Let $X$ be an affine variety defined as a zero set of polynomials $f_1, .., f_s \in k[x_1, .., x_n]$ where $k$ is an algebraically closed field. Let $p \in X$ be regular. Then I have read that $X$ is a local complete intersection at $p$. I was wondering if the following holds in this case, I feel like it should follow easily from the definition though I wasn't too sure. Any reference and comments appreciated.
There exist a Zariski open $p \in U$ and $1 \leq i_1 < .. < i_c \leq s$, $c = n - \dim X$ such that
$$
X\cap U = \{ x \in k^n: f_{i_1}(x) = .. f_{i_c}(x) = 0 \}.
$$
In other words, can I always take the a subset of the defining polynomials for $X$ for the local complete intersection?
Yes, this is true - by smoothness, the Jacobian $\frac{\partial(f_1,\cdots,f_s)}{\partial(x_1,\cdots,x_n)}$ is of rank $n-\dim X=c$ at $p$, and therefore this matrix has a $c\times c$ minor of full rank. Select the $f_i$ corresponding to the rows in this minor, and you have that these $c$ equations generate the ideal of $X$ in the local ring $\mathcal{O}_{\Bbb A^n,p}$ which shows that $X$ is cut out by $c$ equations in a neighborhood of $p$.