A Connection between Decomposability of Ultrafilters and Possible Cofinalities

We introduce the decomposability spectrum KD = {λ ≥ ω|D is λ-decomposable} of an ultrafilter D, and show that Shelah’s pcf theory influences the possible values KD can take. For example, we show that if a is a set of regular cardinals, μ ∈ pcf a, the ultrafilter D is |a|-complete and KD ⊆ a, then μ ∈ KD. As a consequence, we show that if λ is singular and for some λ < λ KD contains all regular cardinals in [λ , λ) then: (a) if cf λ = ω then either λ ∈ KD, or λ ∈ KD; and (b) if D is (cf λ)-complete then λ ∈ KD, and pp(λ) = λ.