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Suppose $Y$ is a smooth variety, $E$ is a rank $d$ bundle on $Y$, $s$ is a regular section of $E$ over $Y$,(i.e.,locally under a trivialization $E|_U\cong O_U^d$, write $s=(s_1,\dots,s_d)$, then $s_i$ form a regular sequence). Suppose $X$ is the zero scheme of $s$, which has codimension $d$ by our assumption,how do we show $N_{Y/X}\cong E|_X$? Is there a global argument?

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    Obvious hint would be `Koszul complex'. – Mohan Aug 27 '15 at 14:21
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    Thanks! Suppose $s\colon O_X\to E$ be a section, we have $E^\vee \to O_Y\to O_X\to 0$, extend it to Koszul complex $\wedge^2E^\vee\to E^\vee\to O_Y$, it is a resolution of $O_X$. As we can calculate by $0\to I\to O_Y\to O_X\to 0$, $I/I^2=Tor_1^{O_Y}(O_X,O_X)$, we can also calculate by Koszul complex, which we get $I/I^2\cong E^\vee$. –  Aug 27 '15 at 14:37

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