Suppose that $\delta$ is a Woodin cardinal and that $\kappa$ is the critical point of the generic embedding $j:V\rightarrow M$ after forcing with the stationary tower ($\kappa$ can be $\omega_1$ or $\omega_2$). I have seen that in some places people mention that $j(\kappa)=\delta$, and in other places that $\delta$ is fixed by $j$. Is this right? Can someone explain the difference, and in which cases we get the first and in which cases the second (for example, does countable or full tower make a difference)? On what conditions do these properties depend?
Unfortunately I have no reference for the tower and my knowlegde comes from some online sources. So perhaps my questions are trivial or vague but I would really appreciate any helpful comments or hints.