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In Hartshorne's Algebraic Geometry we have

Thm. II.8.13. Let $A$ be a ring, let $Y = \mathrm{Spec}(A)$, and let $X = \mathbb{P}_{A}^{n}$. Then there is an exact sequence of sheaves on $X$,

$$0 \rightarrow \Omega_{X / Y} \rightarrow \bigoplus_{i=0}^{n+1} \mathcal{O}_{X}(-1) \rightarrow \mathcal{O}_{X} \rightarrow 0.$$

I would like to know if there is an analogue exact sequence for the smooth locus of a weighted projective space $\mathbb{P} = \mathbb{P}(a_{0}, \ldots, a_{n})$. I followed the proof of that Thm. and it seems that it holds if we replace "-1" by "$-a_{i}$". I want to compute the Chern character of the smooth locus of $\mathbb{P}$, and for that I would like to take the dual of such exact sequence.

Thank you!

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