Liftings of point-wise finite dimensional persistence modules over local commutative Artinian rings

Let k be a field and let V: C → k-Mod be a point-wise finite dimensional persistence modules, where C is a small category. Assume that for all local Artinian k-algebras R with residue field isomorphic to k, there is a generalized persistence module M: C → R-Mod, such that for all x∈ Ob(C), $M(x)$ is free over R with finite rank and k⊗R M(x) V(x). If V is a direct sum of indecomposable persistence modules VI: C→ k-Mod with endomorphism ring isomorphic to k, then M is a direct sum of indecomposables MI:C→ R-Mod with endomorphism ring isomorphic to R