Rigid ideals

Abstract

An ideal $I$ on a cardinal $κ$ is called \emph{rigid} if all automorphisms of $P(κ)/I$ are trivial. An ideal is called \emph{$μ$-minimal} if whenever $G\subseteq P(κ)/I$ is generic and $X\in P(μ)^{V[G]}\setminus V$, it follows that $V[X]=V[G]$. We prove that the existence of a rigid saturated $μ$-minimal ideal on $μ^+$, where $μ$ 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 $μ^+$, where $μ$ is an \emph{uncountable} regular cardinal, is consistent with GCH relative to the existence of an almost-huge cardinal. Addressing the case $μ=ω$, we show that the existence of a rigid \emph{presaturated} ideal on $ω_1$ is consistent with CH relative to the existence of an almost-huge cardinal. The existence of a \emph{precipitous} rigid ideal on $μ^+$ where $μ$ is an uncountable regular cardinal is equiconsistent with the existence of a measurable cardinal.