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Let $Y\subset X$ be smooth schemes and let $\widetilde{X}$ denote the blowup of $X$ along $Y$. In this question it was shown that $$\pi_{*}\mathcal{O}_{\widetilde{X}}(-nE) = I_{Y/X}^{n}$$ for $n \geq 1$. I would like to prove this by induction.

Consider the exact sequence $$0 \longrightarrow \mathcal{O}_{\widetilde{X}}(-E) \longrightarrow \mathcal{O}_{\widetilde{X}} \longrightarrow \mathcal{O}_{E} \longrightarrow 0 \tag{$*$}$$

This handles the case $n = 1 $.

Twisting the exact sequence $(*)$ by $\mathcal{O}_{\widetilde{X}}(-E)$ and applying the direct image by $\pi$, we have:

$$0 \rightarrow \pi_{*}\mathcal{O}_{\widetilde{X}}(-2E) \rightarrow \pi_{*}\mathcal{O}_{\widetilde{X}}(-E) \rightarrow \pi_{*}\mathcal{O}_{E}(-E)\rightarrow R^{1}\pi_{*}\mathcal{O}_{\widetilde{X}}(-2E) \rightarrow \cdots \tag{$**$}$$

Can I use this to prove that $\pi_*\mathcal{O}_\widetilde{X}(-nE)=I^n_{Y/X}$ by induction? If so, how can I put together the inductive step from the exact long sequence $(**)$?

Suggestions are very welcome.

Since already thank you very much.

KReiser
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Emanuell
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    Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. – Shaun Jan 20 '20 at 00:33
  • Thanks for the tip. I will change the title of the question. – Emanuell Jan 20 '20 at 01:31
  • An interesting question. Can the snake's lemma help? – Zlattan R.S Jan 21 '20 at 14:11
  • I don't know ... I'm trying ...what would it be $R^{1}\pi_{*}\mathcal{O}_{\widetilde{X}}(-2E)$? – Emanuell Jan 21 '20 at 21:44
  • I agree. In particular, I will learn a lot from this question. Thank you very much. – Emanuell Jan 22 '20 at 01:34
  • Here's a trivial remark: if you want to have any hope of proving this by induction, you need to identify $\pi_*\mathcal{O}_E(-nE)$ and the kernel of the second map in your LES. Indeed, one can see from the linked question what it has to be, but I chose to avoid a proof by induction in my answer there because it seemed longer and messier than just doing it directly without induction. Good luck! – KReiser Jan 22 '20 at 20:57
  • Hi guys. In relation to the original question (https://mathoverflow.net/q/68014) I just cannot understand how : If $$\pi_{}\mathcal{O}{E} \longrightarrow \mathcal{O}{Y}$$ is an isomorphism, then the map $$p_{}\mathcal{O}{\mathbb{P}}(m) \longrightarrow p{*}\mathcal{O}_{Z}(-mE)$$ will always be surjective. If someone understood, could you please explain it to me? How was this isomorphism used to conclude that the application will always be surjective? Thank you. – Zlattan R.S Jan 23 '20 at 12:56

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