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I am reading this note of Blickle on motivic integration and I am stuck at a technical point. At the beginning of the appendix, we are given a proper birational morphism $f: X' \longrightarrow X$ between smooth $k$-varieties ($k$: a field). In particular, we have a first fundamental sequence $$f^*\Omega^1_{X/k} \overset{df}{\longrightarrow} \Omega^1_{X'/k} \longrightarrow \Omega^1_{X'/X} \longrightarrow 0.$$ It is claimed that this sequence is also exact on the left by the birationality of $f$. It is hard to believe this as birationality just implies that $X$ and $X'$ are isomorphic over some open set but not on the whole spaces.

I tried to google, and in the same situation as the OP of this topic, I find no reference for the sequence in the case of blowing up. Assume that it holds true for blowing up, then can we deduce this from the weak factorization theorem (theorem 3.6, page 19 in Blickle)?

Alexey Do
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1 Answers1

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Both $f^*\Omega_X$ and $\Omega_{X'}$ are locally free sheaves (because $X$ and $X'$ are smooth) and the morphism $df$ is an isomorphism on a dense open subset open (by birationality of $f$). Now the kernel of $df$ is a subsheaf of a locally free sheaf, hence it is torsion free, and it is zero on a dense open subset, hence it is torsion. A combination of these two observations implies it is zero, hence the sequence is left exact.

Sasha
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  • Being zero on a dense open set implies being torsion, do you have a reference for this? I found https://math.stackexchange.com/questions/2197519/must-a-quasicoherent-sheaf-which-is-zero-on-a-dense-open-subset-be-torsion but I think there is other way around to see this. – Alexey Do Nov 15 '22 at 15:58
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    The question is local: if $K = Frac(A)$ and $M \otimes_A K = 0$ then $M$ is torsion. Now $K$ is the colimit of localizations $A_s$ at various nonzero elements $s \in A$, so we may assume $M \otimes_A A_s = 0$. But then every element of $M$ is annihilated by a power of $s$, hence $M$ is torsion. – Sasha Nov 15 '22 at 16:29
  • The idea is true but it needs to be modified a little. Since $X$ is not integral (we only have smoothness here, and smooth + connected = integral), writing $Frac(A)$ does not make sense. It should be $M \otimes_A A_f = M_f = 0$ from the beginning (here the hypothesis of being dense is used). – Alexey Do Nov 15 '22 at 16:53
  • Even in the nonintegral case $Frac(A)$ makes sense --- this is the localization with respect to all nonzero divisors. – Sasha Nov 15 '22 at 17:12