TITLE: collapsing successor of singular
QUESTION [8 upvotes]: Let $\lambda$ be a singular cardinal.  Is it consistent that there is a forcing of size $\lambda^+$ that collapses $\lambda^+$ while preserving all cardinals below $\lambda$?
(Note that even without the size requirement this implies a failure of the Jensen covering property, so such a forcing does not necessarily exist.)

REPLY [5 votes]: The answer is no and it follows easily from the following theorem:

Theorem. Suppose $\kappa$ is a regular uncountable cardinal and $|P|\leq \kappa.$ Then $\Vdash_P cf(\kappa)=|\kappa|.$

For a proof of the theorem see
Singularizing forcing of "small" cardinality?