Destroying Saturation while Preserving Presaturation at an Inaccessible; an Iterated Forcing Argument

Abstract

We prove that nonsaturated, presaturated ideals can exist at inaccessible cardinals, answering both a question of Foreman and of Cox and Eskew. We do so by iterating a generalized version of Baumgartner and Taylor's forcing to add a club with finite conditions along an inaccessible cardinal, and invoking Foreman's Duality Theorem.