TITLE: Precipitous ideals and GCH
QUESTION [11 upvotes]: It is well known that ZFC + "There is a measurable cardinal" is equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$."  Is ZFC + "There is a measurable $\kappa$ such that $2^\kappa > \kappa^+$" equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$" + CH + $2^{\omega_1}>\omega_2$?

REPLY [6 votes]: Just a measurable cardinal is sufficient.
Assume $\kappa$ is a measurable cardinal and force with $Col(\omega, < \kappa)*\dot{Add}(\kappa, \tau^*)$, where $\tau^* \geq \kappa^{++}$
is suitably defined (see Gitik's paper ``On Generic Elementary Embeddings'').
As it is shown in the above cited paper, there is a nomral precipitous ideal on $\omega_1$ in the resulting generic extension.