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# ModuleofdifferentialsGeneralizedEulersequence on a projective scheme

Let $\mathcal{E}$ be a quasi-coherent sheaf on a scheme $S$. Consider the projective scheme $p : \mathbb{P}(\mathcal{E}) \to S$ and the canonical epimorphism $p^*(\mathcal{E}) \to \mathcal{O}_{\mathbb{P}}(1)$. This corresponds to an epimorphism $p^*(\mathcal{E})(-1) \to \mathcal{O}_{\mathbb{P}}$. It is well-known that its kernel is isomorphic to $\Omega^1_{\mathbb{P}/S}$ if $S$ is affine and $\mathcal{E}$ is free of finite rank (Hartshorne, Theorem II.8.13): This is the famous Euler sequence. Of course, this also follows when $S$ is arbitrary and $\mathcal{E}$ is locally free of finite rank. Even for that I don't know a reference in the literature, except for Ravi Vakil's notes, Class 39.

Actually, I have proven that it holds without any assumptions on $\mathcal{E}$, i.e. we always have an exact sequence $0 \to \Omega^1_{\mathbb{P}/S} \to p^*(\mathcal{E})(-1) \to \mathcal{O}_{\mathbb{P}} \to 0$ (generalized Euler sequence). The proof takes some pages, but basically it is a direct coordinate-free generalization of the proof of the special case already mentioned. I don't want to put so much spam in my thesis, and would like to cite this result, which probably has been proven in the 60s. I could not find it in EGA. So my question is: Is this already written down somewhere?

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Let $\mathcal{E}$ be a quasi-coherent sheaf on a scheme $S$. Consider the projective scheme $p : \mathbb{P}(\mathcal{E}) \to S$ and the canonical epimorphism $p^*(\mathcal{E}) \to \mathcal{O}_{\mathbb{P}}(1)$. This corresponds to an epimorphism $p^*(\mathcal{E})(-1) \to \mathcal{O}_{\mathbb{P}}$. Its It is well-known that its kernel is isomorphic to $\Omega^1_{\mathbb{P}/S}$ if $S$ is affine and $\mathcal{E}$ is locally free of finite rank : Reduce to the case where $\mathcal{E}=\mathcal{O}_S^d$ and $S$ is affine, then it's well-known (Hartshorne, Theorem II.8.13); does someone know a direct reference in the literature?.

Question. When Of course, this also follows when $S$ is arbitrary and $\mathcal{E}$ is not assumed to be locally free of finite rank.

Actually, is it still true I have proven that it holds without any assumptions on $\mathcal{E}$, i.e. we always have an exact sequence $0 \to \Omega^1_{\mathbb{P}/S} \to p^*(\mathcal{E})(-1) \to \mathcal{O}_{\mathbb{P}} \to 0$?

If not. The proof takes some pages, how can we describe $\Omega_{\mathbb{P}/S}$ in terms but basically it is a direct generalization of $p^*$ the proof of the special case already mentioned. I don't want to put so much spam in my thesis, and would like to cite this result, which probably has been proven in the Serre twists60s. I could not find it in EGA. So my question is: Is this already written down somewhere?

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# Module of differentials on a projective scheme

Let $\mathcal{E}$ be a quasi-coherent sheaf on a scheme $S$. Consider the projective scheme $p : \mathbb{P}(\mathcal{E}) \to S$ and the canonical epimorphism $p^*(\mathcal{E}) \to \mathcal{O}_{\mathbb{P}}(1)$. This corresponds to an epimorphism $p^*(\mathcal{E})(-1) \to \mathcal{O}_{\mathbb{P}}$. Its kernel is isomorphic to $\Omega^1_{\mathbb{P}/S}$ if $\mathcal{E}$ is locally free of finite rank: Reduce to the case where $\mathcal{E}=\mathcal{O}_S^d$ and $S$ is affine, then it's well-known (Hartshorne, Theorem II.8.13); does someone know a direct reference in the literature?.

Question. When $\mathcal{E}$ is not assumed to be locally free of finite rank, is it still true that we have an exact sequence $0 \to \Omega^1_{\mathbb{P}/S} \to p^*(\mathcal{E})(-1) \to \mathcal{O}_{\mathbb{P}} \to 0$?

If not, how can we describe $\Omega_{\mathbb{P}/S}$ in terms of $p^*$ and the Serre twists?