# 2-Cohomologies of the groups SL (n,q) by Burichenko V.P.

By Burichenko V.P.

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Extra resources for 2-Cohomologies of the groups SL (n,q)

Example text

Str❛t✐♦♥✳ ✖ ▲❛ ❝♦♥❞✐t✐♦♥ ✭❛ ✮ ✐♠♣❧✐q✉❡ q✉❡ Cell(N ) ⊂ C✳ ■❧ s✉✣t ❞♦♥❝ ❞❡ ♠♦♥✲ G L ✉♥ é❧é♠❡♥t ❞❡ C✳ ❖♥ ❝♦♥s✐❞èr❡ ❧✬❡♥s❡♠❜❧❡ E ∗ tr❡r ❧✬❛✉tr❡ ✐♥❝❧✉s✐♦♥✳ ❙♦✐t i : K ❞❡s s♦✉s✲♦❜❥❡ts ❞❡ L q✉✐ s♦♥t ❞❛♥s D q✉❡ ❧✬♦♥ ♠✉♥✐t ❞✬✉♥ ❜♦♥ ♦r❞r❡✳ ❖♥ ♥♦t❡ E ❧✬❡♥✲ s❡♠❜❧❡ ❜✐❡♥ ♦r❞♦♥♥é✱ ❞✬❡♥s❡♠❜❧❡ s♦✉s✲❥❛❝❡♥t E ∗ {0}✱ ♦❜t❡♥✉ ❞❡ E ∗ ❡♥ ❛❞❥♦✐❣♥❛♥t ✉♥ ♥♦✉✈❡❧ é❧é♠❡♥t ✐♥✐t✐❛❧ 0✳ ❖♥ ✈❛ ❝♦♥str✉✐r❡ ✉♥❡ ❛♣♣❧✐❝❛t✐♦♥ ❝r♦✐ss❛♥t❡ ❞❡ E ✈❡rs ❧✬❡♥s❡♠❜❧❡ ❞❡s s♦✉s✲♦❜❥❡ts ❞❡ L ❝♦♥t❡♥❛♥t K ✱ ♦r❞♦♥♥é ♣❛r ✐♥❝❧✉s✐♦♥✱ ❞é✜♥✐ss❛♥t ✉♥ G A t❡❧ q✉❡ F (0) = K ✱ ❡t t❡❧ q✉❡ ♣♦✉r t♦✉t X ∈ E ✱ X = 0✱ X s♦✐t ❢♦♥❝t❡✉r F : E G F (X) s♦✐t ✉♥ é❧é♠❡♥t ❞❡ ❝♦♥t❡♥✉ ❞❛♥s F (X)✱ ❡t ❧❡ ♠♦r♣❤✐s♠❡ −→ ❧✐♠ F (X ) X 0✱ ♦♥ ❛✐t ❝♦♥str✉✐t t♦✉s ❧❡s F (X ) ♣♦✉r X < X ✳ ❖♥ ♣♦s❡ G V ❡st ❞❛♥s Cell(N )✱ ❡t ❡♥ ♣❛rt✐❝✉❧✐❡r V = −→ ❧✐♠ F (X )✳ ❆❧♦rs ❧✬✐♥❝❧✉s✐♦♥ K X

