Collapsing Successors of Singulars

Let κ be a singular cardinal in V , and let W ⊇ V be a model such that κ+V = λ + W for some W -cardinal λ with W |= cf(κ) 6= cf(λ). We apply Shelah’s pcf theory to study this situation, and prove the following results. 1) W is not a κ+-c.c generic extension of V . 2) There is no “good scale for κ” in V , so in particular weak forms of square must fail at κ. 3) If V |= cf(κ) = א0 then V |= “κ is strong limit =⇒ 2κ = κ+”, and also ωκ ∩W ) ωκ ∩ V . 4) If κ = אω then λ ≤ (2א0 )V .