On completeness of the quotient algebras {cal P}(kappa)/I

In this paper, the following are proved: Theorem A. The quotient algebra ${cal P} (kappa )/I$ is complete if and only if the only non-trivial I -closed ideals extending I are of the form $Ilceil A$ for some $Ain I^+$ . Theorem B. If $kappa$ is a stationary cardinal, then the quotient algebra ${cal P} (kappa )/ NS_kappa$ is not complete. Corollary. (1) If $kappa$ is a weak compact cardinal, then the quotient algebra ${cal P} (kappa )/NS_kappa$ is not complete. (2) If $kappa$ bears $kappa$ -saturated ideal, then the quotient algebra ${cal P} (kappa )/NS_kappa$ is not complete. Theorem C. Assume that $kappa$ is a strongly compact cardinal, I is a non-trivial normal $kappa$ -complete ideal on $kappa$ and B is an I -regular complete Boolean algebra. Then if ${cal P} (kappa )/I$ is complete, it is B -valid that for some $Asubseteqcheckkappa$ , ${cal P} (kappa )/({bf J}lceil A)$ is complete, where J is the ideal generated by $check I$ in $V^B$ . Corollary. Let M be a transitive model of ZFC and in M , let $kappa$ be a strongly compact cardinal and $lambda$ a regular uncountable cardinal less than $kappa$ . Then there exists a generic extension M [ G ] in which $kappa =lambda^+$ and $kappa$ carries a non-trivial $kappa$ -complete ideal I which is completive but not $kappa^+$ -saturated. Received: 1 April 1997 / Revised version: 1 July 1998