Suppose that $M$ is a model of $\sf ZFC$, and we add some generic set $G$. Then it is not hard to see that for every $x\in M[G]$ it holds $M\subseteq M[x]\subseteq M[G]$.
Given $x\in M[G]$ such that $x\subseteq M$, can we find a forcing $P\in M[G]$ such that:
- If $H$ is $P$-generic over $M[G]$ then $H\in M[G]$ (triviality).
- If $H$ is $(P\cap M)$-generic over $M$, then $H\notin M$ (not so-triviality).
- $x$ is $(P\cap M)$-generic over $M$ (relevance).
Obviously we're not talking about minimal forcing like Sacks and friends, but on forcing which have a lot of non-trivial intermediate forcings (like Cohen and friends).