Collapsing successors of singulars

Let $\kappa$ be a singular cardinal in $V$, and let $W \supseteq V$ be a model such that $\kappa ^+_V = \lambda ^+_W$ for some $W$-cardinal $\lambda$ with $W \models \operatorname {cf}(\kappa ) \neq \operatorname {cf}(\lambda )$. We apply Shelah's pcf theory to study this situation, and prove the following results. 1) $W$ is not a $\kappa ^+$-c.c generic extension of $V$. 2) There is no ``good scale for $\kappa$'' in $V$, so in particular weak forms of square must fail at $\kappa$. 3) If $V \models \operatorname {cf}(\kappa ) = \aleph _0$ then $V \models \hbox {``$\kappa $ is strong limit $\implies 2^\kappa = \kappa ^+$'',}$ and also ${}^\omega \kappa \cap W \supsetneq {}^\omega \kappa \cap V$. 4) If $\kappa = \aleph _\omega ^V$ then $\lambda \le (2^{\aleph _0})_V$.