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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.

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up vote 6 down vote accepted

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 $\mathfrak p$.

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