On iterated forcing at successors of regular cardinals

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 $\{\delta <{\lambda }^{+}:\text{cf}(\delta )=\lambda \}$ that complements a theorem of Shelah.