Algebraic Groups by Jürgen Müller

By Jürgen Müller

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Hence to show the last assertion, we may assume that G is connected. Letting ϕ be the action morphism, the orbit map ϕx : G → xG is a dominant morphism between irreducible varieties. Hence there is ∅ = U ⊆ xG such that U ⊆ xG, and such that dim(ϕ−1 x (y)) = dim(G) − dim(xG) for all y ∈ U . For any y ∈ U we have ϕ−1 (y) = {h ∈ G; xh = y} = CG (x)g ⊆ G, where g ∈ G is fixed such x that y = xg, implying dim(ϕ−1 x (y)) = dim(CG (x)). 2) Proposition: Closed orbit lemma. Let G be an algebraic group acting morphically on V .

Show that ∂i (f ) = 0 for some i ∈ {1, . . , n}. Proof. 14]. 27) Exercise: Tangent spaces. Let V and W be affine varieties over K, and let x ∈ V and y ∈ W . Show that T[x,y] (V × W ) ∼ = Tx (V ) ⊕K Ty (W ) as K-vector spaces. Proof. 1]. 28) Exercise: Linear spaces. Let K be an algebraically closed field, let V ≤ Kn and W ≤ Km be K-subspaces, and let ϕ : V → W be a K-linear map. a) Show that V is an irreducible affine variety such that dim(V ) = dimK (V ). Show that for any x ∈ V there is a natural identification of Tx (V ) with V .

B= .  ∈ Un , .    0 ··· 0 Eλ Bl−1,l  l−1 0 ··· 0 0 Eλl where Bij ∈ Kλi ×λj for i, j ∈ {1, . . , l}. It is immediate that Uλ ⊆ Un is closed and that Uλ ∼ = KN , for some N ∈ N0 , hence Uλ is irreducible. Moreover, it is k immediate that rkK ((B − En )k ) ≤ n − i=1 λi , for all k ∈ {1, . . , n}. Let     Aλ :=    Eλ1 0 .. 0 0 A12 Eλ2 .. ··· ··· 0 A23 .. 0 0 ··· ··· .. Eλl−1 0 0 0 .. Al−1,l Eλl      ∈ Uλ ,   λ i+1 λi ×λi+1 where Ai,i+1 := for all i ∈ {1, . . , l − 1}, and where j=1 Ejj ∈ K Ejj is the [j, j]-th matrix unit.