# Construction of Jacobian Ideal

In Qing Liu's Algebraic Geometry and Arithmetic Curve, we have the following proposition(6.3.13):

Let $$S$$ be a locally Noetherian scheme. Let $$X$$, $$Y$$ be smooth schemes over $$S$$. Then any immersion $$f:X\rightarrow Y$$ of $$S$$-schemes is a regular immersion, and we have a canonical exact sequence on $$X$$:

$$0\rightarrow \mathcal{C}_{X/Y}=f^*(\mathscr{I}_X/\mathscr{I}_X^2)\rightarrow f^*\Omega_{Y/X}\rightarrow \Omega_{X/S}\rightarrow 0$$

For this question: Suppose $$Y\rightarrow X$$ is a birational morphism of regular schemes. Suppose we have an embedding $$i:Y\hookrightarrow W$$ and $$W$$ is smooth over $$X$$. Then one can show that the sheaf of ideal $$\mathscr{I}_Y$$ of $$\mathcal{O}_W$$ defining $$Y$$ is generated by regular sequence. If we define $$\mathcal{C}_{Y/W}=i^*(\mathscr{I}_Y/\mathscr{I}_Y^2)$$, then $$\mathcal{C}_{Y/W}$$ is a locally free sheaf of rank n on Y, where n is the relative dimension of $$W$$ over $$X$$

I am wondering whether the following sequence is left exact given the fact that $$Y$$ is not smooth over $$X$$. $$0\rightarrow \mathcal{C}_{Y/W}\rightarrow i^*\Omega_{W/X}\rightarrow \Omega_{Y/X}\rightarrow 0$$

If this is exact, then we may define the Jacobian ideal $$\mathcal{J}_{Y/X}$$ as the ideal sheaf $$\bigwedge^n\mathcal{C}_{Y/W}\otimes (\bigwedge^n i^*\Omega_{W/X})^{-1}$$ of $$\mathcal{O}_Y$$. One can show that this definition is independent of $$W$$ but if the above sequence is not exact, then I don't know whether this is still an ideal sheaf of $$\mathcal{O}_Y$$. Thank you.

• I suppose the typical example of the situation : $Y$ blow-up of $k$-smooth $X$ along a smooth subvariety. Now $Y$ is embedded in $\mathbb{P}^N_X$ for some$N$. Here $Y$ is not smooth over $X$. Commented Jul 23, 2021 at 5:50

I think this is true: since $$Y\rightarrow X$$ is birational, the homomorphism $$\mathcal{C}_{Y/W}\rightarrow i^*\Omega_{W/X}$$ is bijective on a dense open subset of $$Y$$. Thus its kernel is torsion, hence zero since $$\mathcal{C}_{Y/W}$$ is locally free.

• Just a small question. What if I drop the birational assumption and keep the fact that $\mathscr{I}_Y$ is generated by a regular sequence, will the result still hold?
• The argument I gave works as soon as the generic fiber of $Y\rightarrow X$ is smooth: then the sheaf homomorphism is generically injective, hence injective because $\mathscr{C}_{Y/W}$ is torsion free. But you need some hypothesis: think of the case $W=X$.