I came across a moduli problem which looks very similar to another one appearing in a little lemma by Rapoport and Zink. I tried to prove that mine is also representable, but as a newbie in this topic I would appreciate some proof verification. Could somebody please confirm if what I wrote is correct ?
Lemma 2.10 in Rapoport and Zink's book "Period spaces for $p$-divisible groups" reads the following
Let $S$ be any scheme and let $\alpha : \mathcal M \to \mathcal L$ be a morphism of $\mathcal O_S$-modules, with $\mathcal L$ being finite and locally free. Let $F$ denote the functor from the category of $S$-schemes to that of sets, sending an $S$-scheme $T$ to the set $$F(T):= \{\phi \in \mathrm{Hom}(T,S) \,|\, \phi^*\alpha = 0\}$$ The functor $F$ is representable by a closed subscheme of $S$.
Here, $\phi^*\alpha$ denotes the morphism $\mathcal M_T \to \mathcal L_T$ induced by base change to $T$.
The proof is not that difficult. Because we have an isomorphism
$$\mathrm{Hom}(\mathcal M,\mathcal L) \simeq \mathrm{Hom}(\mathcal M \otimes \mathcal L ^{*},\mathcal O_S)$$
where $\mathcal L^{*} := \mathrm{Hom}(\mathcal L,\mathcal O_S)$, the morphism $\alpha$ corresponds to some $\tilde{\alpha}: \mathcal M \otimes \mathcal L ^{*} \to \mathcal O_S$. The image of $\tilde{\alpha}$ is a sheaf of ideals in $\mathcal O_S$, which defines the desired closed subscheme.
I am interested in the following similar moduli problem.
Let $R$ be a DVR with maximal ideal $\mathfrak m$. Let $S$ be any scheme over $R$, and denote by $\overline S$ its special fiber, that is the fiber over $\mathfrak m$. It is a closed subscheme of $S$, defined over the field $R/\mathfrak m$.
Let $\overline{\alpha}:\overline{\mathcal M} \to \overline{\mathcal L}$ be a morphism of $\mathcal O_{\overline S}$-modules, with $\overline{\mathcal L}$ being finite and locally free. Let $F$ denote the functor from the category of $S$-schemes to that of sets, sending an $S$-scheme $T$ to the set
$$F(T):= \{\phi \in \mathrm{Hom}(T,S) \,|\, \overline{\phi}^*\alpha = 0\}$$
Here, $\overline{\phi}$ is the induced morphism $\overline T \to \overline S$ on the special fibers (the structure morphism $T\to \mathrm{Spec}(R)$ being given by means of $\phi$).
Is $F$ representable by a closed subscheme of $S$ ?
I believe the answer to be affirmative, for a simple reason : a composition of closed immersions is again a closed immersion.
To be more precise, let $\overline F$ denote the same functor as in Rapoport and Zink's lemma, but over $\overline S$. It is represented by a closed subscheme of $\overline S$. Now, we may see it as a closed subscheme of $S$ as a whole, by composition of closed immersions. This should be our desired closed subscheme.
Is everything ok with my statement and justification ?