Generalized Euler sequence on a projective scheme

2012-12-16T14:54:59Z

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?

Generalized Euler sequence on a projective scheme - MathOverflow

Generalized Euler sequence on a projective scheme