I.1.1-I.1.4

All these problems have height 1.

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
Y = V(x,y) \cup V(x,z) \cup V(x^2 - y, z-1)
 * Use basic rules about products of ideals an intersections to show
 * 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 $$ V \times W$$