# Finitely presentable morphisms in exact sequences

## Michel Hébert

Let $\cal K$ be a locally finitely presentable category. If $\cal K$ is abelian and the sequence
$$0 \to K \to^k X \to^c C \to 0$$
is short exact, we show that 1) $K$ is finitely generated iff $c$ is finitely presentable; 2) $k$ is finitely presentable iff $C$ is finitely presentable. The if" directions fail for semi-abelian varieties. We show that all but (possibly) 2)(if) follow from analogous properties which hold in all locally finitely presentable categories. As for 2)(if), it holds as soon as $\cal K$ is also co-homological, and all its strong epimorphisms are regular. Finally, locally finitely coherent (resp. noetherian) abelian categories are characterized as those for which all finitely presentable morphisms have finitely generated (resp. presentable) kernel objects.

Keywords: finitely presentable morphism, abelian category, Grothendieck category

2000 MSC: 18A20, 18E10, 18C35, 18E15

Theory and Applications of Categories, Vol. 24, 2010, No. 9, pp 209-220.

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