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Let $Y$ be a smooth projective $k$-variety, $D\subset Y$ a smooth (irreducible) divisor and a line bundle such that $L^2=\mathcal O_Y(D)$. Let us call $f:X\rightarrow Y$ the double cover defined by these data. I was trying to compute $\Omega_{X/Y}$ using that $f_*\mathcal O_X=O_Y\oplus L^{-1}$: it seems (locally $\Omega_{B/A}=\oplus_{b\in B}db/$Leibniz rules) that $\Omega_{X/Y}=i_{f^{-1}D,*}\mathcal O_{f^{-1}D}$ but I am not sure.

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$\Omega_{X/Y} \cong i_*O_R(-R)$, where $i:R \hookrightarrow X$ is the ramification divisor.

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  • $\begingroup$ Thank you very much for your answer. May I ask you how do you compute this? $\endgroup$ – pi_1 Jun 14 '16 at 15:52
  • $\begingroup$ Write exact sequence $0 \to f^*\Omega_Y \to \Omega_X \to \Omega_{X/Y} \to 0$, restrict it to the ramification divisor $R$, and compare with the conormal sequences of $R$ and the branch divisor $B \subset Y$. $\endgroup$ – Sasha Jun 14 '16 at 15:57

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