LOCAL DEFINABILITY OF HOD HOD sans serif upper H upper O upper D IN L ( R ) L(R) upper L left parenthesis double struck upper R right parenthesis

We show that in L ( R ) L(R) upper L left parenthesis double struck upper R right parenthesis , assuming large cardinals in V , HOD ∥ η + HOD HOD∥ η +HOD sans serif upper H upper O upper D parallel to eta Superscript plus sans serif upper H upper O upper D is locally definable from HOD ∥ η HOD∥ η sans serif upper H upper O upper D parallel to eta for all HOD HOD sans serif upper H upper O upper D -cardinals η ∈ [ δ 1 2 , Θ ) η ∈ [δ ²₁,Θ ) eta element of left bracket bold italic delta 1 squared comma normal upper Theta right parenthesis . This is a further elaboration of the statement “ HOD L ( R ) HODL(R) sans serif upper H upper O upper D Superscript upper L left parenthesis double struck upper R right parenthesis is a core model below Θ