Grothendieck category

In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957[1] 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 1962 thesis.[2]

To every algebraic variety V {\displaystyle V} one can associate a Grothendieck category  Qcoh ⁡ ( V ) {\displaystyle \operatorname {Qcoh} (V)} , consisting of the quasi-coherent sheaves on  V {\displaystyle V} . This category encodes all the relevant geometric information about  V {\displaystyle V} , and  V {\displaystyle V} can be recovered from  Qcoh ⁡ ( V ) {\displaystyle \operatorname {Qcoh} (V)} (the Gabriel–Rosenberg reconstruction theorem). This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of (certain) Grothendieck categories.[3]

## Definition

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By definition, a Grothendieck category  A {\displaystyle {\mathcal {A}}} is an AB5 category with a generator. Spelled out, this means that

* A {\displaystyle {\mathcal {A}}} is an abelian category;
* every (possibly infinite) family of objects in  A {\displaystyle {\mathcal {A}}} has a coproduct (also known as direct sum) in  A {\displaystyle {\mathcal {A}}} ;
* direct limits of short exact sequences are exact; this means that if a direct system of short exact sequences in  A {\displaystyle {\mathcal {A}}} is given, then the induced sequence of direct limits is a short exact sequence as well. (Direct limits are always right-exact; the important point here is that we require them to be left-exact as well.)
* A {\displaystyle {\mathcal {A}}} possesses a generator, i.e. there is an object  G {\displaystyle G} in  A {\displaystyle {\mathcal {A}}} such that  Hom ⁡ ( G , − ) {\displaystyle \operatorname {Hom} (G,-)} is a faithful functor from  A {\displaystyle {\mathcal {A}}} to the category of sets. (In our situation, this is equivalent to saying that every object  X {\displaystyle X} of  A {\displaystyle {\mathcal {A}}} admits an epimorphism G ( I ) → X {\displaystyle G^{(I)}\rightarrow X} , where  G ( I ) {\displaystyle G^{(I)}} denotes a direct sum of copies of  G {\displaystyle G} , one for each element of the (possibly infinite) set  I {\displaystyle I} .)

The name "Grothendieck category" appeared neither in Grothendieck's Tôhoku paper[1] nor in Gabriel's thesis;[2] it came into use in the second half of the 1960s in the work of several authors, including Jan-Erik Roos, Bo Stenström, Ulrich Oberst, and Bodo Pareigis. (Some authors use a different definition, not requiring the existence of a generator.)

## Examples

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* The prototypical example of a Grothendieck category is the category of abelian groups; the abelian group  Z {\displaystyle \mathbb {Z} } of integers is a generator.
* More generally, given any ring R {\displaystyle R} (associative, with  1 {\displaystyle 1} , but not necessarily commutative), the category  Mod ⁡ ( R ) {\displaystyle \operatorname {Mod} (R)} of all right (or alternatively: left) modules over  R {\displaystyle R} is a Grothendieck category;  R {\displaystyle R} itself is a generator.
* Given a topological space X {\displaystyle X} , the category of all sheaves of abelian groups on  X {\displaystyle X} is a Grothendieck category.[1] (More generally: the category of all sheaves of right  R {\displaystyle R} -modules on  X {\displaystyle X} is a Grothendieck category for any ring  R {\displaystyle R} .)
* Given a ringed space ( X , O X ) {\displaystyle (X,{\mathcal {O}}_{X})} , the category of sheaves of OX-modules is a Grothendieck category.[1]
* Given an (affine or projective) algebraic variety V {\displaystyle V} (or more generally: any scheme or algebraic stack), the category  Qcoh ⁡ ( V ) {\displaystyle \operatorname {Qcoh} (V)} of quasi-coherent sheaves on  V {\displaystyle V} is a Grothendieck category.[4]
* Given a small site (C, J) (i.e. a small category C together with a Grothendieck topology J), the category of all sheaves of abelian groups on the site is a Grothendieck category.

