Let $J$ be an ideal in the ring $\mathbb{C}[u] := \mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$.

The quotient $\mathbb{C}[u] / J$ is naturally isomorphic to the ring of regular functions on the zero set $V(J)$ of $J$.

• Given a subspace $\mathbb{C}^k \subset \mathbb{C}^n$ and $I$ its ideal in $\mathbb{C}[u]$, when can we say that the $n-k$ generators of $I$ form a regular sequence in $\mathbb{C}[u] / J$ ?
• Suppose that the generators of $I$ form indeed a regular sequence in $\mathbb{C}[u] / J$. Is it still true in the ring $\mathbb{C}[u,u^{-1}] / J\mathbb{C}[u,u^{-1}]$ (considered as a $\mathbb{C}[u]$-module) ?