Applications of pcf for mild large cardinals to elementary embeddings

The following pcf results are proved: 1. Assume that κ > א0 is a weakly compact cardinal. Let μ > 2κ be a singular cardinal of cofinality κ. Then for every regular λ < pp+Γ(κ)(μ) there is an increasing sequence ⟨λi | i < κ⟩ of regular cardinals converging to μ such that λ = tcf( ∏ i<κ λi, sup{sup pcfσ∗−complete(a) | a ⊆ Reg ∩ (μ , χ) and |a| < μ}. As an application we show that: if κ is a measurable cardinal and j : V → M is the elementary embedding by a κ– complete ultrafilter over a measurable cardinal κ, then for every τ the following holds: 1. if j(τ) is a cardinal then j(τ) = τ ; 2. |j(τ)| = |j(j(τ))|; 3. for any κ–complete ultrafilter W on κ, |j(τ)| = |jW (τ)|. The first two items provide affirmative answers to questions from [2] and the third to a question of D. Fremlin.