛✐♥ ❘❡❢❡r❡♥❝❡s ✶✳ ❏✳ ❆✳ ●r❡❡♥✱ ❆①✐♦♠❛t✐❝ r❡♣r❡s❡♥t❛t✐♦♥ t❤❡♦r② ❢♦r ✜♥✐t❡ ❣r♦✉♣s ✱ ❏✳ P✉r❡ ❛♥❞ ❆♣♣❧✳ ❆❧❣❡❜r❛ ✶ ✭✶✾✼✶✮✱ ✹✶✕✼✼✳ ✷✳ ❆✳ ❲✳ ▼✳ ❉r❡ss✱ ❈♦♥tr✐❜✉t✐♦♥s t♦ t❤❡ t❤❡♦r② ♦❢ ✐♥❞✉❝❡❞ r❡♣r❡s❡♥t❛t✐♦♥s ✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳ ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✮ ✸✹✷ ✭❆❧❣❡❜r❛✐❝ ❑✲❚❤❡♦r② ■■✮ ✭✶✾✼✸✮✱ ✶✽✸ ✕✷✹✵✳ ✸✳ ❍✳ ▲✐♥❞♥❡r✱ ❆ r❡♠❛r❦ ♦♥ ▼❛❝❦❡② ❢✉♥❝t♦rs ✱ ▼❛♥✉s❝r✐♣t❛ ▼❛t❤✳ ✶✽ ✭✶✾✼✻✮✱ ✷✼✸✕✷✼✽✳ ✹✳ ❙✳ ❇♦✉❝✱ ●r❡❡♥ ❢✉♥❝t♦rs ❛♥❞ ●✲s❡ts ✱ ▲❡❝t✉r❡ ◆♦t❡s ✐♥ ▼❛t❤✳ ✭❙♣r✐♥❣❡r✲❱❡r❧❛❣✱ ❇❡r❧✐♥✮ ✶✻✼✶ ✭✶✾✾✼✮✳ ✷

� � � � ❚❤❡ ❈♦♠♣❛❝t ❝❧♦s❡❞ ❝❛t❡❣♦r② Spn ( E ) • ▲❡t E ❜❡ ❛ ✜♥✐t❡❧② ❝♦♠♣❧❡t❡ ❝❛t❡❣♦r②✳ • ❖❜❥❡❝ts ♦❢ Spn ( E ) ❛r❡ t❤❡ ♦❜❥❡❝ts ♦❢ E ✳ • ▼♦r♣❤✐s♠s U � V ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠s ❝❧❛ss ♦❢ s♣❛♥s ❢r♦♠ U t♦ V ✳ • ❆ s♣❛♥ ❢r♦♠ U t♦ V ✐s ❛ ❞✐❛❣r❛♠✱ S s 1 s 2 � � � ( s 1 , S , s 2 ) : � � � � � � � � U V • ❆♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ t✇♦ s♣❛♥s ( s 1 , S , s 2 ) : U � V ❛♥❞ � S ′ s✉❝❤ ( s ′ 1 , S ′ , s ′ � V ✐s ❛♥ ✐♥✈❡rt✐❜❧❡ ❛rr♦✇ h : S 2 ) : U t❤❛t ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠ ❝♦♠♠✉t❡s✳ S s 1 s 2 � � � ������� � � � � � � ∼ U V h = � ������� � � � � � s ′ s ′ � � S ′ 1 2 ✸

� � � � • ❚❤❡ ❝♦♠♣♦s✐t❡ ♦❢ t✇♦ s♣❛♥s ( s 1 , S , s 2 ) : U � V ❛♥❞ � W ✐s ( s 1 ◦ p 1 , S × V T , t 2 ◦ p 2 ) ( t 1 , T , t 2 ) : V S × V T p 1 � p 2 � ������ � � � � � S T � � s 1 t 2 � ������� � � ������� � � � � � s 2 � � t 1 � � � � U V W • ❚❤❡ ✐❞❡♥t✐t② s♣❛♥ (1, U ,1) : U � U ✐s U 1 � 1 � � � � � � � � � � U U • ❚❤✐s ❞❡✜♥❡s t❤❡ ❝❛t❡❣♦r② Spn ( E ) ✳ • ❲❡ ✇r✐t❡ Spn ( E )( U , V ) ∼ = [ E /( U × V )] ✳ ✹

� � � � • ❚❤❡ ❝❛t❡❣♦r② Spn ( E ) ✐s ♠♦♥♦✐❞❛❧✳ ❚❡♥s♦r ♣r♦❞✉❝t Spn ( E ) × Spn ( E ) × � Spn ( E ) ✐s ❞❡✜♥❡❞ ❜② ( U , V ) � � U × V [ U S � U ′ , V T � V ′ ] � � [ U × V S × T � U ′ × V ′ ]. • ■t ✐s ❛❧s♦ ❝♦♠♣❛❝t ❝❧♦s❡❞✳ ■♥ ❢❛❝t✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✐s♦♠♦r♣❤✐s♠s✿ ⋆ Spn ( E )( U , V ) ∼ = Spn ( E )( V , U ) ⋆ Spn ( E )( U × V , W ) ∼ = Spn ( E )( U , V × W ) ❚❤❡ s❡❝♦♥❞ ✐s♦♠♦r♣❤✐s♠ ❝❛♥ ❜❡ s❤♦✇♥ ❜② t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠ S S S � � � � � � � � � ←�→ ←�→ � � � � � � � � � � � � � � � � � � � � � � � � U × V W U V W U V × W ✺

