By Jürgen Müller

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C) Show that dim(RU ) = sup{di − di−1 ; i ∈ N} ∈ N ∪ {∞}. Proof. 6]. 15) Exercise: Dimension and height. Give an example of a finitely generated K-algebra, where K is a field, which is not a domain, possessing an ideal I R such that dim(I) + ht(I) = dim(R). 16) Exercise: Catenary rings. A finite dimensional Noetherian ring R is called catenary, if for any prime ideals P ⊆ Q R all maximal chains P = P0 ⊂ · · · ⊂ Pr = Q of prime ideals have length r = ht(Q) − ht(P ). Let K be a field, and let R be a finitely generated K-algebra which is a domain.

We may assume that G ≤ GLn closed, for some n ∈ N. Since G is abelian, Gs , Gu ≤ G are subgroups, and µ is a bijective homomorphism of algebraic groups. The set Gu ⊆ G is closed, and we show that Gs ⊆ G also is closed: For any family Λ := {λA ∈ K; A ∈ Gs } let WΛ := A∈Gs Eλ (A) ≤ Kn . r Hence we have Kn ∼ = i=1 WΛr , for some r ∈ N and certain families Λr , where the WΛr are G-invariant. 4). Hence Gs = G ∩ Tn ⊆ G is closed. The morphism Bn → Tn : [aij ] → diag[a11 , . . , ann ] restricts to the morphism G → Gs : g → gs , hence µ−1 : G → Gs × Gu : g → [gs , gs−1 g] is a morphism.

R−1 , λr + 1, λr+1 , . . , λs−1 , λs − 1, λs+1 , . . 26): II Algebraic groups 32 If λ max µ, let r := min{i ∈ {1, . . , n}; λi = µi } and r < s := min{k ∈ k k {r + 1, . . , n}; i=1 λi = i=1 µi } ≤ n. Hence we have λr < µr , and µr ≤ µr−1 = λr−1 if r > 1, as well as λs > µs ≥ µs+1 ≥ λs+1 . This yields λ ν := [λ1 , . . , λr−1 , λr + 1, λr+1 , . . , λs−1 , λs − 1, λs+1 , . . , λn ] µ, hence ν = µ. It remains to show λr = λs whenever s > r + 1: Assume to the contrary that λr > λs , and let r < t := 1 + min{i ∈ {r, .