On iterated forcing at successors of regular cardinals

Abstract

We investigate the problem of when $\le \lambda$--support iterations of $<\lambda$--complete notions of forcing preserve ${\lambda }^{+}$. We isolate a property -- {\em properness over diamonds} -- that implies ${\lambda }^{+}$ is preserved and show that this property is preserved by $\lambda$--support iterations. We close with an application of our technology by presenting a consistency result on uniformizing colorings of ladder systems on $\{δ<λ^+:\cf(δ)=λ\}$ that complements a theorem of Shelah.