### Constructing further Grothendieck categories

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* Any category that's equivalent to a Grothendieck category is itself a Grothendieck category.
* Given Grothendieck categories  A 1 , … , A n {\displaystyle {\mathcal {A_{1}}},\ldots ,{\mathcal {A_{n}}}} , the product category A 1 × … × A n {\displaystyle {\mathcal {A_{1}}}\times \ldots \times {\mathcal {A_{n}}}} is a Grothendieck category.
* Given a small category C {\displaystyle {\mathcal {C}}} and a Grothendieck category  A {\displaystyle {\mathcal {A}}} , the functor category Funct ⁡ ( C , A ) {\displaystyle \operatorname {Funct} ({\mathcal {C}},{\mathcal {A}})} , consisting of all covariant functors from  C {\displaystyle {\mathcal {C}}} to  A {\displaystyle {\mathcal {A}}} , is a Grothendieck category.[1]
* Given a small preadditive category  C {\displaystyle {\mathcal {C}}} and a Grothendieck category  A {\displaystyle {\mathcal {A}}} , the functor category  Add ⁡ ( C , A ) {\displaystyle \operatorname {Add} ({\mathcal {C}},{\mathcal {A}})} of all additive covariant functors from  C {\displaystyle {\mathcal {C}}} to  A {\displaystyle {\mathcal {A}}} is a Grothendieck category.[5]
* If  A {\displaystyle {\mathcal {A}}} is a Grothendieck category and  C {\displaystyle {\mathcal {C}}} is a localizing subcategory of  A {\displaystyle {\mathcal {A}}} , then both  C {\displaystyle {\mathcal {C}}} and the Serre quotient category A / C {\displaystyle {\mathcal {A}}/{\mathcal {C}}} are Grothendieck categories.[2]

## Properties and theorems

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Every Grothendieck category contains an injective cogenerator. For example, an injective cogenerator of the category of abelian groups is the quotient group Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } .

Every object in a Grothendieck category  A {\displaystyle {\mathcal {A}}} has an injective hull in  A {\displaystyle {\mathcal {A}}} .[1][2] This allows to construct injective resolutions and thereby the use of the tools of homological algebra in  A {\displaystyle {\mathcal {A}}} , in order to define derived functors. (Note that not all Grothendieck categories allow projective resolutions for all objects; examples are categories of sheaves of abelian groups on many topological spaces, such as on the space of real numbers.)

In a Grothendieck category, any family of subobjects ( U i ) {\displaystyle (U_{i})} of a given object  X {\displaystyle X} has a supremum (or "sum")  ∑ i U i {\textstyle \sum _{i}U_{i}} as well as an infimum (or "intersection")  ∩ i U i {\displaystyle \cap _{i}U_{i}} , both of which are again subobjects of  X {\displaystyle X} . Further, if the family  ( U i ) {\displaystyle (U_{i})} is directed (i.e. for any two objects in the family, there is a third object in the family that contains the two), and  V {\displaystyle V} is another subobject of  X {\displaystyle X} , we have[6]

∑ i ( U i ∩ V ) = ( ∑ i U i ) ∩ V . {\displaystyle \sum _{i}(U_{i}\cap V)=\left(\sum _{i}U_{i}\right)\cap V.}

Grothendieck categories are well-powered (sometimes called locally small, although that term is also used for a different concept), i.e. the collection of subobjects of any given object forms a set (rather than a proper class).[5]

It is a rather deep result that every Grothendieck category  A {\displaystyle {\mathcal {A}}} is complete,[7] i.e. that arbitrary limits (and in particular products) exist in  A {\displaystyle {\mathcal {A}}} . By contrast, it follows directly from the definition that  A {\displaystyle {\mathcal {A}}} is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in  A {\displaystyle {\mathcal {A}}} . Coproducts in a Grothendieck category are exact (i.e. the coproduct of a family of short exact sequences is again a short exact sequence), but products need not be exact.

A functor  F : A → X {\displaystyle F\colon {\cal {A}}\to {\cal {X}}} from a Grothendieck category  A {\displaystyle {\mathcal {A}}} to an arbitrary category  X {\displaystyle {\mathcal {X}}} has a left adjoint if and only if it commutes with all limits, and it has a right adjoint if and only if it commutes with all colimits. This follows from Peter J. Freyd's special adjoint functor theorem and its dual.[8]

The Gabriel–Popescu theorem states that any Grothendieck category  A {\displaystyle {\mathcal {A}}} is equivalent to a full subcategory of the category  Mod ⁡ ( R ) {\displaystyle \operatorname {Mod} (R)} of right modules over some unital ring  R {\displaystyle R} (which can be taken to be the endomorphism ring of a generator of  A {\displaystyle {\mathcal {A}}} ), and  A {\displaystyle {\mathcal {A}}} can be obtained as a Gabriel quotient of  Mod ⁡ ( R ) {\displaystyle \operatorname {Mod} (R)} by some localizing subcategory.[9]

As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable.[10] Furthermore, Gabriel-Popescu can be used to see that every Grothendieck category is complete, being a reflective subcategory of the complete category  Mod ⁡ ( R ) {\displaystyle \operatorname {Mod} (R)} for some  R {\displaystyle R} .

