A characterization of extenders of HOD

Abstract

Assume $AD+V=L(\mathbb{R})$. Let $\kappa ={\underset{˜}{\delta }}_1^2$, the supremum of all ${\underset{˜}{\Delta }}_1^2$ prewellorderings. We prove that extenders on the sequence of $\H$ that have critical point $\kappa$ are generated by countably complete measures. This provides a partial reversal of Woodin's result that the $<\Theta$-strongness of $\kappa$ in $\H$ is witnessed by $\kappa$-complete ultrafilters on $\k$. The aforementioned characterization of extenders works in a more general setting for all cutpoint measurable cardinals of $\H$ in all models of determinacy where the fine structural analysis of $\H$ has been carried out. For example, it holds in the minimal model of the Largest Suslin Axiom. It also gives a simple proof of a theorem of Steel that the successor members of the Solovay sequence are cutpoints in $\H$ (in models where $\H$ analysis is carried out).