# How to define the Jacobian matrix for all points of a finite-type k-scheme?

Ravi Vakil's notes 13.2.D asks: "Show that if the Jacobian matrix for X = Spec $k[x_1,\ldots,x_n]/(f_1,\ldots f_r)$ has corank d at all closed points, then it has corank d at all points. (Hint: the locus where the Jacobian matrix has corank d can be described in terms of vanishing and nonvanishing of certain explicit matrices.)"

But I can't even see how to define the Jacobian matrix for non-closed points. He defines the Jacobian matrix as the $n$ by $r$ matrix such that $a_{ij} =$ the partial derivative of $f_j$ with respect to $x_i.$ If p is a closed point, then this makes sense; this partial derivative can be evaluated at p. But I don't see how he is defining the Jacobian matrix for all points, not just closed points.

The definition of the Jacobian matrix doesn't make any reference to any points, closed or not. It is just the matrix $a_{ij}$ of partial derivatives of the $f_j$ w.r.t. the $x_i$. It is an $r\times n$ matrix with entries in $k[x_1,\ldots,x_n]$, and you consider its image in $k[x_1,\ldots,x_n]/\mathfrak p$, for any prime ideal $\mathfrak p$ containing $(f_1,\ldots,f_r)$ (or for any prime ideal in $k[x_1,\ldots,x_n]$ at all, for that matter, but we only care about its image modulo the points of $X$, which are those $\mathfrak p$ containing $(f_1,\ldots,f_r)$), which is a matrix with entries in the residue field $\kappa(\mathfrak p)$. Now you can talk about its corank. This is what is meant by the corank of the Jacobian matrix at the point $\mathfrak p$.