# 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 $V \times W$