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21) Exercise: Chevalley’s Theorem (1955). Let V, W be affine varieties and let ϕ : V → W be a morphism. 20). Hint. It suffices to consider ϕ(V ), reduce to the case V, W irreducible, and proceed by induction on dim(W ). Proof. 4]. 22) Exercise: Open morphisms. Let V, W be irreducible affine varieties and let ϕ : V → W be a dominant morphism, such that for all Z ⊆ W closed and irreducible the preimage ϕ−1 (Z) ⊆ V is equidimensional of dimension dim(Z) + dim(V ) − dim(W ). Show that ϕ is an open map, i.

Xn−1 K ≤ Kn / xn K is S-invariant, for all i ∈ {1, . . , n − 1}. If S consists of semisimple matrices, then we again proceed by induction on n, the case n = 1 being trivial. Now we may assume that there is A ∈ S such r that Kn ∼ = i=1 Eλi (A) for some r > 1, where λ1 , . . , λr ∈ K are the pairwise distinct eigenvalues of A, and we are done by induction. b) We first show that G acts irreducibly on Kn if and only if n = 1: Let G act irreducibly, and let A := K G ⊆ Kn×n be the (non-commutative) Ksubalgebra of Kn×n generated by G.

M}, for i, j ∈ j i I such that j = {±i} and t ∈ K let xij (t) := En +t(Eij − |i| · |j| ·E−j,−i ) ∈ SL2m and xi,−i (t) := En + tEi,−i ∈ SL2m be symplectic transvections. Hence for the unipotent root subgroups Uij := {xij (t); t ∈ K} ≤ SL2m the map Ga ∼ = K → Uij : t → xij (t) is an isomorphism of algebraic groups. 186] we have Uij ; i, j ∈ I, i = j = S2m . 7), we may assume b) Since SO1 = {1} and SO2 = n ≥ 3. 51] we have Ωn = [SOn , SOn ] = SOn . Let n = 2m. Indexing rows and columns by I := {−m, .