math 532 – homework set 1
january 15, 2016
1)
(a) Recall that a topological space 𝑋 is called (quasi-)compact if every open cover of
𝑋 has a finite subcover. Show that every Noetherian topological space is quasicompact.
(b) Let 𝑋 be a topological space and assume that 𝑋 = ⋃u�u�=1 𝑈u� is a finite open cover
such that each 𝑈u� is a Noetherian topological space (in the induced topology).
Show that 𝑋 is Noetherian. Conclude that ℙu� is a Noetherian topological space.
2) Show that every morphism of affine algebraic sets is continuous with respect to the
Zariski topology. Give an example of a continuous map of affine algebraic sets that is
not a morphism.
3) Show that any homogeneous ideal of 𝑘[𝑥1 , … , 𝑥u� ] can be generated by finitely many
homogeneous polynomials.
4) Let (𝑈u� )u�=0,…,u� be the standard open cover of ℙu� by affine subsets and write 𝜑u� ∶ 𝑈u� −−→
𝔸u� for the identification discussed in class.
u�
u�
(a) Describe the change-of-coordinates map 𝜑u� ∘ 𝜑−1
u� ∶ 𝔸 → 𝔸 .
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(b) Consider the affine algebraic set 𝑋 = {𝑥22 = 𝑥13 − 𝑥1 } ⊆ 𝔸2 . Describe the projective
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closure 𝑋 of 𝑋 , i.e. the projective algebraic set 𝜑−1
0 (𝑋) ⊆ ℙ . Further give the
equations of 𝜑1 (𝑋 ∩ 𝑈1 ) and 𝜑2 (𝑋 ∩ 𝑈2 ).
5) Consider the morphism 𝑓 ∶ 𝔸1 → 𝔸3 given by 𝑡 ↦ (𝑡, 𝑡 2 , 𝑡 3 ). Show that the image
𝑋 = 𝑓 (𝔸1 ) is a closed subset of 𝔸3 and describe its ideal. Describe the closure of 𝑋 in
ℙ3 and its homogeneous ideal.
6) Show that ℙ1 × ℙ1 can be viewed as a projective algebraic subset of ℙ3 , i.e. give a
bijection of ℙ1 × ℙ1 with a closed subset of ℙ3 .
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