Let X be a smooth projective curve over a basis S of characteristic $ p$. Denote by $\mathscr P^1_{X/S}$ the sheaf of principal parts of degree $≤ 1$, namely the structural sheaf of the second order infinitesimal neighborhood $∆^{(2)}$ of the diagonal in $X ×_S X$ (EGA IV, sect $16$). The left $\mathscr O_X$-module structure is defined by the first projection and the right one by the second projection.Let 1 be the global section of $\mathscr P^1_{X/S}$ defined by the constant function with value 1 on $∆^{(2)}$ . The restriction to the diagonal gives the exact sequence $0 → ω_{X /S} → \mathscr P^1_{X/S} → \mathscr O_X → 0$ of left $\mathscr O_X$-modules.

I am having hard time understanding this paragraph, namely:

  1. $ω_{X /S}$ is supposed to be the canonical line bundle of $X$. I found many references on this notion, for instance here and here but which applies in this context?
  2. From definitions I found that $\mathscr P^1_{X/S} = ∆^* \mathscr O_{X \times_S X}/\mathscr I^2$ for some sheaf of $\mathscr O_X-$ modules $\mathscr I$. But I don't understand the sentence with first and second projections
  3. Finally I don't how we get the exact sequence.

Thank you for your help.

  • 3
    $\begingroup$ If you don’t like the EGA reference, there’s a very nice explanation of this concept in the book 3264 and All That by Eisenbud and Harris in the chapter on contact problems. $\endgroup$ Feb 24, 2020 at 5:23
  • $\begingroup$ Thank you I will check this book $\endgroup$
    – Conjecture
    Feb 24, 2020 at 6:52
  • $\begingroup$ Actually I didn't find any reference to these questions in EGA IV $\endgroup$
    – Conjecture
    Feb 24, 2020 at 7:18


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