Let $M$ be a finitely generated module over a commutative Noetherian ring $R$. Assume that there exists an injective $R$-linear morphism $f: R\to M$. Consider the sets $$U:=\{\mathfrak p\in \text{Spec}(R): R_{\mathfrak p} \text{ is a direct summand of some finite direct sum of copies of } M_{\mathfrak p}\}$$
$$V:=\{\mathfrak p\in \text{Spec}(R): R_{\mathfrak p} \text{ is a direct summand of } M_{\mathfrak p}\}$$
$$W:=\{\mathfrak p\in \text{Spec}(R): f_{\mathfrak p}: R_{\mathfrak p} \to M_{\mathfrak p} \text{ is a split injection}\}$$
Clearly, $W\subseteq V\subseteq U$. I ask you to take me on faith that $U,V$ are Zariski open subsets of $\text{Spec}(R)$. Consider the inclusion maps $i_U: U\to \text{Spec}(R)$, $i_W: W\to \text{Spec}(R)$, and $i_V: V\to \text{Spec}(R)$. By abuse of notation, I will denote the sheaf associated to $M$ on $\text{Spec}(R)$ just by $M$.
My questions are:
(1) If $U$ is non-empty , then is $\mathcal O_U$ a direct summand of a finite direct sum of copies of $i_U^* M$ ?
(2) If $V$ is non-empty , then is $\mathcal O_V$ a direct summand of a finite direct sum of copies of $i_V^* M$ ?
(3) If $W$ is non-empty , then is $\mathcal O_W$ a direct summand of a finite direct sum of copies of $i_W^* M$ ?