In mathematics, the Gabriel–Popescu theorem is an embedding theorem for certain

abelian categories 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 ab ...

, introduced by . It characterizes certain abelian categories (the Grothendieck categories) as quotients of

module categories In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring ...

. There are several generalizations and variations of the Gabriel–Popescu theorem, given by (for an

AB5 category In mathematics, in his " Tôhoku paper" introduced a sequence of axioms of various kinds of categories enriched over the symmetric monoidal category of abelian groups. Abelian categories are sometimes called AB2 categories, according to the axiom ...

with a set of generators), , (for

triangulated categories 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 cate ...

Theorem

Let ''A'' be a

Grothendieck category In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957English translation in order to develop the machinery of homological algebra for modules and for sheaves i ...

(an

with a generator), ''G'' a generator of ''A'' and ''R'' be the

ring of endomorphisms In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...

of ''G''; also, let ''S'' be the

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, an ...

from ''A'' to Mod-''R'' (the category of right ''R''-modules) defined by ''S''(''X'') = Hom(''G'',''X''). Then the Gabriel–Popescu theorem states that ''S'' is full and faithful and has an exact

left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are k ...

. This implies that ''A'' is

equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *''Equivale ...

to the

Serre quotient category In mathematics, the quotient (also called Serre quotient or Gabriel quotient) of an abelian category \mathcal by a Serre subcategory \mathcal is the abelian category \mathcal/\mathcal which, intuitively, is obtained from \mathcal by ignoring (i.e. ...

of Mod-''R'' by a certain

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

''C''. (A localizing subcategory of Mod-''R'' is a full subcategory ''C'' of Mod-''R'', closed under arbitrary

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...

s, such that for any

short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the conte ...

of modules

0\rarr M_1\rarr M_2\rarr M_3\rarr 0

, we have ''M₂'' in ''C'' if and only if ''M₁'' and ''M₃'' are in ''C''. The Serre quotient of Mod-''R'' by any localizing subcategory is a Grothendieck category.) We may take ''C'' to be the

kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine lea ...

of the left adjoint of the functor ''S''. Note that the embedding ''S'' of ''A'' into Mod-''R'' is left-exact but not necessarily right-exact: cokernels of morphisms in ''A'' do not in general correspond to the cokernels of the corresponding morphisms in Mod-''R.''

References

* * emark: "Popescu" is spelled "Popesco" in French.* * * *

External links

* {{DEFAULTSORT:Gabriel-Popescu theorem Category theory Functors Theorems in abstract algebra