Algebras and Representation Theory

, Volume 16, Issue 6, pp 1809–1827 | Cite as

Almost Split Sequences and Approximations

  • Shiping LiuEmail author
  • Puiman Ng
  • Charles Paquette


Let \(\mathcal A\) be an exact category, that is, an extension-closed full subcategory of an abelian category. First, we give new characterizations of an almost split sequence in \(\mathcal{A}\), which yields some necessary and sufficient conditions for \(\mathcal A\) to have almost split sequences. Then, we study when an almost split sequence in \(\mathcal A\) induces an almost split sequence in an exact subcategory \(\mathcal C\) of \(\mathcal A\). In case \(\mathcal A\) has almost split sequences and \(\mathcal C\) is Ext-finite and Krull–Schmidt, we obtain a necessary and sufficient condition for \(\mathcal C\) to have almost split sequences. Finally, we show some applications of these results.


Krull–Schmidt categories Exact categories Almost split sequences Approximations Functorially finite subcategories Non-degenerate bilinear forms 

Mathematics Subject Classifications (2010)

16G20 16G70 18E10 18E40 


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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité de SherbrookeSherbrookeCanada
  2. 2.Department of Mathematics and StatisticsUniversity of New BrunswickFrederictonCanada

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