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 Pierre Gabriel's seminal thesis in 1962.^[2]

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 ${\displaystyle \operatorname {Qcoh} (V)}$ (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

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

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

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 in the work of several authors, including Jan-Erik Roos, Bo Stenström, Ulrich Oberst, and Bodo Pareigis. (Some authors use a different definition in that they don't require the existence of a generator.)

Examples

• The prototypical example of a Grothendieck category is the category of abelian groups; the abelian group ${\displaystyle \mathbb {Z} }$ of integers can serve as a generator.
• More generally, given any ring ${\displaystyle R}$ (associative, with ${\displaystyle 1}$, but not necessarily commutative), the category ${\displaystyle \operatorname {Mod} (R)}$ of all right (or alternatively: left) modules over ${\displaystyle R}$ is a Grothendieck category; ${\displaystyle R}$ itself can serve as a generator.
• Given a topological space ${\displaystyle X}$, the category of all sheaves of abelian groups on ${\displaystyle X}$ is a Grothendieck category.^[1] (More generally: the category of all sheaves of right ${\displaystyle R}$-modules on ${\displaystyle X}$ is a Grothendieck category for any ring ${\displaystyle R}$.)
• Given a ringed space ${\displaystyle (X,{\mathcal {O}}_{X})}$, the category of sheaves of O_X-modules is a Grothendieck category.^[1]
• Given an (affine or projective) algebraic variety ${\displaystyle V}$ (or more generally: any scheme), the category ${\displaystyle \operatorname {Qcoh} (V)}$ of quasi-coherent sheaves on ${\displaystyle V}$ 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

• Any category that's equivalent to a Grothendieck category is itself a Grothendieck category.
• Given Grothendieck categories ${\displaystyle {\mathcal {A_{1}}},\ldots ,{\mathcal {A_{n}}}}$, the product category ${\displaystyle {\mathcal {A_{1}}}\times \ldots \times {\mathcal {A_{n}}}}$ is a Grothendieck category.
• Given a small category ${\displaystyle {\mathcal {C}}}$ and a Grothendieck category ${\displaystyle {\mathcal {A}}}$, the functor category ${\displaystyle \operatorname {Funct} ({\mathcal {C}},{\mathcal {A}})}$, consisting of all covariant functors from ${\displaystyle {\mathcal {C}}}$ to ${\displaystyle {\mathcal {A}}}$, is a Grothendieck category.^[1]
• Given a small preadditive category ${\displaystyle {\mathcal {C}}}$ and a Grothendieck category ${\displaystyle {\mathcal {A}}}$, the functor category ${\displaystyle \operatorname {Add} ({\mathcal {C}},{\mathcal {A}})}$ of all additive covariant functors from ${\displaystyle {\mathcal {C}}}$ to ${\displaystyle {\mathcal {A}}}$ is a Grothendieck category.^[4]
• If ${\displaystyle {\mathcal {A}}}$ is a Grothendieck category and ${\displaystyle {\mathcal {C}}}$ is a localizing subcategory of ${\displaystyle {\mathcal {A}}}$, then both ${\displaystyle {\mathcal {C}}}$ and the Serre quotient category ${\displaystyle {\mathcal {A}}/{\mathcal {C}}}$ are Grothendieck categories.^[2]

Properties and theorems

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 ${\displaystyle {\mathcal {A}}}$.^[1][2] This allows to construct injective resolutions and thereby the use of the tools of homological algebra in ${\displaystyle {\mathcal {A}}}$, in order to define 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 (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 ${\displaystyle X}$. 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^[5]

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

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]

It is a rather deep result that every Grothendieck category ${\displaystyle {\mathcal {A}}}$ is complete,^[6] i.e. that arbitrary limits (and in particular products) exist in ${\displaystyle {\mathcal {A}}}$. By contrast, it follows directly from the definition that ${\displaystyle {\mathcal {A}}}$ is co-complete, i.e. that arbitrary colimits and coproducts (direct sums) exist in ${\displaystyle {\mathcal {A}}}$. 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 ${\displaystyle F\colon {\cal {A}}\to {\cal {X}}}$ from a Grothendieck category ${\displaystyle {\mathcal {A}}}$ to an arbitrary category ${\displaystyle {\mathcal {X}}}$ 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.^[7]

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 Gabriel quotient of ${\displaystyle \operatorname {Mod} (R)}$ by some localizing subcategory.^[8]

As a consequence of Gabriel–Popescu, one can show that every Grothendieck category is locally presentable.^[9] Furthermore, Gabriel-Popescu can be used to see that every Grothendieck category is complete, being a reflective subcategory of the complete category ${\displaystyle \operatorname {Mod} (R)}$ for some ${\displaystyle R}$.

