Timeline for When the sheaf of principal parts is reflexive?
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| Nov 17, 2022 at 19:14 | comment | added | gabriel fazoli | I mean $\otimes $ with the right structure of $\mathcal{P}^n_X$, the one such that $s \cdot f = d^nf \cdot s$. It's at EGA IV, 16.7.2.1. I see what you mean if you are talking about $\mathcal{P}^1_X \otimes \mathcal{L}$ with the left structure of $\mathcal{P}^1_X$, that is, if $\mathcal{L} \otimes \mathcal{P}^1_X \simeq \mathcal{P}^1_X \otimes \mathcal{L}$ then $\mathrm{deg}(L)=0$, but I didn't know the inverse was also true. | |
| Nov 17, 2022 at 16:44 | comment | added | Jason Starr | That is not how the sheaf of principal parts is defined. For an invertible sheaf $\mathcal{L}$ on a smooth projective curve $X$, the locally free sheaf of rank $2$, $\mathcal{P}^1_X(\mathcal{L})$, is isomorphic to $\mathcal{P}^1_X\otimes_{\mathcal{O}_X} \mathcal{L}$ if and only if the degree of $\mathcal{L}$ equals $0$. | |
| Nov 17, 2022 at 14:38 | history | asked | gabriel fazoli | CC BY-SA 4.0 |