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 Peter Gabriel's seminal thèse in 1962.[2]

To every algebraic variety one can associate a Grothendieck category , consisting of the quasi-coherent sheaves on . This category encodes all the relevant geometric information about , and can be recovered from (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[edit]

By definition, a Grothendieck category is an AB5 category with a generator. Spelled out, this means that

  • is an abelian category;
  • every (possibly infinite) family of objects in has a coproduct (a.k.a. direct sum) in ;
  • direct limits of exact sequences are exact; this means that if a direct system of short exact sequences in 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.)
  • possesses a generator, i.e. there is an object in such that is a faithful functor from to the category of sets. (In our situation, this is equivalent to saying that every object of admits an epimorphism , where denotes a direct sum of copies of , one for each element of the (possibly infinite) set .)

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 by authors including J.-E. Roos, B. Stenström, U. Oberst, and B. Pareigis. (Some authors use a different definition in that they don't require the existence of a generator.)

Examples[edit]

  • The prototypical example of a Grothendieck category is the category of abelian groups; the abelian group of integers can serve as a generator.
  • More generally, given any ring (associative, with , but not necessarily commutative), the category of all right (or alternatively: left) modules over is a Grothendieck category; itself can serve as a generator.
  • Given a topological space , the category of all sheaves of abelian groups on is a Grothendieck category.[1] (More generally: the category of all sheaves of right -modules on is a Grothendieck category for any ring .)
  • Given a ringed space , the category of sheaves of OX-modules is a Grothendieck category.[1]
  • Given an (affine or projective) algebraic variety (or more generally: a quasi-compact quasi-separated scheme), the category of quasi-coherent sheaves on 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[edit]

  • Any category that's equivalent to a Grothendieck category is itself a Grothendieck category.
  • Given Grothendieck categories , the product category is a Grothendieck category.
  • Given a small category and a Grothendieck category , the functor category , consisting of all covariant functors from to , is a Grothendieck category.[1]
  • Given a small preadditive category and a Grothendieck category , the functor category of all additive covariant functors from to is a Grothendieck category.[4]
  • If is a Grothendieck category and is a localizing subcategory of , then both and the Serre quotient category are Grothendieck categories.[2]

Properties and theorems[edit]

Every Grothendieck category contains an injective cogenerator. For example, an injective cogenerator of the category of abelian groups is the quotient group .

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

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]

An object in a Grothendieck category is called finitely generated if the sum of every directed family of proper subobjects of is again a proper subobject of . (In the case of module categories, this notion is equivalent to the familiar notion of finitely generated modules.) 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. In such a category, every object is the sum of its finitely generated subobjects.[4]

It is a rather deep result that every Grothendieck category is complete, i.e. that arbitrary limits (and in particular products) exist in . By contrast, it follows directly from the definition that is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in . 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 from a Grothendieck categories to an arbitrary category 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 Freyd's special adjoint functor theorem and its dual.[5]

The Gabriel–Popescu theorem states that any Grothendieck category is equivalent to a full subcategory of the category of right modules over some unital ring (which can be taken to be the endomorphism ring of a generator of ), and can be obtained as a Gabriel quotient of by some localizing subcategory.[6]

As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable.[7]

Every small abelian category can be embedded in a Grothendieck category, in the following fashion. The category of left-exact additive (covariant) functors (where denotes the category of abelian groups) is a Grothendieck category, and the functor , with , is full, faithful and exact. A generator of is given by the coproduct of all , with .[2] The category is equivalent to the category of ind-objects of and the embedding corresponds to the natural embedding .

References[edit]

  1. ^ a b c d e f Grothendieck, A. (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, P. (1962), "Des catégories abéliennes" (PDF), Bull. Soc. Math. Fr., 90: 323–448
  3. ^ Izuru Mori (2007). "Quantum Ruled Surfaces" (PDF).
  4. ^ a b c Faith, Carl (1973). Algebra: Rings, Modules and Categories I. Springer. pp. 486–498. ISBN 9783642806346.
  5. ^ Lane, Saunders Mac (1978). Categories for the Working Mathematician, 2nd edition. Springer. p. 130.
  6. ^ 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)
  7. ^ 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.
  • N. Popescu (1973). Abelian categories with applications to rings and modules. Academic Press.
  • Jara, Pascual; Verschoren, Alain; Vidal, Conchi (1995), Localization and sheaves: a relative point of view, Pitman Research Notes in Mathematics Series, 339, Longman, Harlow.

External links[edit]