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. The theory of these categories was further developed in Peter Gabriel's seminal thèse.^[2]

To every algebraic variety $V$ one can associate a Grothendieck category $\operatorname {Qcoh}(V)$, consisting of the quasi-coherent sheaves on $V$. This category encodes all the relevant geometric information about $V$, and $V$ can be recovered from $\operatorname {Qcoh}(V)$. 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.^[3]

Definition

By definition, a Grothendieck category ${\mathcal {A}}$ is an AB5 category with a generator. Spelled out, this means that

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

The name "Grothendieck category" neither appeared in^[1] nor in;^[2] only in the second half of the 1960s authors began to use this name, e.g. J.-E. Roos, B. Stenström, U. Oberst, B. Pareigis. For some authors the existence of generators is not part of the definition.

Examples

• The prototypical example of a Grothendieck category is the category of abelian groups; the abelian group $\mathbb {Z}$ of integers can serve as a generator.
• More generally, given any ring $R$ (associative, with $1$, but not necessarily commutative), the category $\operatorname {Mod}(R)$ of all right (or alternatively: left) modules over $R$ is a Grothendieck category; $R$ itself can serve as a generator.
• Given a topological space $X$, the category of all sheaves of abelian groups on $X$ is a Grothendieck category.^[1] (More generally: the category of all sheaves of left $R$-modules on $X$ is a Grothendieck category for any ring $R$.)
• Given a ringed space $(X,{\mathcal {O}}_{X})$, the category of sheaves of O_X-modules is a Grothendieck category.^[1]
• Given an (affine or projective) algebraic variety $V$ (or more generally: a quasi-compact quasi-separated scheme), the category $\operatorname {Qcoh}(V)$ of quasi-coherent sheaves on $V$ is a Grothendieck category.
• Any category that's equivalent to a Grothendieck category is itself a Grothendieck category.
• Given a small category ${\mathcal {C}}$ and a Grothendieck category ${\mathcal {A}}$, the functor category $\operatorname{Funct}(\mathcal{C},\mathcal{A})$, consisting of all covariant (or alternatively: all contravariant) functors from ${\mathcal {C}}$ to ${\mathcal {A}}$, is a Grothendieck category.^[1]
• Given a small preadditive category ${\mathcal {C}}$, the functor category ${\displaystyle \operatorname {Add} ({\mathcal {C}},\mathrm {Ab} )}$ of all additive covariant (or alternatively: contravariant) functors from ${\mathcal {C}}$ to the category ${\displaystyle \mathrm {Ab} }$ of all abelian groups is a Grothendieck category.
• Let ${\mathcal {C}}$ be a small abelian category. Then the category ${\displaystyle {\mathcal {A}}:=\operatorname {Lex} ({\mathcal {C}}^{op},\mathrm {Ab} )}$ of left-exact (covariant) functors ${\displaystyle {\mathcal {C}}^{op}\rightarrow \mathrm {Ab} }$ is a Grothendieck category,^[2] and the functor ${\displaystyle h\colon {\mathcal {C}}\rightarrow {\mathcal {A}}}$, with ${\displaystyle C\mapsto h_{C}=\operatorname {Hom} (-,C)}$, is full, faithful and exact.^[2] A generator of ${\mathcal {A}}$ is given by^[2] the coproduct of all $h_{C}$, with ${\displaystyle C\in {\mathcal {C}}}$.
• If ${\mathcal {A}}$ is a Grothendieck category and ${\mathcal {C}}$ is a localizing subcategory of ${\mathcal {A}}$, we can form the Serre quotient category ${\mathcal {A}}/{\mathcal {C}}$. Then ${\mathcal {C}}$ and ${\mathcal {A}}/{\mathcal {C}}$ are again Grothendieck categories.^[2]

Properties

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

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

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

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 ${\mathcal {A}}$ is equivalent to a full subcategory of the category $\operatorname {Mod}(R)$ of right modules over some unital ring $R$ (which can be taken to be the endomorphism ring of a generator of ${\mathcal {A}}$), and ${\mathcal {A}}$ can be obtained as a Serre quotient of $\operatorname {Mod}(R)$ by some localizing subcategory.^[4]

References

1. Grothendieck, A. (1957), "Sur quelques points d'algèbre homologique", Tôhoku Mathematical Journal, (2), 9: 119–221, doi:10.2748/tmj/1178244839, MR 0102537. English translation.
2. Gabriel, P. (1962), "Des catégories abéliennes", Bull. Soc. math. France, 90: 323–448
3. Izuru Mori (2007). "Quantum Ruled Surfaces" (PDF).
4. 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.

• 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

• Tsalenko, M.Sh. (2001), "Grothendieck category", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Abelian Categories, notes by Daniel Murfet. Section 2.3 covers Grothendieck categories.