I am trying to practice calculating categorical quotients and I ran into this example. I am unable to get the answer and was wondering if someone can help?
Let $G = Z/3Z =$ $\{1, \omega, \omega^2\}$, where $\omega$ is a cube root of unity. Let $G$ act on $^{2}$ via $\omega(x, y) = (\omega x, \omega y)$. Find the categorical quotient of $^{2}$ by G.
The category of affine schemes is the opposite of the category of commutative rings, and taking opposites switches limits and colimits, so in general the categorical quotient $\left( \text{Spec } R \right) / G$ is the spectrum $\text{Spec } R^G$ of the fixed point subring. Hence you want to compute the subring of $R = k[x, y]$ ($k$ a field containing $\omega$, and in particular not of characteristic $3$) fixed under the action of $\mathbb{Z}/3\mathbb{Z}$ generated by
$$x \mapsto \omega x, y \mapsto \omega y.$$
It shouldn't be hard to convince yourself that this subring is $k[x^3, x^2 y, x y^2, y^3]$; equivalently, it's the subalgebra of $k[x, y]$ of polynomials with total degree $3$.
If you want to write $\text{Spec } k[x^3, x^2 y, xy^2, y^3]$ as a closed subscheme of affine space then you need a presentation. You can take the generators to be $a = x^3, b = x^2 y, c = x y^2, d = y^3$ (exercise: you need at least this many generators), and the relations they satisfy are (I think) generated by
$$ac = b^2, bd = c^2, ad = bc.$$