On the Splitting Number at Regular Cardinals

Let $\kappa$,$\lambda$ be regular uncountable cardinals such that $\lambda > \kappa^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(\kappa) = \lambda$ starting from a ground model in which $o(\kappa) = \lambda$ and prove that assuming $\neg 0^{\P}$, $s(\kappa) = \lambda$ implies that $o(\kappa) \geq \lambda$ in the core model.