Math 480, Spring 2014, Assignment 15

Carefully state the following theorems (you need not prove them):

1. Theorem relating dimension to the transcendence degree of the coordinate ring.

Solve the following problems:

1. Working in $\mathsf{k}^4$, let $V=\mathbb{V}(x_1x_2-x_3x_4)$. Compute the dimension of $V$.
2. Let $V$ be as above. Find three algebraically independent regular functions on $V$. (Hint: some answer is probably obvious, but it's still instructive to choose a graded term order and consider the monomials lying outside the leading term ideal. What are the largest coordinate subspaces of this set?)