2
$\begingroup$

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.

$\endgroup$
1
  • $\begingroup$ 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 $X$. $\endgroup$ Jul 23, 2021 at 5:50

1 Answer 1

3
$\begingroup$

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.

$\endgroup$
4
  • $\begingroup$ Thank you for your answer $\endgroup$
    – Eva Zhuang
    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? $\endgroup$
    – Eva Zhuang
    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
    Jul 23, 2021 at 9:52
  • $\begingroup$ Thank you very much $\endgroup$
    – Eva Zhuang
    Jul 23, 2021 at 16:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.