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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 $J_{X/S} = J_{X/V}$.

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Isn't this the Fitting ideal of the sheaf of differentials (or smth like that) ? In that case, another reference is the appendix to the paper by Mazur-Wiles "Class fields...". – Damian Rössler Feb 2 '13 at 23:00
This is indeed the fitting ideal of the sheaf of differentials. Thanks for the reference, I'll check that out. – Karl Schwede Feb 2 '13 at 23:48
A student of Huneke's named Hsin-Ju Wang wrote some papers about the Jacobian, and always cited Lipman-Sathaye for the definition and basic properties. Here's one of Wang's papers: – Graham Leuschke Feb 3 '13 at 0:01
Hi Graham, thanks! – Karl Schwede Feb 3 '13 at 0:57
There are some old summer school notes of B. Teissier, where §1-2 are about fitting ideals and differential forms, available here: – Blup Feb 3 '13 at 3:25

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