Blowing up the power of a singular cardinal of uncountable cofinality with collapses

Abstract: The {\em Singular Cardinal Hypothesis} (SCH) is one of the most classical combinatorial principles in set theory. It says that if $\kappa$ is singular strong limit, then $2^{\kappa}=\kappa^+$. We prove that given a singular cardinal $\kappa$ of {\em cofinality} $\eta$ in the ground model, which is a limit of suitable large cardinals, and $\eta^+=\aleph_{\gamma}$, then there is a forcing extension which preserves cardinals and cofinalities up to and including $\eta$, such that $\kappa$ becomes $\aleph_{\gamma+\eta}$, and SCH fails at $\kappa$. Furthermore, if $\eta$ is not an $\aleph$-fixed point, then in our model, SCH fails at $\aleph_{\eta}$. Our large cardinal assumption is below the existence of a Woodin cardinal. In our model we also obtain a very good scale.