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 omega(3),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 kappa which is kappa(+)-saturated. The second author had established the sharp;result that it is consistent by forcing to have such an ideal over kappa = omega(1) relative 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: 2(omega 1) = omega(3) and there is an ideal ideal I over omega(1) such that P(omega(1))/I is complete. (The cardinality assertion implies that there is no ideal over omega(1) which is omega(2)-saturated, and so completeness of the Boolean algebra and saturation of the ideal has been separated.)