, Volume 16, Issue 6, pp 1809-1827

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.