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Let $i: D\hookrightarrow X$ be a closed subscheme which we can think of as an effective Cartier divisor. Then Hartshorne 6.18 claims that $\mathscr{J}_D\cong \mathscr{O}(-D)$, where the LHS is the ideal sheaf of the closed embedding $D\hookrightarrow X$, and the RHS is the invertible sheaf associated to the divisor $-D$. But Vakil 21.2.H claims that in fact $\mathscr{N}^\vee_{D/X}\cong \mathscr{O}(-D)|_D$, where the LHS is the conormal sheaf, namely $i^*(\mathscr{J}_D/\mathscr{J}_D^2)$.

Doesn't this imply that $i^*(\mathscr{J}_D) \cong i^*(\mathscr{J}_D/\mathscr{J}_D^2)$? (Which doesn't make sense because it would imply $i^*(\mathscr{J}_D^2)=0$).

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The claimed equality is true and does make sense. The point is that $i^*$ tensors with the structure sheaf $\mathcal{O}_X / \mathscr{J}_D$! Now let me just consider the affine situation, i.e. a ring $A$ with an ideal $I$, then of course $$I \otimes_A A/I \cong I/I^2 \otimes_A A/I$$ which does not mean $I^2 =0$ because $A /I$ is not a faithfully flat $A$-algebra.

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