By Tamás Szamuely
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Additional info for Lectures on Linear Algebraic Groups
3. Let G be an aﬃne algebraic group, H ⊂ G a closed subgroup. Then the quotient of G by H exists. Moreover, it is a homogeneous space for G such that H is the stabilizer of a point. Proof. 4. 2. It is enough to check this property for a connected aﬃne open subset U ⊂ X. Pick f ∈ O(ρ−1 (U )) constant on the left cosets of H, and (ρ,f ) consider the composite map ρ−1 (U ) −→ U × A1 → U , where the last map is the natural projection. Note that ρ−1 (U ) is a ﬁnite disjoint union of connected open sets, each one dense in a component of G.
So h ∈ U U −1 ⊂ Im (ϕn )Im (ϕn ). This shows H = [G, G]. 1. 7. Let ϕ : X → Y be an injective morphism of irreducible quasi-projective varieties with Zariski dense image. If the induced ﬁeld extension k(X)|k(Y ) is separable, then it is an isomorphism. 1, we see that the proposition is a consequence of the following lemma. 8. Assume that ϕ : X → Y is a morphism of aﬃne varieties and ϕ∗ induces an isomorphism AX ∼ = AY [f ] with f separable over k(Y ). Then there is an open subset V ⊂ Y such that each point of V has exactly [k(X) : ϕ∗ k(Y )] preimages in X.
2 we ﬁnd a representation of G on some ﬁnite-dimensional V with GP stabilizing a one-dimensional subspace in V , hence ﬁxing a point Q in the induced action of G on the projective space P(V ). Let Y be the orbit of Q in P(V ) and Z that of (P, Q) in X × P(V ) (equipped with the product action). The natural projections Z → X and Z → Y are bijective G-morphisms, so it is enough to ﬁnd a ﬁxed point in Y (which must then be the whole of Y ). 5 below, but this was not used in the proof of the proposition).
Lectures on Linear Algebraic Groups by Tamás Szamuely