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 , and is in a certain sense the most general abelian category with this property.

Forming Serre quotients of abelian categories is thus formally akin to forming quotients of groups. Serre quotients are somewhat similar to quotient categories, the difference being that with Serre quotients all involved categories are abelian and all functors are exact. Serre quotients also often have the character of localizations of categories, especially if the Serre subcategory is localizing.

Definition

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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)

 

where the limit is taken over subobjects   and   such that   and  . (Here,   and   denote quotient objects computed in  .) These pairs of subobjects are ordered by  .

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.

An alternative, equivalent construction of the quotient category uses what is called a "calculus of fractions" to define the morphisms of  . Here, one starts with the class   of those morphisms in   whose kernel and cokernel both belong to  . This is a multiplicative system in the sense of Gabriel-Zisman, and one can localize the category   at the system   to obtain  .[1]

Examples

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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 Serre 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. This example shows that the Serre quotient can behave like a quotient category.

For another example, take the abelian category Ab of all abelian groups and the Serre subcategory of all torsion abelian groups. The Serre quotient here is equivalent to the category   of all vector spaces over the rationals, with the canonical functor   given by tensoring with  . Similarly, the Serre quotient of the category of finitely generated abelian groups by the subcategory of finitely generated torsion groups is equivalent to the category of finite-dimensional vectorspaces over  .[2] Here, the Serre quotient behaves like a localization.

Properties

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The Serre quotient   is an abelian category, and the canonical functor   is exact and surjective on objects. The kernel of   is  , i.e.,   is zero in   if and only if   belongs to  .

The Serre 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 in   for each object  , then there is a unique exact functor   such that  .[3]

Given three abelian categories  ,  ,  , we have

 

if and only if

there exists an exact and essentially surjective functor   whose kernel is   and such that for every morphism   in   there exist morphisms   and   in   so that   is an isomorphism and  .

Theorems involving Serre quotients

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Serre's description of coherent sheaves on a projective scheme

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According to a theorem by Jean-Pierre Serre, the category   of coherent sheaves on a projective scheme   (where   is a commutative noetherian graded ring, graded by the non-negative integers and generated by degree-0 and finitely many degree-1 elements, and   refers to the Proj construction) can be described as the Serre quotient

 

where   denotes the category of finitely-generated graded modules over   and   is the Serre subcategory consisting of all those graded modules   which are 0 in all degrees that are high enough, i.e. for which there exists   such that   for all  .[4][5]

A similar description exists for the category of quasi-coherent sheaves on  , even if   is not noetherian.

Gabriel–Popescu theorem

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The Gabriel–Popescu theorem states that any Grothendieck category   is equivalent to a Serre quotient of the form  , where   denotes the abelian category of right modules over some unital ring  , and   is some localizing subcategory of  .[6]

Quillen's localization theorem

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Daniel Quillen's algebraic K-theory assigns to each exact category   a sequence of abelian groups  , and this assignment is functorial in  . Quillen proved that, if   is a Serre subcategory of the abelian category  , there is a long exact sequence of the form[7]

 

References

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  1. ^ Section 12.10 The Stacks Project
  2. ^ "109.76 The category of modules modulo torsion modules". The Stacks Project.
  3. ^ Gabriel, Pierre, Des categories abeliennes, Bull. Soc. Math. France 90 (1962), 323-448.
  4. ^ Görtz, Ulrich; Wedhorn, Torsten (2020). "Remark 13.21". Algebraic Geometry I: Schemes: With Examples and Exercises (2nd ed.). Springer Nature. p. 381. ISBN 9783658307332.
  5. ^ "Proposition 30.14.4". The Stacks Project.
  6. ^ 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.
  7. ^ Quillen, Daniel (1973). "Higher algebraic K-theory: I" (PDF). Higher K-Theories. Lecture Notes in Mathematics. 341. Springer: 85–147. doi:10.1007/BFb0067053. ISBN 978-3-540-06434-3.