Grothendieck category

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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 seminal thesis in 1962.[2] To every algebraic variety [math]\displaystyle{ V }[/math] one can associate a Grothendieck category [math]\displaystyle{ \operatorname{Qcoh}(V) }[/math], consisting of the quasi-coherent sheaves on [math]\displaystyle{ V }[/math]. This category encodes all the relevant geometric information about [math]\displaystyle{ V }[/math], and [math]\displaystyle{ V }[/math] can be recovered from [math]\displaystyle{ \operatorname{Qcoh}(V) }[/math] (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 [math]\displaystyle{ \mathcal{A} }[/math] is an AB5 category with a generator. Spelled out, this means that

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

The name "Grothendieck category" neither appeared 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 in that they don't require the existence of a generator.)

Examples

  • The prototypical example of a Grothendieck category is the category of abelian groups; the abelian group [math]\displaystyle{ \Z }[/math] of integers can serve as a generator.
  • More generally, given any ring [math]\displaystyle{ R }[/math] (associative, with [math]\displaystyle{ 1 }[/math], but not necessarily commutative), the category [math]\displaystyle{ \operatorname{Mod}(R) }[/math] of all right (or alternatively: left) modules over [math]\displaystyle{ R }[/math] is a Grothendieck category; [math]\displaystyle{ R }[/math] itself can serve as a generator.
  • Given a topological space [math]\displaystyle{ X }[/math], the category of all sheaves of abelian groups on [math]\displaystyle{ X }[/math] is a Grothendieck category.[1] (More generally: the category of all sheaves of right [math]\displaystyle{ R }[/math]-modules on [math]\displaystyle{ X }[/math] is a Grothendieck category for any ring [math]\displaystyle{ R }[/math].)
  • Given a ringed space [math]\displaystyle{ (X,\mathcal{O}_X) }[/math], the category of sheaves of OX-modules is a Grothendieck category.[1]
  • Given an (affine or projective) algebraic variety [math]\displaystyle{ V }[/math] (or more generally: any scheme), the category [math]\displaystyle{ \operatorname{Qcoh}(V) }[/math] of quasi-coherent sheaves on [math]\displaystyle{ V }[/math] is a Grothendieck category.
  • 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 [math]\displaystyle{ \mathcal{A_1},\ldots,\mathcal{A_n} }[/math], the product category [math]\displaystyle{ \mathcal{A_1}\times\ldots\times\mathcal{A_n} }[/math] is a Grothendieck category.
  • Given a small category [math]\displaystyle{ \mathcal{C} }[/math] and a Grothendieck category [math]\displaystyle{ \mathcal{A} }[/math], the functor category [math]\displaystyle{ \operatorname{Funct}(\mathcal{C},\mathcal{A}) }[/math], consisting of all covariant functors from [math]\displaystyle{ \mathcal{C} }[/math] to [math]\displaystyle{ \mathcal{A} }[/math], is a Grothendieck category.[1]
  • Given a small preadditive category [math]\displaystyle{ \mathcal{C} }[/math] and a Grothendieck category [math]\displaystyle{ \mathcal{A} }[/math], the functor category [math]\displaystyle{ \operatorname{Add}(\mathcal{C},\mathcal{A}) }[/math] of all additive covariant functors from [math]\displaystyle{ \mathcal{C} }[/math] to [math]\displaystyle{ \mathcal{A} }[/math] is a Grothendieck category.[4]
  • If [math]\displaystyle{ \mathcal{A} }[/math] is a Grothendieck category and [math]\displaystyle{ \mathcal{C} }[/math] is a localizing subcategory of [math]\displaystyle{ \mathcal{A} }[/math], then both [math]\displaystyle{ \mathcal{C} }[/math] and the Serre quotient category [math]\displaystyle{ \mathcal{A}/\mathcal{C} }[/math] 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 [math]\displaystyle{ \mathbb{Q}/\mathbb{Z} }[/math].

