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Algebra

Extended modules

Profile image of Roger WiegandRoger Wiegand

2009, Journal of Commutative Algebra

https://doi.org/10.1216/JCA-2009-1-3-481
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17 pages

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Abstract

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 → R and the Henselization R → R h 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 h-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. 1. Two out of three: direct sums Our goal in this section is to prove the following theorem, which generalizes Proposition 3.1 of [FSW]: The research of W. Hassler was supported by the Fonds zur Förderung der Wissenschaftlichen Forschung, project number P20120-N18. Wiegand's research was partially supported by NSA Grant H98230-05-1-0243.

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