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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.

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  • $\begingroup$ 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 $X$. $\endgroup$
    – Evans Gambit
    Commented Jul 23, 2021 at 5:50

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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.

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  • $\begingroup$ Thank you for your answer $\endgroup$
    – user294943
    Commented Jul 23, 2021 at 6:38
  • $\begingroup$ 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? $\endgroup$
    – user294943
    Commented Jul 23, 2021 at 6:47
  • $\begingroup$ 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 $W=X$. $\endgroup$
    – abx
    Commented Jul 23, 2021 at 9:52
  • $\begingroup$ Thank you very much $\endgroup$
    – user294943
    Commented Jul 23, 2021 at 16:58

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