Cardinal sequences of LCS spaces under GCH

Let C(α) denote the class of all cardinal sequences of length α associated with compact scattered spaces. Also put Cλ(α) = {f ∈ C(α) : f(0) = λ = min[f(β) : β < α]}. If λ is a cardinal and α < λ++ is an ordinal, we define Dλ(α) as follows: if λ = ω, Dω(α) = {f ∈ {ω, ω1} : f(0) = ω}, and if λ is uncountable, Dλ(α) = {f ∈ {λ, λ} : f(0) = λ, f−1{λ} is < λ-closed and successor-closed in α}. We show that for each uncountable regular cardinal λ and ordinal α < λ++ it is consistent with GCH that Cλ(α) is as large as possible, i.e. Cλ(α) = Dλ(α). This yields that under GCH for any sequence f of regular cardinals of length α the following statements are equivalent: (1) f ∈ C(α) in some cardinal preserving and GCH-preserving generic-extension of the ground model. (2) for some natural number n there are infinite regular cardinals λ0 > λ1 > · · · > λn−1 and ordinals α0, . . . , αn−1 such that α = α0 + · · · + αn−1 and f = f0 _ f1 _· · · fn−1 where each fi ∈ Dλi(αi). Preprint submitted to Elsevier 8 February 2010 The proofs are based on constructions of universal locally compact scattered spaces.