Rigid ideals

Abstract

An ideal $I$ on a cardinal $\kappa$ is called \emph{rigid} if all automorphisms of $P(\kappa )/I$ are trivial. An ideal is called \emph{$\mu$-minimal} if whenever $G\subseteq P(\kappa )/I$ is generic and $X\in P(\mu {)}^{V[G]}\setminus V$, it follows that $V[X]=V[G]$. We prove that the existence of a rigid saturated $\mu$-minimal ideal on ${\mu }^{+}$, where $\mu$ is a regular cardinal, is consistent relative to the existence of large cardinals. The existence of such an ideal implies that GCH fails. However, we show that the existence of a rigid saturated ideal on ${\mu }^{+}$, where $\mu$ is an \emph{uncountable} regular cardinal, is consistent with GCH relative to the existence of an almost-huge cardinal. Addressing the case $\mu =\omega$, we show that the existence of a rigid \emph{presaturated} ideal on ${\omega }_1$ is consistent with CH relative to the existence of an almost-huge cardinal. The existence of a \emph{precipitous} rigid ideal on ${\mu }^{+}$ where $\mu$ is an uncountable regular cardinal is equiconsistent with the existence of a measurable cardinal.