Every small abelian category  C {\displaystyle {\mathcal {C}}} can be embedded in a Grothendieck category, in the following fashion. The category  A := Lex ⁡ ( C o p , A b ) {\displaystyle {\mathcal {A}}:=\operatorname {Lex} ({\mathcal {C}}^{op},\mathrm {Ab} )} of left-exact additive (covariant) functors  C o p → A b {\displaystyle {\mathcal {C}}^{op}\rightarrow \mathrm {Ab} } (where  A b {\displaystyle \mathrm {Ab} } denotes the category of abelian groups) is a Grothendieck category, and the functor  h : C → A {\displaystyle h\colon {\mathcal {C}}\rightarrow {\mathcal {A}}} , with  C ↦ h C = Hom ⁡ ( − , C ) {\displaystyle C\mapsto h_{C}=\operatorname {Hom} (-,C)} , is full, faithful and exact. A generator of  A {\displaystyle {\mathcal {A}}} is given by the coproduct of all  h C {\displaystyle h_{C}} , with  C ∈ C {\displaystyle C\in {\mathcal {C}}} .[2] The category  A {\displaystyle {\mathcal {A}}} is equivalent to the category  Ind ( C ) {\displaystyle {\text{Ind}}({\mathcal {C}})} of ind-objects of  C {\displaystyle {\mathcal {C}}} and the embedding  h {\displaystyle h} corresponds to the natural embedding  C → Ind ( C ) {\displaystyle {\mathcal {C}}\to {\text{Ind}}({\mathcal {C}})} . We may therefore view  A {\displaystyle {\mathcal {A}}} as the co-completion of  C {\displaystyle {\mathcal {C}}} .

### Special kinds of objects and Grothendieck categories

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An object  X {\displaystyle X} in a Grothendieck category is called finitely generated if, whenever  X {\displaystyle X} is written as the sum of a family of subobjects of  X {\displaystyle X} , then it is already the sum of a finite subfamily. (In the case  A = Mod ⁡ ( R ) {\displaystyle {\cal {A}}=\operatorname {Mod} (R)} of module categories, this notion is equivalent to the familiar notion of finitely generated modules.) Epimorphic images of finitely generated objects are again finitely generated. If  U ⊆ X {\displaystyle U\subseteq X} and both  U {\displaystyle U} and  X / U {\displaystyle X/U} are finitely generated, then so is  X {\displaystyle X} . The object  X {\displaystyle X} is finitely generated if, and only if, for any directed system  ( A i ) {\displaystyle (A_{i})} in  A {\displaystyle {\cal {A}}} in which each morphism is a monomorphism, the natural morphism  lim → ⁡ H o m ( X , A i ) → H o m ( X , lim → ⁡ A i ) {\displaystyle \varinjlim \mathrm {Hom} (X,A_{i})\to \mathrm {Hom} (X,\varinjlim A_{i})} is an isomorphism.[11] A Grothendieck category need not contain any non-zero finitely generated objects.

A Grothendieck category is called locally finitely generated if it has a set of finitely generated generators (i.e. if there exists a family  ( G i ) i ∈ I {\displaystyle (G_{i})_{i\in I}} of finitely generated objects such that to every object  X {\displaystyle X} there exist  i ∈ I {\displaystyle i\in I} and a non-zero morphism  G i → X {\displaystyle G_{i}\rightarrow X} ; equivalently:  X {\displaystyle X} is epimorphic image of a direct sum of copies of the  G i {\displaystyle G_{i}} ). In such a category, every object is the sum of its finitely generated subobjects.[5] Every category  A = Mod ⁡ ( R ) {\displaystyle {\cal {A}}=\operatorname {Mod} (R)} is locally finitely generated.

