Suppose that $f : X \to V$ is a flat equidimensional (of dimension $h$) morphism of schemes of finite type and $V$ is excellent (or a variety) For this one can formulate something called the Jacobian ideal of $f$ which measures where $f$ is not smooth. One reference is Section 4.4 of the book by Swanson-Huneke You can download an earlier draft of the book

Let me work this out in an explicit affine case. Suppose that $V = \text{Spec} A$ and $X = \text{Spec} A[x_1, \ldots, x_n]/(g_1, \ldots, g_m)$. One can then form the $m \times n$ Jacobian matrix $M_{X/V}$ whose $ij$th entry is ${\partial g_i \over \partial x_j}$. Then the ideal generated by the $h \times h$ minors of $M_{X/V}$ is called the Jacobian ideal of $X$ over $V$, and denote $J_{X/V}$.

Does anyone know any good references for this object (I know about the Lipman-Satheye papers, some notes of Hochster, and the above book, but not much else).

In particular, I'd love to have references to the following.

**Question:** Base change for $J_{X/V}$. (ok, this is essentially obvious but a reference would still be great, it also follows from the fitting ideal of the sheaf of differentials description of the Jacobian ideal which is described briefly in the above book).

**Question:** Say that $V$ is flat, equidimensional and finite type over another excellent scheme $S$ (for example, $S = \text{Spec} k$ for some field $k$). I'd like to relate the Jacobian ideals $J_{X/S}$, $J_{X/V}$ and $J_{V/S}$. In particular, if $V$ is smooth over $S$, I'd love to say that $J_{X/S} = J_{X/V}$.