Cardinal sequences of LCS spaces under GCH

Let C(α)C(α) denote the class of all cardinal sequences of length αα associated with compact scattered spaces. Also put Cλ(α)={f∈C(α):f(0)=λ=min[f(β):β<α]}.Cλ(α)={f∈C(α):f(0)=λ=min[f(β):β<α]}.If λλ is a cardinal and α<λ++α<λ++ is an ordinal, we define Dλ(α)Dλ(α) as follows: if λ=ωλ=ω, Dω(α)={f∈α{ω,ω1}:f(0)=ω},Dω(α)={f∈α{ω,ω1}:f(0)=ω}, and if λλ is uncountable, Dλ(α)={f∈α{λ,λ+}:f(0)=λ,f−1{λ} is <λ-closed and successor-closed in α}. We show that for each uncountable regular cardinal λλ and ordinal α<λ++α<λ++ it is consistent with GCH that Cλ(α)Cλ(α) is as large as possible, i.e. Cλ(α)=Dλ(α).Cλ(α)=Dλ(α). This yields that under GCH for any sequence ff of regular cardinals of length αα the following statements are equivalent: (1)f∈C(α)f∈C(α) in some cardinal-preserving and GCH-preserving generic-extension of the ground model.(2)for some natural number nn there are infinite regular cardinals λ0>λ1>⋯>λn−1λ0>λ1>⋯>λn−1 and ordinals α0,…,αn−1α0,…,αn−1 such that α=α0+⋯+αn−1α=α0+⋯+αn−1 and f=f0⌢f1⌢⋯⌢fn−1 where each fi∈Dλi(αi)fi∈Dλi(αi).The proofs are based on constructions of universal locally compact scattered spaces.