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Following EGA IV, we can construct the sheaf of principal parts of a sheaf $\mathcal{E}$ over a scheme $X$ by $\mathcal{P}_X^n(\mathcal{E}) = \mathcal{P}_X^n \otimes \mathcal{E}$, where $\otimes$ is at the right structure of $\mathcal{P}_X^n$. If $X$ is non-singular and $\mathcal{E}$ is coherent, I can see that $\mathcal{P}_X^n(\mathcal{E})$ is also coherent.

We can put a structure of $\mathcal{O}_X$-module at $\mathcal{Diff}_X^{\le n}(\mathcal{E}, \mathcal{O}_X)$ using the isomorphism of additive group sheaves $\mathcal{Diff}_X^{\le n}(\mathcal{E}, \mathcal{O}_X) \simeq \mathcal{Hom}(\mathcal{P}_X^n(\mathcal{E}), \mathcal{O}_X)$. If $\mathcal{E}$ is coherent, then $\mathcal{Diff}_X^{\le n}(\mathcal{E}, \mathcal{O}_X)$ is reflexive.

Under what conditions $\mathcal{P}_X^n(\mathcal{E})$ is also reflexive? That is, when $\mathcal{P}_X^n(\mathcal{E}) = \mathcal{Hom}(\mathcal{Diff}_X^{\le n}(\mathcal{E}, \mathcal{O}_X),\mathcal{O}_X)$? I can see this under strong assumptions, such as $\mathcal{E}$ locally free, but I expect this to be true under weaker ones(maybe $\mathcal{E}$ reflexive?).

Thanks in advance.

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  • $\begingroup$ 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 $0$. $\endgroup$
    – Jason Starr
    Commented Nov 17, 2022 at 16:44
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    $\begingroup$ 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. $\endgroup$
    – gabriel fazoli
    Commented Nov 17, 2022 at 19:14

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