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 $V$ one can associate a Grothendieck category $\operatorname {Qcoh} (V)$ , consisting of the quasi-coherent sheaves on $V$ . This category encodes all the relevant geometric information about $V$ , and $V$ can be recovered from $\operatorname {Qcoh} (V)$ (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

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

${\mathcal {A}}$ is an abelian category;
every (possibly infinite) family of objects in ${\mathcal {A}}$ has a coproduct (also known as direct sum) in ${\mathcal {A}}$ ;
direct limits of short exact sequences are exact; this means that if a direct system of short exact sequences in ${\mathcal {A}}$ 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.)
${\mathcal {A}}$ possesses a generator, i.e. there is an object $G$ in ${\mathcal {A}}$ such that $\operatorname {Hom} (G,-)$ is a faithful functor from ${\mathcal {A}}$ to the category of sets. (In our situation, this is equivalent to saying that every object $X$ of ${\mathcal {A}}$ admits an epimorphism $G^{(I)}\rightarrow X$ , where $G^{(I)}$ denotes a direct sum of copies of $G$ , one for each element of the (possibly infinite) set $I$ .)

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

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

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

Properties and theorems

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

Every object in a Grothendieck category ${\mathcal {A}}$ has an injective hull in ${\mathcal {A}}$ .^[1]^[2] This allows to construct injective resolutions and thereby the use of the tools of homological algebra in ${\mathcal {A}}$ , 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 $(U_{i})$ of a given object $X$ has a supremum (or "sum") ${\textstyle \sum _{i}U_{i}}$ as well as an infimum (or "intersection") $\cap _{i}U_{i}$ , both of which are again subobjects of $X$ . Further, if the family $(U_{i})$ is directed (i.e. for any two objects in the family, there is a third object in the family that contains the two), and $V$ is another subobject of $X$ , we have^[6]

\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 ${\mathcal {A}}$ is complete,^[7] i.e. that arbitrary limits (and in particular products) exist in ${\mathcal {A}}$ . By contrast, it follows directly from the definition that ${\mathcal {A}}$ is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in ${\mathcal {A}}$ . 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 $F\colon {\cal {A}}\to {\cal {X}}$ from a Grothendieck category ${\mathcal {A}}$ to an arbitrary category ${\mathcal {X}}$ 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 ${\mathcal {A}}$ is equivalent to a full subcategory of the category $\operatorname {Mod} (R)$ of right modules over some unital ring $R$ (which can be taken to be the endomorphism ring of a generator of ${\mathcal {A}}$ ), and ${\mathcal {A}}$ can be obtained as a Gabriel quotient of $\operatorname {Mod} (R)$ 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 $\operatorname {Mod} (R)$ for some $R$ .

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

Special kinds of objects and Grothendieck categories

An object $X$ in a Grothendieck category is called finitely generated if, whenever $X$ is written as the sum of a family of subobjects of $X$ , then it is already the sum of a finite subfamily. (In the case ${\cal {A}}=\operatorname {Mod} (R)$ 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 $U\subseteq X$ and both $U$ and $X/U$ are finitely generated, then so is $X$ . The object $X$ is finitely generated if, and only if, for any directed system $(A_{i})$ in ${\cal {A}}$ in which each morphism is a monomorphism, the natural morphism $\varinjlim \mathrm {Hom} (X,A_{i})\to \mathrm {Hom} (X,\varinjlim A_{i})$ 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 $(G_{i})_{i\in I}$ of finitely generated objects such that to every object $X$ there exist $i\in I$ and a non-zero morphism $G_{i}\rightarrow X$ ; equivalently: $X$ is epimorphic image of a direct sum of copies of the $G_{i}$ ). In such a category, every object is the sum of its finitely generated subobjects.^[5] Every category ${\cal {A}}=\operatorname {Mod} (R)$ is locally finitely generated.

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

An object $X$ in a Grothendieck category ${\cal {A}}$ 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 ${\cal {A}}$ is abelian and the inclusion functor is exact.^[13]

An object $X$ in a Grothendieck category is called Noetherian if the set of its subobjects satisfies the ascending chain condition, i.e. if every sequence $X_{1}\subseteq X_{2}\subseteq \cdots$ of subobjects of $X$ eventually becomes stationary. This is the case if and only if every subobject of X is finitely generated. (In the case ${\cal {A}}=\operatorname {Mod} (R)$ , 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

^ ^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.
^ ^a ^b ^c ^d ^e Gabriel, Pierre (1962), "Des catégories abéliennes" (PDF), Bull. Soc. Math. Fr., 90: 323–448, doi:10.24033/bsmf.1583
^ 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
^ Stacks Project, Tag 077P; Stacks Project, Tag 0781.
^ ^a ^b ^c Faith, Carl (1973). Algebra: Rings, Modules and Categories I. Springer. pp. 486–498. ISBN 9783642806346.
^ Stenström, Prop. V.1.1
^ Stenström, Cor. X.4.4
^ Mac Lane, Saunders (1978). Categories for the Working Mathematician, 2nd edition. Springer. p. 130.
^ 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.
^ 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.
^ Stenström, Prop. V.3.2
^ Stenström, Prop. V.3.4
^ ^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

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

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

[tohoku-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.

[gabriel-these-2] 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.

[:0-5] 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

[herzog-13] 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.

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