Ascent of Module Structures, Vanishing of Ext, and Extended Modules

Let (R;m) and (S;n) be commutative Noetherian local rings, and let ' : R ! S be a ∞at local homomorphism such that mS = n and the induced map on residue flelds R=m ! S=n is an isomorphism. Given a flnitely 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 i (S;M) is flnitely generated as an R-module for each i ‚ 1. We say that an S-module N is extended if there is a flnitely generated R-module M such that N » S ›R M. Given a short exact sequence 0 ! N1 ! N ! N2 ! 0 of flnitely 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 flnitely 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.

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