� � � � � ❉✐r❡❝t s✉♠s ✐♥ Spn ( E ) • ▲❡t E ❜❡ ❛ ❧❡①t❡♥s✐✈❡ ❝❛t❡❣♦r②✳ • ❆ ❝❛t❡❣♦r② E ✐s ❝❛❧❧❡❞ ❧❡①t❡♥s✐✈❡ ✇❤❡♥ ✐t ❤❛s ✜♥✐t❡ ❧✐♠✐ts ❛♥❞ ✜♥✐t❡ ❝♦♣r♦❞✉❝ts s✉❝❤ t❤❛t t❤❡ ❢✉♥❝t♦r X Y X + Y E / A × E / B � E / A + B ; g f , f + g A B A + B ✐s ❛♥ ❡q✉✐✈❛❧❛♥❝❡ ♦❢ ❝❛t❡❣♦r✐❡s ❢♦r ❛❧❧ ♦❜❥❡❝ts A ❛♥❞ B ✳ • ■♥ ❛ ❧❡①t❡♥s✐✈❡ ❝❛t❡❣♦r②✱ ❝♦♣r♦❞✉❝ts ❛r❡ ❞✐s❥♦✐♥t ❛♥❞ ✉♥✐✈❡rs❛❧ ❛♥❞ 0 ✐s str✐❝t❧② ✐♥✐t✐❛❧✳ ❆❧s♦ ✇❡ ❤❛✈❡ t❤❛t t❤❡ ❝❛♥♦♥✐❝❛❧ ♠♦r♣❤✐s♠ � A × ( B + C ) ( A × B ) + ( A × C ) ✐s ✐♥✈❡rt✐❜❧❡✳ ✻

� � � � � � • ■♥ Spn ( E ) t❤❡ ♦❜❥❡❝t U + V ✐s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ U ❛♥❞ V . ❚❤✐s ❝❛♥ ❜❡ s❤♦✇♥ ❛s ❢♦❧❧♦✇s✿ Spn ( E )( U + V , W ) ∼ = [ E /(( U + V ) × W )] ∼ = [ E /(( U × W ) + ( V × W ))] ≃ [ E /( U × W )] × [ E /( V × W )] ∼ = Spn ( E )( U , W ) × Spn ( E )( V , W ); ❛♥❞ s♦ Spn ( E )( W , U + V ) ∼ = Spn ( E )( W , U ) × Spn ( E )( W , V ) ✳ • ❚❤❡ ❛❞❞✐t✐♦♥ ♦❢ t✇♦ s♣❛♥s ( s 1 , S , s 2 ) : U � V ❛♥❞ � V ✐s ❣✐✈❡♥ ❜② ( t 1 , T , t 2 ) : U S + T � s 1 + t 1 s 2 + t 2 � � ������ [ s 1 , t 1 ] [ s 2 , t 2 ] � � S T � � s 1 s 2 t 1 t 2 � � � � � � + = U + U V + V � � � � � � � � � � � � � � � � � � � ������ U V U V ∇ ∇ � � � � U V . • Spn ( E ) ✐s ❛ ♠♦♥♦✐❞❛❧ ❝♦♠♠✉t❛t✐✈❡✲♠♦♥♦✐❞✲❡♥r✐❝❤❡❞ ❝❛t❡❣♦r②✳ ✼

� � � � � � � ▼❛❝❦❡② ❢✉♥❝t♦rs ♦♥ E • ❆ ▼❛❝❦❡② ❢✉♥❝t♦r M : E � Mod k ❝♦♥s✐sts ♦❢ t✇♦ ❢✉♥❝t♦rs M ∗ : ( E ) ♦♣ � Mod k , � Mod k s✉❝❤ t❤❛t M ∗ : E ⋆ M ∗ ( U ) = M ∗ ( U ) ( = M ( U )) ❢♦r ❛❧❧ U ✐♥ E ✳ ⋆ ❋♦r ❛❧❧ ♣✉❧❧❜❛❝❦s q P V p s W , U r ✐♥ E ✱ t❤❡ sq✉❛r❡✭▼❛❝❦❡② sq✉❛r❡✮ M ∗ ( q ) � M ( P ) M ( V ) M ∗ ( p ) M ∗ ( s ) M ( U ) M ( W ) M ∗ ( r ) ❝♦♠♠✉t❡s✳ ✽

