Grothendieck category
In mathematics, a Grothendieck category is a certain kind of abelian category, introduced in Alexander Grothendieck's Tôhoku paper of 1957 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.
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 . This example gives rise to one approach to noncommutative algebraic geometry: the study of "non-commutative varieties" is then nothing but the study of Grothendieck categories.
Definition
By definition, a Grothendieck category is an AB5 category with a generator. Spelled out, this means that- is an abelian category;
- every family of objects in has a coproduct in ;
- direct limits of short 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.
- possesses a generator, i.e. there is an object in such that is a faithful functor from to the category of sets.
Examples
- 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 , the category of all right 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.
- Given a ringed space, the category of sheaves of OX-modules is a Grothendieck category.
- Given an algebraic variety , the category of quasi-coherent sheaves on is a Grothendieck category.
- Given a small site , 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 , 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.
- Given a small preadditive category and a Grothendieck category, the functor category of all additive covariant functors from to is a Grothendieck category.
- If is a Grothendieck category and is a localizing subcategory of, then both and the Serre quotient category are Grothendieck categories.
Properties and theorems
Every object in a Grothendieck category has an injective hull in. This allows to construct injective resolutions and thereby the use of the tools of homological algebra in, in order to define derived functors.
In a Grothendieck category, any family of subobjects of a given object has a supremum as well as an infimum , both of which are again subobjects of. Further, if the family is directed, and is another subobject of, we have
Grothendieck categories are well-powered, i.e. the collection of subobjects of any given object forms a set.
It is a rather deep result that every Grothendieck category is complete, i.e. that arbitrary limits exist in. By contrast, it follows directly from the definition that is co-complete, i.e. that arbitrary colimits and coproducts exist in. Coproducts in a Grothendieck category are exact, but products need not be exact.
A functor from a Grothendieck categoriy 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 Peter J. Freyd's special adjoint functor theorem and its dual.
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 , and can be obtained as a Gabriel quotient of by some localizing subcategory.
As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable.
Every small abelian category can be embedded in a Grothendieck category, in the following fashion. The category of left-exact additive functors is a Grothendieck category, and the functor, with, is full, faithful and exact. A generator of is given by the coproduct of all, with. The category is equivalent to the category of ind-objects of and the embedding corresponds to the natural embedding . We may therefore view as the co-completion of.
Special kinds of objects and Grothendieck categories
An object in a Grothendieck category is called finitely generated if, whenever is written as the sum of a family of subobjects of, then it is already the sum of a finite subfamily. Epimorphic images of finitely generated objects are again finitely generated. If and both and are finitely generated, then so is. The object is finitely generated if, and only if, for any directed system in in which each morphism is a monomorphism, the natural morphism is an isomorphism. 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. Every category is locally finitely generated.
An object in a Grothendieck category is called finitely presented if it is finitely generated and if every epimorphism with finitely generated domain has a finitely generated kernel. Again, this generalizes the notion of finitely presented modules. If and both and are finitely presented, then so is. In a locally finitely generated Grothendieck category, the finitely presented objects can be characterized as follows: in is finitely presented if, and only if, for every directed system in, the natural morphism is an isomorphism.
An object in a Grothendieck category is called coherent if it is finitely presented and if each of its finitely generated subobjects is also finitely presented. The full subcategory of all coherent objects in is abelian and the inclusion functor is exact.
An object in a Grothendieck category is called Noetherian if the set of its subobjects satisfies the ascending chain condition, i.e. if every sequence of subobjects of eventually becomes stationary. This is the case if and only if every subobject of X is finitely generated. 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.