Homework 2

Ideals can have many bases, but these different bases inevitably yield the same variety.  To explore this, consider the following:

a) Let $I \subset k[x_1,\ldots,x_n]$ be an ideal and let $f_1,\ldots,f_s\in k[x_1,\ldots,x_n]$.  Prove that $f_1,\ldots,f_s \in I$ if and only if $\langle f_1,\ldots,f_s\rangle\subset I$.

b) Show that $\langle 2x^2 + 3y^2 - 11, x^2 - y^2 -3\rangle = \langle x^2 - 4, y^2 -1 \rangle$ in the polynomial ring $\mathbb{Q}[x,y]$.

c) Let $\{f_1,\ldots,f_s\}$ and $\{g_1,\ldots,g_t\}$ be two bases for the ideal $I \subset k[x_1,\ldots,x_n]$.  Show that $V(f_1,\ldots,f_s) = V(g_1,\ldots,g_t)$.

d) Show that $V(x+xy, y+xy, x^2,y^2) = V(x,y)$.