Suppose (R, m) and (S, n) are commutative Noetherian local rings and ϕ: R→ S is a flat local homomorphism with the property that the induced homomorphism R/m→ S/mS is bijective. We consider natural questions of ascent and descent of modules between R and S:(i) Given a finitely generated R-module M, when does M have an S-module structure that is compatible with the R-module structure via ϕ?(ii) Given a finitely generated S-module N, is there a finitely generated R-module M such that N is S-isomorphic to S⊗ R M or (iii) S-isomorphic to a direct summand of S⊗ R M?

In Section 1 we make some general observations about homomorphisms R→ S satisfying the condition R/m= S/mS. We show that if a compatible S-module structure exists, then it arises in an obvious way: The natural map M→ S⊗ R M is an isomorphism.(One example to keep in mind is that of a finite-length module M when S= ˆR, the m-adic completion.) Moreover, if R→ S is flat, then M has a compatible S-module structure if and only if S⊗ R M is finitely generated as an R-module.