Following Larson's "The Stationary Tower", let $\mathbb{P}_{<\delta}$ be the full stationary tower on $\delta$, and for a stationary $S\subset \mathcal{P}_\delta(V_\delta)$, $\mathbb{P}_{<\delta}^S$ is the restriction to $S$.

It is stated that "under fairly general assumptions on $S$, much of the basic theory of $\mathbb{P}_{<\delta}$ carries over to $\mathbb{P}_{<\delta}^S$", but the book then deals only with the case $S=\mathcal{P}_{\omega_1}(V_\delta)$. My questions are:

- What are the general assumptions? Is there a reference for the general case?
- More specifically, if we let $S=\mathcal{P}_{\kappa}(V_\delta)$ for $\kappa<\delta$, will the same theorems still apply? I suppose we need $\kappa$ regular, does it need to be successor cardinal?
- When $S=\mathcal{P}_{\omega_1}(V_\delta)$ and $\delta$ is Woodin, we get in the extension an embedding $j$ such that $crit(j)=\omega_1$ and $j(\omega_1)=\delta$. If $S=\mathcal{P}_{\kappa}(V_\delta)$, will we get $crit(j)=\kappa$ and $j(\kappa)=\delta$? If not, how (and for which $\kappa$) can we get such an embedding?