# 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 ${\displaystyle V}$ one can associate a Grothendieck category ${\displaystyle \operatorname {Qcoh} (V)}$, consisting of the quasi-coherent sheaves on ${\displaystyle V}$. This category encodes all the relevant geometric information about ${\displaystyle V}$, and ${\displaystyle V}$ can be recovered from ${\displaystyle \operatorname {Qcoh} (V)}$. 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 ${\displaystyle {\mathcal {A}}}$ is an AB5 category with a generator. Spelled out, this means that

## Properties

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

Every object in a Grothendieck category ${\displaystyle {\mathcal {A}}}$ has an injective hull in ${\displaystyle {\mathcal {A}}}$. This allows to construct injective resolutions and thereby the use of the tools of homological algebra in ${\displaystyle {\mathcal {A}}}$, 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 ${\displaystyle (U_{i})}$ of a given object ${\displaystyle X}$ has a supremum ${\displaystyle \sum _{i}U_{i}}$ which is again a subobject of ${\displaystyle X}$. (Note that an infimum need not exist.) Further, if the family ${\displaystyle (U_{i})}$ is directed (i.e. for any two objects in the family, there is a third object in the family that contains the two), and ${\displaystyle V}$ is another subobject of ${\displaystyle X}$, we have

${\displaystyle \sum _{i}(U_{i}\cap V)=\left(\sum _{i}U_{i}\right)\cap V.}$

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 ${\displaystyle {\mathcal {A}}}$ is equivalent to a full subcategory of the category ${\displaystyle \operatorname {Mod} (R)}$ of right modules over some unital ring ${\displaystyle R}$ (which can be taken to be the endomorphism ring of a generator of ${\displaystyle {\mathcal {A}}}$), and ${\displaystyle {\mathcal {A}}}$ can be obtained as a Serre quotient of ${\displaystyle \operatorname {Mod} (R)}$ by some localizing subcategory.[3]

## References

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