$\mathsf{AD}_\mathbb{R}$ and Elementary Embeddings

Suppose $\mathsf{AD}_\mathbb{R} + V = L(\mathscr{P}(\mathbb{R})) + \mathsf{DC}$ holds. (We can use more if it is helpful.)

I believe under $\mathsf{AD}_\mathbb{R}$, every $A \subseteq \mathbb{R}$ is homogeneous Suslin. Let $T$ be some tree so that $A = p[T]$. Suppose $\mathbb{P}$ is a small forcing in $V_{\omega + \omega}$. (I am most interested by forcing whose conditions are reals like Sacks forcing, Cohen forcing, or Random forcing.) Let $G \subseteq \mathbb{P}$ be $\mathbb{P}$-generic for $V$. Let $A_G = p[T]^{V[G]}$. The question is

Is there an elementary embedding $j : L(A,\mathbb{R}) \rightarrow L(A_G,\mathbb{R}^{V[G]})$?

Ultimately, I would like, under $\mathsf{AD}_\mathbb{R}$, to have some generic absoluteness about statements involving $A$ (as defined as $p[T]$).

This statement also resemble results that are proved in choice setting assuming large cardinals.

The only ideas I have are: In Solovay's paper about independence of DC, he showed that under $\mathsf{AD}_\mathbb{R}$, for all $A \subseteq \mathbb{R}$, $(A,R)^\sharp$ exists. In $\mathsf{AD}_\mathbb{R}$, $(A,\mathbb{R})^\sharp$ is homogeneously Suslin. So $(A,\mathbb{R})^\sharp = p[U]$ for some homogeneously Suslin tree $U$. I am hoping perhaps one can show that $p[U]^{V[G]} = (A_G,\mathbb{R}^{V[G]})^\sharp$. Therefore, $(A,\mathbb{R})^\sharp \subseteq (A_G, \mathbb{R}^{V[G]})^\sharp$. I believe this will imply that there is an elementary embedding from $L(A,\mathbb{R})$ into $L(A_G,\mathbb{R}^{V[G]})$.

I would like to know if the embedding above can exists under $\mathsf{AD}_\mathbb{R}$. Can my vague argument possibly work? Are there other ways to obtain this embedding or the desired generic absoluteness. Thanks for any information and references.

Yes, you can show this using your assumption that every set of reals in $L(A,\mathbb{R})$ is $\delta$-weakly homogeneously Suslin (Woodin's Pmax book, theorem 2.30). In that case $(A,\mathbb{R})^{\#}$ exists by closure of pointclass under countable unions. Alternatively you can assume that all sets of reals in $L(A,\mathbb{R})$ are $\delta$-universally Baire and $(A,\mathbb{R})^{\#}$ exists, see theorem 6.3 in http://www.math.yorku.ca/~ifarah/Ftp/2005l14-extender.pdf
• Thanks for your answer. Is it true that $(A,\mathbb{R})^\sharp \subseteq (A_G, \mathbb{R}^{V[G]})^\sharp$? The proof of Theorem 6.3 in Farah seems to suggest this. Also it seems that in the Woodin and Farah reference, they are working with choice and perhaps $\delta$ is a some limit of large cardinals. In the $\mathsf{AD}_\mathbb{R}$ setting, a theorem of Martin says that any set of reals is $\kappa$-homogeneously Suslin for $\kappa < \Theta$. Do you know if the results mentioned in the sources still hold in these settings? Apr 15, 2016 at 20:59