# Question about the definition of the Jacobian ideal

Let $$R=\frac{k[[x_1,\ldots,x_n]]}{(f_1,\ldots,f_c)}$$ (or $$\frac{k[x_1,\ldots,x_n]}{(f_1,\ldots,f_c)}$$), where $$k$$ is a field with characteristic $$0$$ and $$0\ne(f_1,\ldots,f_c)\subseteq (x_1,\ldots,x_c)$$.

In the paper On the fitting ideal in free resolution (see also Jacobian ideal reference), there is a definition of the Jacobian ideal of $$R$$. Assume that the height of $$(f_1,\ldots,f_c)$$ in $$k[[x_1,\ldots,x_n]]$$ is $$h$$, i.e. $$h=\operatorname{inf}\{\operatorname{height}(p)\mid p\in \operatorname{Spec}(k[[x_1,\ldots,x_n]]), (f_1,\ldots,f_c)\subseteq p\}$$. Then the Jacobian ideal of $$R$$ is defined to be the ideal of $$R$$ generated by $$h\times h$$ minors of the Jacobian matrix $$\frac{\partial(f_1,\ldots,f_c)}{\partial(x_1,\ldots,x_n)}$$.

I have two questions about the definition.

Must the Jacobian ideal of $$R$$ be non-zero? I am confused that why the definition considers $$h\times h$$ minors of the Jacobian matrix. Why not higher minors?

Consider the example $$R=\frac{k[[x,y,z]]}{(xy,xz)}$$, the height of $$(xy,xz)$$ in $$k[[x,y,z]]$$ is $$1$$. We can check easily that there is a $$2\times 2$$ minor of the Jacobian matrix that is non-zero in $$R$$. So it seems that it is reasonable to consider higher minors.

Thank you in advance.

The ideal can be zero, but only if $$R$$ is non-reduced: take for example $$(x^2,xy,y^2)$$ which has Jacobian ideal $$(2x^2,2y^2,4xy)=0$$. If $$R$$ is reduced, then $$\operatorname{Spec} R$$ is either a reduced variety (in the case you're taking $$k[x_1,\cdots,x_n]$$) or the germ of a reduced variety (in the case you're taking $$k[[x_1,\cdots,x_n]]$$) and such things are generically smooth if the characteristic of $$k$$ is zero. This is covered in most introductory algebraic geometry texts if you're looking for a reference.

The reason for considering $$h\times h$$ minors is that you're trying to look at nonsingularity of $$V(f_1,\cdots,f_c)$$, which happens when the Jacobian matrix is of rank $$n-\dim_p V(f_1,\cdots,f_c)$$ at each point of $$p\in V(f_1,\cdots,f_c)$$.

• Nice. Thank you. Cool, I found the last sentence you said in Eisenbud's book Corollary 16.20. Under the last condition you said, the Jacobian ideal defines the singular locus of $R$.
– Jian
Nov 24, 2021 at 1:07
• If $R$ is reduced, the ideal is non-zero. Can you give me a reference? I didn't find it. Also, do you think the ideal is also non-zero when $f_1,\ldots,f_c$ is a regular sequence in $k[[x_1,\ldots,x_n]]$ or $k[x_1,\ldots,x_n]$?
– Jian
Nov 24, 2021 at 1:31
• It's not as direct as that: the vanishing locus of the Jacobian ideal is the singular locus, which is known to be a proper subvariety when the characteristic is zero (see Hartshorne I.5 for a proof, for instance), hence the Jacobian ideal cannot be zero. The regular sequence idea can't work: if $f_1,\cdots,f_c$ is a regular sequence, $f_1^r,\cdots,f_c^r$ is also a regular sequence. Nov 24, 2021 at 1:32
• Yes, the power of a regular sequence is still a regular sequence. In the regular sequence case, do you mean the $c\times c$ minor can be zero? Can you give an example here? Thank you. I don't find any example that the Jacobian ideal is zero when $f_1,\ldots,f_c$ is a regular sequence.
– Jian
Nov 24, 2021 at 1:45
• Sorry, perhaps I was hasty - at present I'm not totally sure about the claim either way. Nov 24, 2021 at 2:52