Let $\widetilde{X}$ denote the blow-up of a scheme $X$ with respect to a sheaf of ideals $\mathcal{I}$. Let $Y$ be a closed subscheme of $X$, such that $\mathcal{I}\mathcal{O}_Y$ (the inverse image ideal sheaf) is invertible. Does there exist a map $f: BL_Y(X)\rightarrow \widetilde{X}$ such that $\pi_1f=\pi_2$, where $\pi_1:\widetilde{X} \rightarrow X $ and $\pi_2:BL_Y(X) \rightarrow X$ are the blow-up maps?
To get such a map we have to show $\mathcal{I}\mathcal{O}_{BL_Y(X)}$ is invertible via the map $\pi_2$ which I am unable to show.