Y2 ❈♦♠♠❡ i2 ❡st ❞❛♥s F ✱ ❧❛ st❛❜✐❧✐té ❞❡ F ♣❛r ✐♠❛❣❡s ❞✐r❡❝t❡s ✐♠♣❧✐q✉❡ q✉❡ i2 ❡st ❞❛♥s G Y1 F ✳ ❖r ❧❛ ✢è❝❤❡ ❝❛♥♦♥✐q✉❡ X1 X0 X2 Y0 Y2 ❡st ❧❡ ❝♦♠♣♦sé X1 X0 i2 X2 G X1 X0 X2 X2 l Y2 G Y1 Y0 Y2 . ❈♦♠♠❡ k ❡st ❞❛♥s F ✱ ✐❧ rés✉❧t❡ ❞✉ ❧❡♠♠❡ ✶✳✶✳✻ q✉❡ l ❡st ❞❛♥s F ✱ ❡t ❧✬❛ss❡rt✐♦♥ rés✉❧t❡ ❞❡ ❧❛ st❛❜✐❧✐té ❞❡ F ♣❛r ❝♦♠♣♦s✐t✐♦♥✳ ✭❜ ✮ ◆♦t♦♥s 0 ❧❡ ♣❧✉s ♣❡t✐t é❧é♠❡♥t ❞❡ I ✱ ❡t ❝♦♥s✐❞ér♦♥s ❧❡ ❞✐❛❣r❛♠♠❡ ❝♦♠♠✉t❛t✐❢ G ❧✐♠ X −→ X0 α0  Y0 G Y0  α0 ❧✐♠ −→ α ❧✐♠w X X0 −→ www wwmw www 8 " E ❧✐♠ Y −→ . ❊♥ ✈❡rt✉ ❞✉ ❧❡♠♠❡ ✶✳✶✳✻✱ ❧❡ ♠♦r♣❤✐s♠❡ m ❡st ❞❛♥s F ✱ ❡t ♣❛r ❤②♣♦t❤ès❡✱ ❧❡ ♠♦r♣❤✐s♠❡ G Y0 ✮ ❡st ❛✉ss✐ ❞❛♥s F ✳ ▲❛ Yj ) (❧✐♠ Xj ) X0 α0 ✭q✉✐ s✬✐❞❡♥t✐✜❡ à ❧❛ ✢è❝❤❡ (−→ ❧✐♠ j<0 −→j<0 st❛❜✐❧✐té ♣❛r ✐♠❛❣❡s ❞✐r❡❝t❡s ✐♠♣❧✐q✉❡ q✉❡ α0 ❡st ❞❛♥s F ✱ ❡t ✐❧ ❡♥ ❡st ❞♦♥❝ ❞❡ ♠ê♠❡ ❞❡ −→ ❧✐♠ α✳ ▲❡♠♠❡ ✶✳✶✳✶✶✳ ✖ ❙♦✐❡♥t C ✉♥❡ ❝❛té❣♦r✐❡ ❛❞♠❡tt❛♥t ❞❡s ❧✐♠✐t❡s ✐♥❞✉❝t✐✈❡s✱ ❡t E ✱ F ❞❡✉① ❝❧❛ss❡s ❞❡ ✢è❝❤❡s ❞❡ C s❛t✐s❢❛✐s❛♥t à ❧❛ ♣r♦♣r✐été ❝✐✲❞❡ss♦✉s✳ G Z ❞❛♥s G Y✱ g : Y ✭P✮ P♦✉r t♦✉t ❝♦✉♣❧❡ ❞❡ ♠♦r♣❤✐s♠❡s ❝♦♠♣♦s❛❜❧❡s f : X C ✱ s✐ f ❡t gf s♦♥t ❞❛♥s F ✱ ❡t g ❞❛♥s E ✱ ❛❧♦rs g ❡st ❞❛♥s F ✳ ❆❧♦rs ♦♥ ❛ ❧❡s ❛ss❡rt✐♦♥s s✉✐✈❛♥t❡s✳ ✭❛✮ ❙✐ ❧❛ ❝❧❛ss❡ F ❡st st❛❜❧❡ ♣❛r ❝♦♠♣♦s✐t✐♦♥ ❡t ✐♠❛❣❡s ❞✐r❡❝t❡s✱ ♣♦✉r t♦✉t ❞✐❛✲ ❣r❛♠♠❡ ❝♦♠♠✉t❛t✐❢ X1 o i1  Y1 o X0 i0  Y0 G X2 i2  G Y2 , G Y1 ❞❛♥s E ✱ ❛❧♦rs s✐ ❧❡s ♠♦r♣❤✐s♠❡s i0 , i1 , i2 s♦♥t ❞❛♥s F ❡t k : X1 X0 Y0 G ❧❡ ♠♦r♣❤✐s♠❡ ❝❛♥♦♥✐q✉❡ X1 X0 X2 Y1 Y0 Y2 ❡st ❞❛♥s F ✳ ✭❜✮ ❙✐ ❧❛ ❝❧❛ss❡ F ❡st st❛❜❧❡ ♣❛r ✐♠❛❣❡s ❞✐r❡❝t❡s ❡t ❝♦♠♣♦s✐t✐♦♥s tr❛♥s✜♥✐❡s✱ ♣♦✉r t♦✉t ❡♥s❡♠❜❧❡ ❜✐❡♥ ♦r❞♦♥♥é I ✱ ❞❡ ♣❧✉s ♣❡t✐t é❧é♠❡♥t 0✱ t♦✉t ❝♦✉♣❧❡ ❞❡ ❢♦♥❝t❡✉rs G Y ✱ s✐ ♣♦✉r t♦✉t i G C ✱ ❡t t♦✉t ♠♦r♣❤✐s♠❡ ❞❡ ❢♦♥❝t❡✉rs α : X X, Y : I ❞❛♥s I ✱ ❧❛ ✢è❝❤❡ αi ❡st ❞❛♥s F ✱ ❡t s✐ ♣♦✉r t♦✉t i > 0✱ ❧❛ ✢è❝❤❡ (−→ ❧✐♠ Yj ) j

I0 ❡st ❞❛♥s F ✱ ❧❛ ✢è❝❤❡ i0 ❛✉ss✐✳ ❖r ❧❡ ♠♦r♣❤✐s♠❡ i1 s❡ ❞é❝♦♠♣♦s❡ ❡♥ X1 i0 G X1 X0 Y0 k G S1 . ❱✉ q✉❡ i1 ❡st ❞❛♥s F ❡t k ❞❛♥s E ✱ ✐❧ rés✉❧t❡ ❞❡ ❧❛ ♣r♦♣r✐été ✭P✮ q✉❡ k ❡st ❞❛♥s F ✳ ▲❡ ♠♦r♣❤✐s♠❡ i2 ét❛♥t ❞❛♥s F ✱ ❧✬❛ss❡rt✐♦♥ rés✉❧t❡ ❞✉ ❧❡♠♠❡ ✶✳✶✳✶✵✳ ✭❜ ✮ ❊♥ ✈❡rt✉ ❞✉ ❧❡♠♠❡ ✶✳✶✳✶✵✱ ✐❧ s✉✣t ❞❡ ♠♦♥tr❡r q✉❡ ♣♦✉r t♦✉t i ∈ I ✱ ❧❡ ♠♦r✲ G Yi ❡st ❞❛♥s F ✳ ❖♥ r❛✐s♦♥♥❡ ♣❛r ré❝✉rr❡♥❝❡ Yj ) (❧✐♠ Xj ) Xi ♣❤✐s♠❡ mi : (−→ ❧✐♠ j 0✱ ❡t s✉♣♣♦s♦♥s q✉❡ ♣♦✉r t♦✉t j < i✱ ❧❡ ♠♦r♣❤✐s♠❡ mj s♦✐t ❞❛♥s F ✳ ▲❡ ❧❡♠♠❡ ✶✳✶✳✶✵ ✐♠♣❧✐q✉❡ ❛❧♦rs q✉❡ ❧❛ ✢è❝❤❡ G ❧✐♠ Xj α : ❧✐♠ ❧✐♠ −→ −→j