# Canonical scheme structure on the singular locus of a variety

I first asked this question on Math StackExchange but no answers were given.

Let $X$ be subvariety of affine space $\mathbb{A}_{k}^n$, where $k$ is a field, and suppose $X$ is given by equations

$$X:(F_1=\cdots = F_m=0)\subset \mathbb{A}^n.$$

Then $X$ is singular at $p\in X$ if

$$\text{rank}(a_{ij}(p))<d,$$

where $d=n-\text{dim}(X)$ (i.e., the codimension of $X$ in $\mathbb{A}^n$), and $(a_{ij})$ is the matrix valued function

$$(a_{ij})=\frac{\partial F_i}{\partial x_j}.$$ Thus the ideal generated by the defining equations for $X$ along with the $d\times d$ minors of the matrix $(a_{ij})$ define a natural scheme structure on the singular locus of $X$ (as a subscheme of $\mathbb{A}^n$). Let's call this scheme $S_X$. My question then is the following:

Is $S_X$ intrinsic to $X$?

The answer is yes. Let $X$ be a scheme of finite type over the field $k$, of pure dimension $r$; then $S_X$ can be defined as the closed subscheme of $X$ defined by the $r$th Fitting ideal of the sheaf $\Omega^1_{X/k}$.
This construction of $S_X$ coincides with your definition when $X$ is a closed subscheme of $\mathbb{A}^n_k$. Indeed, consider the exact sequence $$I/I^2 \longrightarrow \Omega^1_{\mathbb{A}^n_k/k} \vert_X \longrightarrow \Omega^1_{X/k} \longrightarrow 0$$ where $I$ is the ideal sheaf of $X$ in $\mathbb{A}^n_k$. If $I$ is generated by the polynomials $f_1, \dots, f_m$ one can obtain another exact sequence of $\mathcal{O}_X$-modules $$\mathcal{O}_X^{\oplus m} \overset{J}\longrightarrow \mathcal{O}_X^{\oplus n} \longrightarrow \Omega^1_{X/k} \longrightarrow 0$$ where $J$ is the Jacobian matrix of the polynomials $f_i$. Then the $r$th Fitting ideal of $\Omega^1_{X/k}$ is the ideal in $\mathcal{O}_X$ generated by the $(n-r) \times (n-r)$ minors of the matrix $J$.