# 2017 seminar talk: Rigid Ideals

Talk held by Brent M. Cody (Virginia Commonwealth University, Richmond, USA) at the KGRC seminar on 2017-03-02.

### Abstract

An ideal $I$ on a cardinal $\kappa$ is called rigid if all automorphisms of $P(\kappa)/I$ are trivial. Woodin proved that if $MA_{\omega_1}$ holds, then every saturated ideal on $\omega_1$ is rigid. In all previously known models containing rigid saturated ideals, GCH fails. In this talk I will discuss recent joint work with Monroe Eskew in which we prove that the existence of a rigid saturated ideal on $\mu^+$, where $\mu$ is an uncountable regular cardinal, is consistent with GCH, relative to the existence of an almost huge cardinal. Our proof involves adapting the Friedman-Magidor coding forcing (from the number of normal measures paper) to code a generic for a universal collapsing poset which forces an almost huge cardinal $\kappa$ to become the successor of an uncountable regular $\mu$. Our forcing is ${<}\mu$-distributive and in the resulting forcing extension, GCH holds and there is a saturated ideal $I$ on $\mu^+$ such that in any forcing extension by $\mathbb{P}=P(\mu^+)/I$ there is a unique generic filter for $\mathbb{P}$, hence $I$ is rigid.