Rigid Ideals

An ideal I on a cardinal κ is called rigid if all automorphisms of P (κ)/I are trivial. An ideal is called μ-minimal if whenever G ⊆ P (κ)/I is generic and X ∈ P (μ)V [G] \ 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 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 presaturated ideal on ω1 is consistent with CH relative to the existence of an almost-huge cardinal. The existence of a precipitous rigid ideal on μ+ where μ is an uncountable regular cardinal is equiconsistent with the existence of a measurable cardinal.