Local mantles of $L[x]$

Abstract

Assume ZFC. Let $\kappa$ be a cardinal. Recall that a ${<\kappa}$-ground is a transitive proper class $W$ modelling ZFC such that $V$ is a generic extension of $W$ via a forcing $\mathbb{P}\in W$ of cardinality ${<\kappa}$, and the $\kappa$-mantle is the intersection of all ${<\kappa}$-grounds. Assume there is a Woodin cardinal and a proper class of measurables, and let $x$ be a real of sufficiently high Turing degree. Let $\kappa$ be a limit cardinal of $L[x]$ of uncountable cofinality in $L[x]$. Using methods from Woodin's analysis of $\mathrm{HOD}^{L[x,G]}$, we analyze the $\kappa$-mantle of $L[x]$, and show that it models ZFC + GCH + "There is a Woodin cardinal". Moreover, we show that it is a fully iterable strategy mouse (analogous to $\mathrm{HOD}^{L[x,G]}$). We also analyze another form of "local mantle", partly assuming also a weak form of Turing determinacy. We also compute bounds on how much iteration strategy can be added to $M_1$ before $M_1^\#$ is added.