Every object in a Grothendieck category [math]\displaystyle{ \mathcal{A} }[/math] has an injective hull in [math]\displaystyle{ \mathcal{A} }[/math].[1][2] This allows to construct injective resolutions and thereby the use of the tools of homological algebra in [math]\displaystyle{ \mathcal{A} }[/math], 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 [math]\displaystyle{ (U_i) }[/math] of a given object [math]\displaystyle{ X }[/math] has a supremum (or "sum") [math]\displaystyle{ \sum_i U_i }[/math] as well as an infimum (or "intersection") [math]\displaystyle{ \cap_i U_i }[/math], both of which are again subobjects of [math]\displaystyle{ X }[/math]. Further, if the family [math]\displaystyle{ (U_i) }[/math] is directed (i.e. for any two objects in the family, there is a third object in the family that contains the two), and [math]\displaystyle{ V }[/math] is another subobject of [math]\displaystyle{ X }[/math], we have[5]

[math]\displaystyle{ \sum_{i}(U_i\cap V) = \left(\sum_{i}U_i\right) \cap V. }[/math]

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).[4]

It is a rather deep result that every Grothendieck category [math]\displaystyle{ \mathcal{A} }[/math] is complete,[6] i.e. that arbitrary limits (and in particular products) exist in [math]\displaystyle{ \mathcal{A} }[/math]. By contrast, it follows directly from the definition that [math]\displaystyle{ \mathcal{A} }[/math] is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in [math]\displaystyle{ \mathcal{A} }[/math]. 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 [math]\displaystyle{ F \colon {\cal A} \to {\cal X} }[/math] from a Grothendieck category [math]\displaystyle{ \mathcal{A} }[/math] to an arbitrary category [math]\displaystyle{ \mathcal{X} }[/math] 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.[7]

The Gabriel–Popescu theorem states that any Grothendieck category [math]\displaystyle{ \mathcal{A} }[/math] is equivalent to a full subcategory of the category [math]\displaystyle{ \operatorname{Mod}(R) }[/math] of right modules over some unital ring [math]\displaystyle{ R }[/math] (which can be taken to be the endomorphism ring of a generator of [math]\displaystyle{ \mathcal{A} }[/math]), and [math]\displaystyle{ \mathcal{A} }[/math] can be obtained as a Gabriel quotient of [math]\displaystyle{ \operatorname{Mod}(R) }[/math] by some localizing subcategory.[8]

As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable.[9] Furthermore, Gabriel-Popescu can be used to see that every Grothendieck category is complete, being a reflective subcategory of the complete category [math]\displaystyle{ \operatorname{Mod}(R) }[/math] for some [math]\displaystyle{ R }[/math].

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

Special kinds of objects and Grothendieck categories

An object [math]\displaystyle{ X }[/math] in a Grothendieck category is called finitely generated if, whenever [math]\displaystyle{ X }[/math] is written as the sum of a family of subobjects of [math]\displaystyle{ X }[/math], then it is already the sum of a finite subfamily. (In the case [math]\displaystyle{ {\cal A} = \operatorname{Mod}(R) }[/math] 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 [math]\displaystyle{ U\subseteq X }[/math] and both [math]\displaystyle{ U }[/math] and [math]\displaystyle{ X/U }[/math] are finitely generated, then so is [math]\displaystyle{ X }[/math]. The object [math]\displaystyle{ X }[/math] is finitely generated if, and only if, for any directed system [math]\displaystyle{ (A_i) }[/math] in [math]\displaystyle{ {\cal A} }[/math] in which each morphism is a monomorphism, the natural morphism [math]\displaystyle{ \varinjlim \mathrm{Hom}(X,A_i)\to \mathrm{Hom}(X,\varinjlim A_i) }[/math] is an isomorphism.[10] 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 [math]\displaystyle{ (G_i)_{i\in I} }[/math] of finitely generated objects such that to every object [math]\displaystyle{ X }[/math] there exist [math]\displaystyle{ i\in I }[/math] and a non-zero morphism [math]\displaystyle{ G_{i}\rightarrow X }[/math]; equivalently: [math]\displaystyle{ X }[/math] is epimorphic image of a direct sum of copies of the [math]\displaystyle{ G_{i} }[/math]). In such a category, every object is the sum of its finitely generated subobjects.[4] Every category [math]\displaystyle{ {\cal A} = \operatorname{Mod}(R) }[/math] is locally finitely generated.

