Small u(kappa) at singular kappa with compactness at kappa++

Abstract

We show that the tree property, stationary reflection and the failure of approachability at $\kappa^{++}$ are consistent with $\mathfrak{u}(\kappa) = \kappa^+ < 2^\kappa$, where $\kappa$ is a singular strong limit cardinal with the countable or uncountable cofinality. As a by-product, we show that if $\lambda$ is a regular cardinal, then stationary reflection at $\lambda^+$ is indestructible under all $\lambda$-cc forcings (out of general interest, we also state a related result for the preservation of club stationary reflection).