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In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it has no nontrivial automorphisms. He states that this is well-known, so my question is, where else does it appear in the literature? I would like to find a different proof.

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    $\begingroup$ This seems to be a refinement of what one finds in the Martin's Maximum paper (see around Theorem 18). I do not know of an actual source for Lemma 5.100, though. $\endgroup$
    – Andrés E. Caicedo
    Commented Aug 18, 2015 at 23:47
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    $\begingroup$ It also appears in my paper 'A uniqueness theorem for iterations' (maybe only for the nonstationary ideal, but I think the proof works in general). $\endgroup$
    – Paul Larson
    Commented Aug 20, 2015 at 19:46

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