� � � � ⋆ ❋♦r ❛❧❧ ❝♦♣r♦❞✉❝t ❞✐❛❣r❛♠s j i U � U + V V ✐♥ E ✱ t❤❡ ❞✐❛❣r❛♠ M ∗ j M ∗ i M ( U ) � M ( U + V ) M ( V ) M ∗ i M ∗ j ✐s ❛ ❞✐r❡❝t s✉♠ s✐t✉❛t✐♦♥ ✐♥ Mod k ✳ ✭❚❤✐s ✐♠♣❧✐❡s M ( U + V ) ∼ = M ( U ) ⊕ M ( V ) ✳✮ • ❆ ♠♦r♣❤✐s♠ θ : M � N ♦❢ ▼❛❝❦❡② ❢✉♥❝t♦rs ✐s ❛ ❢❛♠✐❧② θ U : M ( U ) � N ( U ) ♦❢ ♠♦r♣❤✐s♠s ❢♦r U ✐♥ E ✳ ❚❤✐s ❣✐✈❡s ♥❛t✉r❛❧ tr❛♥s❢♦r♠❛t✐♦♥s θ ∗ : M ∗ � N ∗ ❛♥❞ θ ∗ : M ∗ � N ∗ ✳ • Pr♦♣♦s✐t✐♦♥✿ ✭❉✉❡ t♦ ▲✐♥❞♥❡r✮ ❚❤❡ ❝❛t❡❣♦r② Mky ( E , Mod k ) ♦❢ ▼❛❝❦❡② ❢✉♥❝t♦rs ✐s ❡q✉✐✈❛❧❡♥t t♦ [ Spn ( E ), Mod k ] + ♦❢ t❤❡ ❝❛t❡❣♦r② ♦❢ ❝♦♣r♦❞✉❝t✲♣r❡s❡r✈✐♥❣ ❢✉♥❝t♦rs✳ ❚❤❛t ✐s✿ Mky ( E , Mod k ) ≃ [ Spn ( E ), Mod k ] + ✾

� � � � • Pr♦♦❢ ▲❡t M : E � Mod k ❜❡ ❛ ▼❛❝❦❡② ❢✉♥❝t♦r✳ ❲❡ ❉❡✜♥❡ ❛ ♠♦r♣❤✐s♠ M : Spn ( E ) � Mod k ❜② M ( U ) = M ∗ ( U ) = M ∗ ( U ) ❛♥❞ � � S M ∗ ( s 1 ) � M ∗ ( s 2 ) � s 1 s 2 � � � � ����� � M M ( U ) M ( S ) M ( V ) . = � � � U V ❈♦♥✈❡rs❡❧②✱ ❧❡t M : Spn ( E ) � Mod k ❜❡ ❛ ❢✉♥❝t♦r✳ ❚❤❡♥ ✇❡ ❝❛♥ ❞❡✜♥❡ t✇♦ ❢✉♥❝t♦rs M ∗ ❛♥❞ M ∗ ✱ ( − ) ∗ M Spn ( E ) Mod k , E � � � � � � � � ( − ) ∗ � � � � E ♦♣ ❜② ♣✉tt✐♥❣ M ∗ = M ◦ ( − ) ∗ ❛♥❞ M ∗ = M ◦ ( − ) ∗ ✳ • ❉❡♥♦t❡ Mky = Mky ( E , Mod k ) ≃ [ Spn ( E ), Mod k ] + ✶✵

❚❡♥s♦r ♣r♦❞✉❝ts ✐♥ Mky • ▲❡t T ❜❡ ❣❡♥❡r❛❧ ❝♦♠♣❛❝t ❝❧♦s❡❞✱ ❝♦♠♠✉t❛t✐✈❡✲ ♠♦♥♦✐❞✲❡♥r✐❝❤❡❞ ❝❛t❡❣♦r②✳ ✭❚❤❡ ♠❛✐♥ ❡①❛♠♣❧❡ ✐s Spn ( E ) ✮✳ • ❚❤❡ t❡♥s♦r ♣r♦❞✉❝t ♦❢ ▼❛❝❦❡② ❢✉♥❝t♦rs ❝❛♥ ❜❡ ❞❡✜♥❡❞ ❜② ❝♦♥✈♦❧✉t✐♦♥ ✐♥ [ T , Mod k ] + s✐♥❝❡ T ✐s ❛ ♠♦♥♦✐❞❛❧ ❝❛t❡❣♦r②✳ • ❚❤❡ t❡♥s♦r ♣r♦❞✉❝t ✐s✿ � X , Y ( M ∗ N )( Z ) = T ( X ⊗ Y , Z ) ⊗ M ( X ) ⊗ k N ( Y ) � X , Y T ( Y , X ∗ ⊗ Z ) ⊗ M ( X ) ⊗ k N ( Y ) ∼ = � X M ( X ) ⊗ k N ( X ∗ ⊗ Z ) ∼ = � Y ∼ M ( Z ⊗ Y ∗ ) ⊗ k N ( Y ). = ✶✶