From angular mechanics I am trying to figure out where an equation used by my textbook comes from

$$\Delta \omega = \frac{r \times J}{I}$$

where $J$ is the linear impulse. [This](https://physics.stackexchange.com/questions/302833/impulse-and-angular-velocity) answer derives that formula but it uses $\tau = I\omega$ to get there. I am not sure how this can be reconciled with the usual definition that I have seen $\tau=I\alpha$ and I am also not sure about the definition of $J$ used in the answer. Thanks for any pointers to help me figure this out!