An object  X {\displaystyle X} in a Grothendieck category is called finitely presented if it is finitely generated and if every epimorphism  W → X {\displaystyle W\to X} with finitely generated domain  W {\displaystyle W} has a finitely generated kernel. Again, this generalizes the notion of finitely presented modules. If  U ⊆ X {\displaystyle U\subseteq X} and both  U {\displaystyle U} and  X / U {\displaystyle X/U} are finitely presented, then so is  X {\displaystyle X} . In a locally finitely generated Grothendieck category  A {\displaystyle {\cal {A}}} , the finitely presented objects can be characterized as follows:[12] X {\displaystyle X} in  A {\displaystyle {\cal {A}}} is finitely presented if, and only if, for every directed system  ( A i ) {\displaystyle (A_{i})} in  A {\displaystyle {\cal {A}}} , the natural morphism  lim → ⁡ H o m ( X , A i ) → H o m ( X , lim → ⁡ A i ) {\displaystyle \varinjlim \mathrm {Hom} (X,A_{i})\to \mathrm {Hom} (X,\varinjlim A_{i})} is an isomorphism.

An object  X {\displaystyle X} in a Grothendieck category  A {\displaystyle {\cal {A}}} is called coherent if it is finitely presented and if each of its finitely generated subobjects is also finitely presented.[13] (This generalizes the notion of coherent sheaves on a ringed space.) The full subcategory of all coherent objects in  A {\displaystyle {\cal {A}}} is abelian and the inclusion functor is exact.[13]

An object  X {\displaystyle X} in a Grothendieck category is called Noetherian if the set of its subobjects satisfies the ascending chain condition, i.e. if every sequence  X 1 ⊆ X 2 ⊆ ⋯ {\displaystyle X_{1}\subseteq X_{2}\subseteq \cdots } of subobjects of  X {\displaystyle X} eventually becomes stationary. This is the case if and only if every subobject of X is finitely generated. (In the case  A = Mod ⁡ ( R ) {\displaystyle {\cal {A}}=\operatorname {Mod} (R)} , this notion is equivalent to the familiar notion of Noetherian modules.) A Grothendieck category is called locally Noetherian if it has a set of Noetherian generators; an example is the category of left modules over a left-Noetherian ring.

## Notes

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1. ^ a b c d e f Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique", Tôhoku Mathematical Journal, (2), 9 (2): 119–221, doi:10.2748/tmj/1178244839, MR 0102537. English translation.
2. ^ a b c d e Gabriel, Pierre (1962), "Des catégories abéliennes" (PDF), Bull. Soc. Math. Fr., 90: 323–448, doi:10.24033/bsmf.1583
3. ^ Van, Hoang Dinh; Liu, Liyu; Lowen, Wendy (2016), "Non-commutative deformations and quasi-coherent modules", Selecta Mathematica, 23: 1061–1119, arXiv:1411.0331, doi:10.1007/s00029-016-0263-9, MR 3624905
4. ^ Stacks Project, Tag 077P; Stacks Project, Tag 0781.
5. ^ a b c Faith, Carl (1973). Algebra: Rings, Modules and Categories I. Springer. pp. 486–498\. ISBN 9783642806346.
6. ^ Stenström, Prop. V.1.1
7. ^ Stenström, Cor. X.4.4
8. ^ Mac Lane, Saunders (1978). Categories for the Working Mathematician, 2nd edition. Springer. p. 130.
9. ^ Popescu, Nicolae; Gabriel, Pierre (1964). "Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes". Comptes rendus de l'Académie des Sciences. 258: 4188–4190.
10. ^ Šťovíček, Jan (2013-01-01). "Deconstructibility and the Hill Lemma in Grothendieck categories". Forum Mathematicum. 25 (1). arXiv:1005.3251. Bibcode:2010arXiv1005.3251S. doi:10.1515/FORM.2011.113. S2CID 119129714.
11. ^ Stenström, Prop. V.3.2
12. ^ Stenström, Prop. V.3.4
13. ^ a b Herzog, I. (1997). "The Ziegler Spectrum of a Locally Coherent Grothendieck Category". Proceedings of the London Mathematical Society. 74 (3): 503–558\. doi:10.1112/S002461159700018X. S2CID 121827768.

## References

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* Popescu, Nicolae (1973). Abelian categories with applications to rings and modules. Academic Press.
* Stenström, Bo T. (1975). Rings of Quotients: An Introduction to Methods of Ring Theory. Springer-Verlag. ISBN 978-0-387-07117-6.

## External links

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* Abelian Categories, notes by Daniel Murfet. Section 2.3 covers Grothendieck categories.
* The Stacks Project Authors, The Stacks Project
* Tsalenko, M.Sh. (2001) [1994], "Grothendieck category", Encyclopedia of Mathematics, EMS Press