FINITELY PRESENTABLE MORPHISMS IN EXACT SEQUENCES

Let K be a locally finitely presentable category. If K is abelian and the sequence 0 -> K (k)-> X (c)-> C -> 0 is short exact, we show that 1) K is finitely generated double left right arrow c is finitely presentable; 2) k is finitely presentable double left right arrow C is finitely presentable. The "double left right arrow" directions fail for semi-abelian varieties. We show that all but (possibly) 2)(double left arrow) follow from analogous properties which hold in all locally finitely presentable categories. As for 2)(double left arrow), it holds as soon as 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.