Quotient of an abelian category

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In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category by a Serre subcategory is the abelian category which, intuitively, is obtained from by ignoring (i.e. treating as zero) all objects from . There is a canonical exact functor whose kernel is .

Contents

Definition

Formally, is the category whose objects are those of and whose morphisms from X to Y are given by the direct limit (of abelian groups) over subobjects and such that and . (Here, and denote quotient objects computed in .) Composition of morphisms in is induced by the universal property of the direct limit.

The canonical functor sends an object X to itself and a morphism to the corresponding element of the direct limit with X′ = X and Y′ = 0.

Examples

Let be a field and consider the abelian category of all vector spaces over . Then the full subcategory of finite-dimensional vector spaces is a Serre-subcategory of . The quotient has as objects the -vector spaces, and the set of morphisms from to in is

(which is a quotient of vector spaces). This has the effect of identifying all finite-dimensional vector spaces with 0, and of identifying two linear maps whenever their difference has finite-dimensional image.

Properties

The quotient is an abelian category, and the canonical functor is exact. The kernel of is , i.e., is a zero object of if and only if belongs to .

The quotient and canonical functor are characterized by the following universal property: if is any abelian category and is an exact functor such that is a zero object of for each object , then there is a unique exact functor such that . [1]

Gabriel–Popescu

The Gabriel–Popescu theorem states that any Grothendieck category is equivalent to a quotient category , where denotes the abelian category of right modules over some unital ring , and is some localizing subcategory of . [2]

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References

  1. Gabriel, Pierre, Des categories abeliennes , Bull. Soc. Math. France 90 (1962), 323-448.
  2. N. Popesco, P. Gabriel (1964). "Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes". Comptes Rendus de l'Académie des Sciences. 258: 4188–4190.CS1 maint: uses authors parameter (link)