TITLE: Consistency of the nonrigidity of $P(\omega_1)/NS$
QUESTION [6 upvotes]: Is it consistent with ZFC that there exists an automorphism of $P(\omega_1)/\mathrm{NS}_{\omega_1}$ which is not the identity?

REPLY [6 votes]: It is consistent relative to large cardinals that $P(\omega_1)/NS \cong \mathcal B(\mathrm{Col}(\omega,\omega_1))$ and many other homogeneous algebras.
It almost looks like the answer is yes in $L$, but I'm not sure.  In $L$ there are no precipitous ideals, and GCH holds.  A theorem of Balcar and Franek implies that therefore $\mathcal P(\omega_1)/NS$ is forcing-equivalent to $\mathrm{Col}(\omega,\omega_2)$.  Moreover, by general facts about collapsing, there is a dense embedding of $\mathrm{Col}(\omega,\omega_2)$ into $\mathcal P(\omega_1)/NS$.  $\mathrm{Col}(\omega,\omega_2)$ carries many nontrivial isomorphisms, but as Yair pointed out, we may not necessarily be able to lift this to an algebra strictly smaller than the completion.