, Volume 16, Issue 6, pp 1809-1827
Date: 27 Oct 2012

Almost Split Sequences and Approximations

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Abstract

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.

Dedicated to the memory of Dieter Happel.
Presented by Claus Michael Ringel.