TITLE: centeredness in forcing iterations
QUESTION [8 upvotes]: Suppose $A$ is a complete subalgebra of a complete boolean algebra $B$, and $B$ is $\kappa$-centered.  Let $G \subset A$ be a generic ultrafilter.  Is $B/G$ $\kappa$-centered in $V[G]$?
Naively, we might attempt to prove it as follows.  Let $\{ F_\alpha : \alpha < \kappa \}$ be a collection of filters and in $V[G]$ let $F^*_\alpha = \{ [b]_G : b \in F_\alpha, b \not=_G 0 \}$.  But for $b_1, b_2 \in F_\alpha$, maybe $b_1 \wedge b_2 =_G 0$, so that $F^*_\alpha$ is not a filter.
An example where the centeredness cardinal goes up when modding out by a filter is given by comparing $\scr P(\omega)$ and $\scr P(\omega)/ \mathrm{fin}$.  But maybe the generic filter case is different.

REPLY [6 votes]: Let $P$ be the forcing for adding a Suslin tree $\mathring{T}$ by finite conditions. Then both $P$ and $P \star \mathring{T}$ are forcing isomorphic to adding $\omega_1$ Cohen reals (so $P \star \mathring{T}$ is sigma-centered). See lemma 5.6 here.