Suppose $Y$ is a singular projective variety defined over (for simplicity) $\mathbb{Q}$, then from Prop 7.5 of Hartshorne, the dualizing sheaf of $Y$ is defined and let's denote it by $\omega_Y$.
Let $Z$ be the singular locus of $Y$ and for simplicity, let's assume $Z$ is irreducible. Now let $X$ be the blow-up of $Y$ at $Z$ with exceptional divisor $D$ \begin{equation} \pi:X \rightarrow Y \end{equation} What is the relation between the dualizing sheaf $\omega_X$ (which is a line bundle as $X$ is smooth) and the pull-back of $\omega_Y$? e.g is it still true that \begin{equation} \omega_X=\pi^* \omega_Y+D \end{equation} just like the smooth case?