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I.1.1-I.1.4

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All these problems have height 1.

HAPPY =

I.1.1

a) use  x \mapsto z and  y \mapsto z^2
b) notice x,y map to units  \subset k
c) use the result that after a linear change of variable any conic can be expressed as a hyperbola, ellipse, or parabola.

I.1.2

  • Show  I(Y) = (z - x^3, y - x^2) then find an isomorphism with  A(Y) and  k[t] to show the dimension is 1.

I.1.3

  • Use basic rules about products of ideals an intersections to show
Y = V(x,y) \cup V(x,z) \cup V(x^2 - y, z-1)
  • Check each term in the union if the vanishing of a prime ideal.

I.1.4

  • A base for the topology of  X = \mathbb{A} \times \mathbb{A} consists of sets  V \times W where each are open in the cofinite topology of  \mathbb{A}
  • Compare an open set like U = \mathbb{A}^2 - V(xy - 1) with a open set like  V \times W

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