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Algebra pp 300-321 | Cite as

Abelian Categories

  • Carl Faith
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 190)

Abstract

An exact additive category is said to be abelian (6.1). If C is an abelian category, then so is C L , for any (small) category L. Furthermore (Proposition 6.17), C L inherits the following properties from C: (4) (finitely) complete; (2) (finite) products; (3) kernels or equalizers; (4) images, sums, or (finite) intersections; (5) normality with epic images; (6) additive; (7) exact; (8) abelian; (9) exact and locally small (assuming L is small). Moreover (Proposition 6.25), C L has exact direct limits when C has (cf. Exercise 6.13). When this is true, then (Corollary 6.24) for any family {F (X)} XM of (left) exact functors LC, the direct limit \({\underrightarrow {\lim }_{X \in M}}F(X)\) is a (left) exact functor LC. This implies that a direct limit of a directed family of flat modules in mod-R is flat, for any ring R. A colimit of colimit preserving functors is always colimit preserving (6.22).

Keywords

Natural Transformation Full Subcategory Additive Category Abelian Category Finite Product 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag, Berlin · Heidelberg 1973

Authors and Affiliations

  • Carl Faith
    • 1
  1. 1.Rutgers UniversityNew BrunswickUSA

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