Complete quotient boolean algebras

For I a proper, countably complete ideal on the power set p(X) for some set X, can the quotient Boolean algebra p(X)/I be complete? We first show that, if the cardinality of X is at least É₃, then having completeness implies the existence of an inner model with a measurable cardinal. A well-known situation that entails completeness is when the ideal I is a (nontrivial) ideal over a cardinal k which is k⁺-saturated. The second author had established the sharp result that it is consistent by forcing to have such an ideal over k = É₁relative to the existence of a Woodin cardinal. Augmenting his proof by interlacing forcings that adjoin Boolean suprema, we establish, relative to the same large cardinal hypothesis, the consistency of: 2É₁= É₃and there is an ideal ideal I over É₁suchthat p((É₁)I is complete. (The cardinality assertion implies that there is no ideal over É₁which is É₂- saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.).