Of Module Structures , Vanishing of Ext , and Extended Modules

Let (R, m) and (S, n) be commutative Noetherian local rings, and let φ : R→ S be a flat local homomorphism such that mS = n and the induced map on residue fields R/m→ S/n is an isomorphism. Given a finitely generated R-module M , we show that M has an S-module structure compatible with the given R-module structure if and only if Ext R (S,M) = 0 for each i ≥ 1. We say that an S-module N is extended if there is a finitely generated R-module M such that N ∼= S⊗R M . Given a short exact sequence 0→ N1 → N → N2 → 0 of finitely generated S-modules, with two of the three modules N1, N, N2 extended, we obtain conditions forcing the third module to be extended. We show that every finitely generated module over the Henselization of R is a direct summand of an extended module, but that the analogous result fails for the m-adic completion.