# 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.

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.^{[2]}

## Definition

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 (a.k.a. filtered colimits) 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 .)

## 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 (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. (More generally: the category of all sheaves of left -modules on is a Grothendieck category for any ring .)
- Given a ringed space , the category of sheaves of
*O*-modules is a Grothendieck category._{X} - 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.
- Any category that's equivalent to a Grothendieck category is itself a Grothendieck category.
- Given a small category and a Grothendieck category , the functor category is a Grothendieck category; if is preadditive, then the functor category of all additive functors from to is a Grothendieck category as well.
- If is a Grothendieck category and is a localizing subcategory of , we can form the Serre quotient category . This quotient is again a Grothendieck category.

## Properties

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 . 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 which is again a subobject of . (Note that an infimum need not exist.) 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

In a Grothendieck category, arbitrary limits (and in particular products) exist. It follows directly from the definition that arbitrary colimits and coproducts (direct sums) exist as well. We can thus say that every Grothendieck category is complete and co-complete. 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.

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 Serre quotient of by some localizing subcategory.^{[3]}

## References

- ↑ {{#invoke:citation/CS1|citation |CitationClass=citation }}. English translation.
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## External links

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- Abelian Categories, notes by Daniel Murfet. Section 2.3 covers Grothendieck categories.