Local Definability of $\mathsf{HOD}$ in $L(\mathbb{R})$

Abstract

We show that in $L(\mathbb{R})$, assuming large cardinals, $\mathsf{HOD} {\parallel}\eta^{+\mathsf{HOD}}$ is locally definable from $\mathsf{HOD} {\parallel}\eta$ for all $\mathsf{HOD}$-cardinals $\eta\in [\boldsymbol{\delta}^2_1,\Theta)$. This is a further elaboration of the statement "$\mathsf{HOD}^{L(\mathbb{R})}$ is a core model below $\Theta$" made by John Steel.