Applications of pcf for mild large cardinals to elementary embeddings

The following pcf results are proved:1. Assume that κ>ℵ0is a weakly compact cardinal. Let μ>2κbe a singular cardinal of cofinality κ. Then for every regular λsup{suppcfσ⁎-complete(a)|a⊆Reg∩(μ+,χ)and|a|<μ}.As an application we show that:if κ is a measurable cardinal and j:V→Mis the elementary embedding by a κ-complete ultrafilter over 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 Gitik and Shelah (1993) [2] and the third to a question of D. Fremlin.