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.



Construction of Jacobian Ideal
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.


