11 research outputs found
Ascent of module structures, vanishing of Ext, and extended modules
Let (R,\m) and (S,\n) be commutative Noetherian local rings, and let
be a flat local homomorphism such that \m S = \n and the
induced map on residue fields R/\m \to S/\n is an isomorphism. Given a
finitely generated -module , we show that has an -module structure
compatible with the given -module structure if and only if \Ext^i_R(S,M)=0
for each .
We say that an -module is {\it extended} if there is a finitely
generated -module such that . Given a short exact
sequence of finitely generated -modules, with
two of the three modules extended, we obtain conditions forcing the
third module to be extended. We show that every finitely generated module over
the Henselization of is a direct summand of an extended module, but that
the analogous result fails for the \m-adic completion.Comment: 16 pages, AMS-TeX; final version to appear in Michigan Math. J.;
corrected proof of Main Theorem and made minor editorial changes; v3 has
dedication to Mel Hochste