How to define a smooth subvariety as the vanishing of local coordinates I keep stumbling upon this fact, and would like to see or get an idea for the proof: _An ideal of a smooth subvariety at a point of a smooth variety can be generated by a subset of a suitably chosen system of local coordinates._ This is similar to the definition of a submanifold, but how do we show this in the algebraic geometry case? I believe the Cohen Structure Theorem comes into play, but I want to understand this connection better. An explicit example of how this choice of local coordinates can be made would also be helpful.

A variety $X$ is nonsingular at a point $p$ iff the local ring $\mathcal{O}_{X,p}$ is a regular local ring, i.e. there exists a regular system of parameters ($=$ a sequence of elements $x_1, \ldots, x_n \in \mathcal{O}_{X,p}$ such that $(x_1, \ldots, x_n) = \mathfrak{m}_p$ (the maximal ideal) and $n = \dim \mathcal{O}_{X,p}$), which are precisely the local coordinates at $p$. The corresponding commutative algebra fact (much easier than Cohen Structure Theorem) is:

**Proposition:** Let $R$ be a regular local ring, and $I$ an $R$-ideal. Then $R/I$ is regular local iff $I$ is generated by part of a regular system of parameters.

The proof uses only Nakayama's lemma, and the fact that a regular local ring is a domain. For full details see e.g. Bruns-Herzog, Proposition 2.2.4.