Homework assignments
... Please study the following Problems 1-10 by January 18 (Friday). In Problems 1-10, let X be a compact Hausdorff space and let A be the ring of all C-valued continuous functions on X. We consider how X is reflected in A and how A is reflected in X like the relation of a flower and its image in the wa ...
... Please study the following Problems 1-10 by January 18 (Friday). In Problems 1-10, let X be a compact Hausdorff space and let A be the ring of all C-valued continuous functions on X. We consider how X is reflected in A and how A is reflected in X like the relation of a flower and its image in the wa ...
Model Answers 4
... Now suppose that there is a dominant morphism of schemes Z −→ X, where Z is normal. This induces a dominant morphism ZU −→ U , where U is an open affine subscheme and ZU is the inverse image of U Thus it suffices to prove the universal property of X in the case when X is affine. Covering Z by open a ...
... Now suppose that there is a dominant morphism of schemes Z −→ X, where Z is normal. This induces a dominant morphism ZU −→ U , where U is an open affine subscheme and ZU is the inverse image of U Thus it suffices to prove the universal property of X in the case when X is affine. Covering Z by open a ...
Invertible and nilpotent elements in the group algebra of a
... ng ∈ Nil(k) for every g ∈ G is an ideal of A consisting of nilpotent elements. Indeed, as an element of A, an n ∈ N has only finitely many non-zero components, say n1 ug1 , . . . , np ugp . Hence there exists q ∈ N such that nqi = 0 for all 1 ≤ i ≤ p. r Then npq is a sum of terms nr11 · · · npp ug w ...
... ng ∈ Nil(k) for every g ∈ G is an ideal of A consisting of nilpotent elements. Indeed, as an element of A, an n ∈ N has only finitely many non-zero components, say n1 ug1 , . . . , np ugp . Hence there exists q ∈ N such that nqi = 0 for all 1 ≤ i ≤ p. r Then npq is a sum of terms nr11 · · · npp ug w ...
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