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Ext-Tor Vanishing and Completeness Criteria

  • Peter Schenzel
  • Anne-Marie Simon
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

Abstract

As a classical result, Jensen proved that, if \(\mathfrak a=\mathfrak m\) is the maximal ideal of a Noetherian local ring R, then a finitely generated R-module M is \(\mathfrak m\)-complete if and only if \(\text {Ext}_R^1(F, M)=0\) for all countably generated flat modules F if and only if \(\text {Ext}_R^i(F, M)=0\) for all \(i\ge 1\) and any flat module F (see Jensen in J. Algebra 15:151–166, 1970 [1] and Les foncteurs dérivés de lim et leurs applications en théorie des modules, Springer, Berlin, Heidelberg, New York [2] [Proposition 8.2]).

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für InformatikMartin-Luther-Universität Halle-WittenbergHalleGermany
  2. 2.Service de Geometrie DifferentielleUniversité Libre de BruxellesBrusselsBelgium

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