Consider $(f_1,\dots,f_n), (g_1,\dots,g_n)\in \mathbb{C}[z_1,\dots,z_n]\ $ such that:
i) $\{f_1=\dots=f_n=0\}= \{g_1=\dots=g_n=0\}=\{0\}\in \mathbb{C}^n\ $ and
ii) $f_1g_1+\dots+f_ng_n\equiv0$.
What is the relation between
$ \displaystyle\dim \frac{\mathbb{C}[z_1,\dots,z_n]}{(f_1,\dots,f_n)} $ and $ \displaystyle\dim \frac{\mathbb{C}[z_1,\dots,z_n]}{(g_1,\dots,g_n)} $ ?
Put $S=\mathbb{C}[z_1,\ldots ,z_n]$. Since $V(f_1,\ldots ,f_n)=\{0\} $, $(f_1,\ldots ,f_n)$ is a $S$-regular sequence, so the (partial) Koszul complex $$\wedge^2S^n\xrightarrow{\ u\ } S^n\xrightarrow{(f_1,\ldots ,f_n)} S$$is exact, with $u(e_i\wedge e_j)=f_ie_j-f_je_i$. This implies $(g_1,\ldots ,g_n)\subset (f_1,\ldots ,f_n)$. Since the $f_i$ and $g_j$ play a symmetric role, we have in fact $\ (g_1,\ldots ,g_n)=(f_1,\ldots ,f_n)$.
