A model of Cummings and Foreman revisited

This paper concerns the model of Cummings and Foreman where from ω supercompact cardinals they obtain the tree property at each ℵn for 2≤n<ω. We prove some structural facts about this model. We show that the combinatorics at ℵω+1 in this model depend strongly on the properties of ω1 in the ground model. From different ground models for the Cummings-Foreman iteration we can obtain either ℵω+1∈I[ℵω+1] and every stationary subset of ℵω+1 reflects or there are a bad scale at ℵω and a non-reflecting stationary subset of ℵω+1∩cof(ω1). We also prove that regardless of the ground model a strong generalization of the tree property holds at each ℵn for n≥2.