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A criterion for completeness


Author: Peter Schenzel
Journal: Proc. Amer. Math. Soc. 143 (2015), 2387-2394
MSC (2010): Primary 13J10; Secondary 13C11, 13D07
Published electronically: January 22, 2015
MathSciNet review: 3326021
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Abstract: Let $ (R,\mathfrak{m})$ denote a local ring with $ E = E_R(k)$ the injective hull of $ k = R/\mathfrak{m}$, its residue field. Let $ M$ denote a finitely generated $ R$-module. By Jensen's result it follows that $ \mathrm {Ext}^1_R(F,M) = 0$ for any flat $ R$-module $ F$ if and only if $ M$ is complete. Let $ \underline {x} = x_1,\ldots ,x_r$ be a system of elements of $ R$ such that $ \mathrm {Rad}\underline {x}R = \mathfrak{m}$. In the main result it is shown that the vanishing of $ \mathrm {Ext}_R^1(F,M), F = \bigoplus _{i = 1}^r R_{x_i},$ implies that $ M$ is complete. It is known from work of Enochs and Jenda that $ \mathrm {Hom}_R(E_R(R/\mathfrak{p}), E) \simeq \widehat {R_{\mathfrak{p}}^{\mu _{\mathfrak{p}}}}$ for a certain finite or infinite number $ \mu _{\mathfrak{p}}$. We discuss which $ \mu _{\mathfrak{p}}$ might occur for certain primes with $ \dim R/\mathfrak{p} = 1$.


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Additional Information

Peter Schenzel
Affiliation: Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, D — 06 099 Halle (Saale), Germany
Email: peter.schenzel@informatik.uni-halle.de

DOI: https://doi.org/10.1090/S0002-9939-2015-12470-7
Keywords: Completion, flat module, injective hull, localization
Received by editor(s): August 1, 2013
Received by editor(s) in revised form: January 25, 2014
Published electronically: January 22, 2015
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society