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By Jürgen Müller

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V ) ⊆ W is dense. b) Show that ϕ∗ is surjective if and only if ϕ is a closed embedding, i. e. ϕ(V ) ⊆ W is closed and ϕ : V → ϕ(V ) is an isomorphism of affine varieties. Proof. 1]. 8) Exercise: Morphisms. Let K be an algebraically closed field. a) Let ϕ : K2 → K2 : [x, y] → [xy, y]. Show that ϕ(K2 ) ⊆ K2 is neither open nor closed. b) Let ψ : K → K2 : x → [x2 , x3 ]. Show that ψ(K) ⊆ K2 is closed, and that ψ : K → ψ(K) is bijective, but not an isomorphism of affine varieties. c) Give an example of a continuous map between affine varieties which is not a morphism.

N}, we also write λ = [1a1 , . . , nan ] n. b) Let λ = [λ1 , . . , λn ] n, and let λi := |{j ∈ {1, . . , n}; λj ≥ i}| ∈ N0 n for i ∈ {1, . . , n}. Hence we have λ1 ≥ · · · ≥ λn ≥ 0 as well as i=1 λi = n n n, j=1 |{i ∈ {1, . . , n}; i ≤ λj }| = j=1 λj = n. Thus λ = [λ1 , . . , λn ] being called the associated conjugate partition. n Hence we have λi = j=i aj (λ), for i ∈ {1, . . , n}. Moreover, we have Yλ = {[i, j] ∈ N2 ; i ∈ {1, . . , n}, j ∈ {1, . . , λi }} = {[i, j] ∈ N2 ; j ∈ {1, .

N Hence we have λi = j=i aj (λ), for i ∈ {1, . . , n}. Moreover, we have Yλ = {[i, j] ∈ N2 ; i ∈ {1, . . , n}, j ∈ {1, . . , λi }} = {[i, j] ∈ N2 ; j ∈ {1, . . , n}, i ∈ {k ∈ {1, . . , n}; λk ≥ j}} = {[i, j] ∈ N2 ; j ∈ {1, . . , n}, i ∈ {1, . . , λj }}, implying that Yλ = {[i, j] ∈ N2 ; [j, i] ∈ Yλ }, and thus (λ ) = λ. c) Let λ = [λ1 , . . , λn ] n and µ = [µ1 , . . , µn ] n. Then µ is called to k k dominate λ, if for all k ∈ {1, . . , n} we have i=1 λi ≤ i=1 µi ; we write λ µ. 25). d) We have λ max µ if and only if µ = [λ1 , .

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