# Quotient of an abelian category

Last updated

In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category ${\mathcal {A}}$ by a Serre subcategory ${\mathcal {B}}$ is the abelian category ${\mathcal {A}}/{\mathcal {B}}$ which, intuitively, is obtained from ${\mathcal {A}}$ by ignoring (i.e. treating as zero) all objects from ${\mathcal {B}}$ . There is a canonical exact functor $Q\colon {\mathcal {A}}\to {\mathcal {A}}/{\mathcal {B}}$ whose kernel is ${\mathcal {B}}$ .

## Definition

Formally, ${\mathcal {A}}/{\mathcal {B}}$ is the category whose objects are those of ${\mathcal {A}}$ and whose morphisms from X to Y are given by the direct limit (of abelian groups) $\varinjlim \mathrm {Hom} _{\mathcal {A}}(X',Y/Y')$ over subobjects $X'\subseteq X$ and $Y'\subseteq Y$ such that $X/X'\in {\cal {B}}$ and $Y'\in {\cal {B}}$ . (Here, $X/X'$ and $Y/Y'$ denote quotient objects computed in ${\mathcal {A}}$ .) Composition of morphisms in ${\mathcal {A}}/{\mathcal {B}}$ is induced by the universal property of the direct limit.

The canonical functor $Q\colon {\mathcal {A}}\to {\mathcal {A}}/{\mathcal {B}}$ sends an object X to itself and a morphism $f\colon X\to Y$ to the corresponding element of the direct limit with X′ = X and Y′ = 0.

## Examples

Let $k$ be a field and consider the abelian category ${\rm {Mod}}(k)$ of all vector spaces over $k$ . Then the full subcategory ${\rm {mod}}(k)$ of finite-dimensional vector spaces is a Serre-subcategory of ${\rm {Mod}}(k)$ . The quotient ${\cal {{C}={\rm {Mod}}(k)/{\rm {mod}}(k)}}$ has as objects the $k$ -vector spaces, and the set of morphisms from $X$ to $Y$ in ${\cal {C}}$ is

$\{k{\text{-linear maps from }}X{\text{ to }}Y\}/\{k{\text{-linear maps from }}X{\text{ to }}Y{\text{ with finite-dimensional image}}\}$ (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 ${\mathcal {A}}/{\mathcal {B}}$ is an abelian category, and the canonical functor $Q\colon {\mathcal {A}}\to {\mathcal {A}}/{\mathcal {B}}$ is exact. The kernel of $Q$ is ${\mathcal {B}}$ , i.e., $Q(X)$ is a zero object of ${\mathcal {A}}/{\mathcal {B}}$ if and only if $X$ belongs to ${\mathcal {B}}$ .

The quotient and canonical functor are characterized by the following universal property: if ${\mathcal {C}}$ is any abelian category and $F\colon {\mathcal {A}}\to {\mathcal {C}}$ is an exact functor such that $F(X)$ is a zero object of ${\mathcal {C}}$ for each object $X\in {\mathcal {B}}$ , then there is a unique exact functor ${\overline {F}}\colon {\mathcal {A}}/{\mathcal {B}}\to {\mathcal {C}}$ such that $F={\overline {F}}\circ Q$ . 

## Gabriel–Popescu

The Gabriel–Popescu theorem states that any Grothendieck category ${\mathcal {A}}$ is equivalent to a quotient category $\operatorname {Mod} (R)/{\cal {B}}$ , where $\operatorname {Mod} (R)$ denotes the abelian category of right modules over some unital ring $R$ , and ${\cal {B}}$ is some localizing subcategory of $\operatorname {Mod} (R)$ . 

## Related Research Articles

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Abelian categories are named after Niels Henrik Abel.

In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels.

In mathematics, a direct limit is a way to construct a object from many objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by .

In mathematics, specifically category theory, a subcategory of a category C is a category S whose objects are objects in C and whose morphisms are morphisms in C with the same identities and composition of morphisms. Intuitively, a subcategory of C is a category obtained from C by "removing" some of its objects and arrows.

In algebraic geometry, motives is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety.

In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.

In mathematics, the derived categoryD(A) of an abelian category A is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on A. The construction proceeds on the basis that the objects of D(A) should be chain complexes in A, with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described by complicated spectral sequences.

In mathematics, especially in category theory, a closed monoidal category is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.

In mathematics, the Grothendieck group construction constructs an abelian group from a commutative monoid M in the most universal way, in the sense that any abelian group containing a homomorphic image of M will also contain a homomorphic image of the Grothendieck group of M. The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.

In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object.

In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of categories, analogous to a quotient group or quotient space, but in the categorical setting.

In algebraic geometry, a Fourier–Mukai transformΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K ∈ D(X×Y). Most natural functors, including basic ones like pushforwards and pullbacks, are of this type.

In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 in order to develop the machinery of homological algebra for modules and for sheaves in a unified manner. The theory of these categories was further developed in Pierre Gabriel's seminal thesis in 1962.

In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra.

This is a glossary of algebraic geometry.

In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category.

In mathematics, a semiorthogonal decomposition is a way to divide a triangulated category into simpler pieces. One way to produce a semiorthogonal decomposition is from an exceptional collection, a special sequence of objects in a triangulated category. For an algebraic variety X, it has been fruitful to study semiorthogonal decompositions of the bounded derived category of coherent sheaves, .

In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety, , called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted . For instance, the derived category of coherent sheaves on a smooth projective variety can be used as an invariant of the underlying variety for many cases. Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.

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)