Extended Modules

Suppose R and S are local rings and (R,m) → (S,n) is a flat local homomorphism. Given a finitely generated S-module N , we say N is extended (from R) provided there is an R-module M such that S ⊗R M is isomorphic to N as an S-module. If such a module M exists, it is unique up to isomorphism (cf. [EGA, (2.5.8)], and it is necessarily finitely generated. The m-adic completion R → R̂ and the Henselization R → R are particularly important examples. One reason is that the Krull-Remak-Schmidt uniqueness theorem holds for directsum decompositions of finitely generated modules over a Henselian local ring. Indeed, failure of uniqueness for general local rings stems directly from the fact that some modules over the Henselization (or completion) are not extended. Understanding which R-modules are extended is the key to unraveling the direct-sum behavior of R-modules. Throughout, we assume that (R,m) and (S,n) are Noetherian local rings and that R → S is a flat local homomorphism. Many of our results generalize easily to a mildly noncommutative setting. Moreover, it is not always necessary to assume that our rings are local. Thus, we assume that A is a commutative ring, that B is a faithfully flat commutative A-algebra, and that Λ is a module-finite A-algebra. Given a finitely generated left B ⊗A Λ-module N , we say that N is extended (from Λ) provided there is a finitely generated left Λ-module M such that B ⊗A M is isomorphic to N as a B ⊗A Λ-module. In Sections 1 and 2 of the paper, we examine how the extended modules sit inside the family of all finitely generated modules. In Sections 3 – 4 we consider rings of dimension 2 and 1, respectively, find criteria for a module to be extended , and show how the extendedness problem for one-dimensional rings reduces to the Artinian case. In Section 5, we find situations where every finitely generated B-module is a direct summand of an extended module, and in Section 6 we make a few observations about the Artinian case.