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B) If V is irreducible, then any open subset ∅ = U ⊆ V is dense and irreducible. c) If V is irreducible and ϕ : V → W is a continuous map, where W is a topological space, then ϕ(V ) ⊆ W is irreducible. d) V is Hausdorff if and only if the diagonal {[v, v] ∈ V × V ; v ∈ V } ⊆ V × V is closed, where V × V is endowed with the product topology. e) If V is Noetherian, then V is Hausdorff if and only if V discrete. f ) If V = ∅ is Noetherian, then it is a finite union V = V1 ∪ · · · ∪ Vr , where the Vi ⊆ V are closed and irreducible.

Thus it suffices to show that Cµ Cλ , i. e. Cµ ⊆ Cλ , since then Cµ × {B} ⊆ Cλ × {B} = Cλ × {B} ⊆ Cλ , implying Cµ ⊆ Cλ . Hence we may assume that λ = [n − k, k] n and µ = [n − k − 1, k + 1] n, for some k ∈ {0, . . , n2 − 1}. We have λ = [1n−2k , 2k ] n and µ = [1n−2k−2 , 2k+1 ] n, which immediately implies Uµ ⊆ Uλ . Hence we get Aµ ∈ Cµ ∩ Uµ ⊆ Cµ ∩ Uλ ⊆ Cµ ∩ Cλ , where Aµ is as above, implying Cµ ⊆ Cλ . 7) Corollary. For G := SLn , where n ∈ N, we have dim(Gu ) = n(n − 1). Proof. We have λ [n] for all λ n.

Let V be an affine variety over K a) Show that dim(V ) = 0 if and only if V is a finite set. Which are the irreducible varieties amongst them? b) Let V be irreducible. Show that dim(V ) is the maximum of the lengths d ∈ N0 of chains ∅ = V0 ⊂ · · · ⊂ Vd = V of closed irreducible subsets. III Exercises and references 49 c) Let V be irreducible such that K[V ] is a factorial domain. Show that any closed subset W ⊂ V having equidimension dim(W ) = dim(V ) − 1 is of the form W = V(f ) for some f ∈ K[V ].

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Algebraic Groups by Jürgen Müller

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