Finitely Presentable Morphisms in Exact Sequences

Let K be a locally finitely presentable category. If K is abelian and the sequence 0 K // X // k // C c // // 0 // is short exact, we show that 1) K is finitely generated⇔ c is finitely presentable; 2) k is finitely presentable⇔ C is finitely presentable. The “⇐” directions fail for semi-abelian varieties. We show that all but (possibly) 2)(⇐) follow from analogous properties which hold in all locally finitely presentable categories. As for 2)(⇐), 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. 1. Finitely presentable morphisms Recall (from [GU, 71] or [AR, 94]) that an object X in a category K is finitely presentable (finitely generated) if the hom-functor K(X,−) : X −→ Set preserves filtered colimits (resp. colimits of filtered diagrams made of monomorphisms). Then, K is finitely accessible if it has a (small) set of finitely presentable objects whose closure under filtered colimits is all of K. Finally, K is locally finitely presentable if it is finitely accessible and cocomplete. 1.1. Definition. Let f : X // Y be a morphism in K. (a) f is finitely presentable (resp. finitely generated) if it is a finitely presentable (resp. finitely generated) object of the slice category (X ↓ K). (b) f is finitary if X and Y are finitely presentable. The finitary morphisms of K are actually the finitely presentable objects of the category of morphisms K→. They are precisely those finitely presentable morphisms with a finitely presentable domain (see below). Finitely presentable morphisms have been first considered in algebraic geometry, where they play an important role (for example in the Chevalley Theorem; see [GD, 64] and [D, 92]). In fact, in the category CRng of commutative rings, the definition above Received by the editors 2010-11-13 and, in revised form, 2010-05-03. Transmitted by J. Rosicky. Published on 2010-05-06. 2000 Mathematics Subject Classification: 18A20, 18E10, 18C35, 18E15.