An object [math]\displaystyle{ X }[/math] in a Grothendieck category is called finitely presented if it is finitely generated and if every epimorphism [math]\displaystyle{ W\to X }[/math] with finitely generated domain [math]\displaystyle{ W }[/math] has a finitely generated kernel. Again, this generalizes the notion of finitely presented modules. If [math]\displaystyle{ U\subseteq X }[/math] and both [math]\displaystyle{ U }[/math] and [math]\displaystyle{ X/U }[/math] are finitely presented, then so is [math]\displaystyle{ X }[/math]. In a locally finitely generated Grothendieck category [math]\displaystyle{ {\cal A} }[/math], the finitely presented objects can be characterized as follows:[11] [math]\displaystyle{ X }[/math] in [math]\displaystyle{ {\cal A} }[/math] is finitely presented if, and only if, for every directed system [math]\displaystyle{ (A_i) }[/math] in [math]\displaystyle{ {\cal A} }[/math], the natural morphism [math]\displaystyle{ \varinjlim \mathrm{Hom}(X,A_i)\to \mathrm{Hom}(X,\varinjlim A_i) }[/math] is an isomorphism.

An object [math]\displaystyle{ X }[/math] in a Grothendieck category [math]\displaystyle{ {\cal A} }[/math] is called coherent if it is finitely presented and if each of its finitely generated subobjects is also finitely presented.[12] (This generalizes the notion of coherent sheaves on a ringed space.) The full subcategory of all coherent objects in [math]\displaystyle{ {\cal A} }[/math] is abelian and the inclusion functor is exact.[12]

An object [math]\displaystyle{ X }[/math] in a Grothendieck category is called Noetherian if the set of its subobjects satisfies the ascending chain condition, i.e. if every sequence [math]\displaystyle{ X_1\subseteq X_2 \subseteq \cdots }[/math] of subobjects of [math]\displaystyle{ X }[/math] eventually becomes stationary. This is the case if and only if every subobject of X is finitely generated. (In the case [math]\displaystyle{ {\cal A} = \operatorname{Mod}(R) }[/math], 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

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 Grothendieck, Alexander (1957), "Sur quelques points d'algèbre homologique", Tôhoku Mathematical Journal, (2) 9 (2): 119–221, doi:10.2748/tmj/1178244839, http://projecteuclid.org/euclid.tmj/1178244839 . English translation.
  2. ↑ 2.0 2.1 2.2 2.3 2.4 Gabriel, Pierre (1962), "Des catégories abéliennes", Bull. Soc. Math. Fr. 90: 323–448, doi:10.24033/bsmf.1583, http://www.maths.ed.ac.uk/~aar/papers/gabriel.pdf 
  3. ↑ Izuru Mori (2007). "Quantum Ruled Surfaces". http://mathsoc.jp/section/algebra/algsymp_past/algsymp07_files/mouri.pdf. 
  4. ↑ 4.0 4.1 4.2 Faith, Carl (1973) (in en). Algebra: Rings, Modules and Categories I. Springer. pp. 486–498. ISBN 9783642806346. https://books.google.com/books?id=vsfyCAAAQBAJ&pg=PA487. 
  5. ↑ Stenström, Prop. V.1.1
  6. ↑ Stenström, Cor. X.4.4
  7. ↑ Mac Lane, Saunders (1978) (in en). Categories for the Working Mathematician, 2nd edition. Springer. pp. 130. 
  8. ↑ Popesco, Nicolae; Gabriel, Pierre (1964). "Caractérisation des catégories abéliennes avec générateurs et limites inductives exactes". Comptes rendus de l'Académie des Sciences 258: 4188–4190. 
  9. ↑ Šťovíček, Jan (2013-01-01). "Deconstructibility and the Hill Lemma in Grothendieck categories" (in en). Forum Mathematicum 25 (1). doi:10.1515/FORM.2011.113. Bibcode: 2010arXiv1005.3251S. 
  10. ↑ Stenström, Prop. V.3.2
  11. ↑ Stenström, Prop. V.3.4
  12. ↑ 12.0 12.1 Herzog, I. (1997). "The Ziegler Spectrum of a Locally Coherent Grothendieck Category" (in en). Proceedings of the London Mathematical Society 74 (3): 503–558. doi:10.1112/S002461159700018X. https://www.researchgate.net/publication/231890405. 

References

External links



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