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Suppose $P$ is a variety over $\mathbb{C}$(in my case $P$ might have slight singularities), and $V=\cap_{i}^{r} D_i$ is a complete intersection. Then, it is claimed exists the so called "Standard exact sequence for a complete intersection":

$$0 \to O_{V}(-D_1) \oplus \cdots \oplus O_{V}(-D_r) \to \Omega_{P}^{1}|\_V \to \Omega_{V}^1 \to 0.$$

Does anyone know how to show the above exact sequence? Or point out some reference having discussiong about such things? Moreover, what are the corresponding sequence for the $p$-forms (i.e. the sequence tells something about $\Omega\_{P}^{p}|_V $ and $\Omega\_{V}^p $ )?

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Sorry about the appearance of my post, but why it can code through in but not (which I think it might more suitable to put it here)? – Li Zhan Aug 5 '12 at 3:10
This is a consequence of the adjunction exact sequence. Each term corresponds to a generator of the ideal. – Will Sawin Aug 5 '12 at 3:11
Could do say something more about your answer? What do you mean by adjunction exact sequence? – Li Zhan Aug 5 '12 at 3:15
If you're worried about the appearance of your post, you should not revert my changes, which make the TeX work. There is a bug in the parser here that requires that you escape underscores and superscripts sometimes; if math doesn't render, just try putting "\_"in place of "_" and so on. – Ryan Reich Aug 5 '12 at 3:43

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