Every small abelian category ${\displaystyle {\mathcal {C}}}$ can be embedded in a Grothendieck category, in the following fashion. The category ${\displaystyle {\mathcal {A}}:=\operatorname {Lex} ({\mathcal {C}}^{op},\mathrm {Ab} )}$ of left-exact additive (covariant) functors ${\displaystyle {\mathcal {C}}^{op}\rightarrow \mathrm {Ab} }$ (where ${\displaystyle \mathrm {Ab} }$ denotes the category of abelian groups) is a Grothendieck category, 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. A generator of ${\displaystyle {\mathcal {A}}}$ is given by the coproduct of all ${\displaystyle h_{C}}$, with ${\displaystyle C\in {\mathcal {C}}}$.^[2] The category ${\displaystyle {\mathcal {A}}}$ is equivalent to the category ${\displaystyle {\text{Ind}}({\mathcal {C}})}$ of ind-objects of ${\displaystyle {\mathcal {C}}}$ and the embedding ${\displaystyle h}$ corresponds to the natural embedding ${\displaystyle {\mathcal {C}}\to {\text{Ind}}({\mathcal {C}})}$. We may therefore view ${\displaystyle {\mathcal {A}}}$ as the co-completion of ${\displaystyle {\mathcal {C}}}$.

Special kinds of objects and Grothendieck categories

An object ${\displaystyle X}$ in a Grothendieck category is called finitely generated if, whenever ${\displaystyle X}$ is written as the sum of a family of subobjects of ${\displaystyle X}$, then it is already the sum of a finite subfamily. (In the case ${\displaystyle {\cal {A}}=\operatorname {Mod} (R)}$ of module categories, this notion is equivalent to the familiar notion of finitely generated modules.) Epimorphic images of finitely generated objects are again finitely generated. If ${\displaystyle U\subseteq X}$ and both ${\displaystyle U}$ and ${\displaystyle X/U}$ are finitely generated, then so is ${\displaystyle X}$. The object ${\displaystyle X}$ is finitely generated if, and only if, for any directed system ${\displaystyle (A_{i})}$ in ${\displaystyle {\cal {A}}}$ in which each morphism is a monomorphism, the natural morphism ${\displaystyle \varinjlim \mathrm {Hom} (X,A_{i})\to \mathrm {Hom} (X,\varinjlim A_{i})}$ is an isomorphism.^[10] 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 (i.e. if there exists a family ${\displaystyle (G_{i})_{i\in I}}$ of finitely generated objects such that to every object ${\displaystyle X}$ there exist ${\displaystyle i\in I}$ and a non-zero morphism ${\displaystyle G_{i}\rightarrow X}$; equivalently: ${\displaystyle X}$ is epimorphic image of a direct sum of copies of the ${\displaystyle G_{i}}$). In such a category, every object is the sum of its finitely generated subobjects.^[4] Every category ${\displaystyle {\cal {A}}=\operatorname {Mod} (R)}$ is locally finitely generated.

An object ${\displaystyle X}$ in a Grothendieck category is called finitely presented if it is finitely generated and if every epimorphism ${\displaystyle W\to X}$ with finitely generated domain ${\displaystyle W}$ has a finitely generated kernel. Again, this generalizes the notion of finitely presented modules. If ${\displaystyle U\subseteq X}$ and both ${\displaystyle U}$ and ${\displaystyle X/U}$ are finitely presented, then so is ${\displaystyle X}$. In a locally finitely generated Grothendieck category ${\displaystyle {\cal {A}}}$, the finitely presented objects can be characterized as follows:^[11] ${\displaystyle X}$ in ${\displaystyle {\cal {A}}}$ is finitely presented if, and only if, for every directed system ${\displaystyle (A_{i})}$ in ${\displaystyle {\cal {A}}}$, the natural morphism ${\displaystyle \varinjlim \mathrm {Hom} (X,A_{i})\to \mathrm {Hom} (X,\varinjlim A_{i})}$ is an isomorphism.

An object ${\displaystyle X}$ in a Grothendieck category ${\displaystyle {\cal {A}}}$ is called coherent if it is finitely presented and if each of its finitely generated subobjects is also finitely presented.^[12] (This generalizes the notion of coherent sheaves on a ringed space.) The full subcategory of all coherent objects in ${\displaystyle {\cal {A}}}$ is abelian and the inclusion functor is exact.^[12]

An object ${\displaystyle X}$ in a Grothendieck category is called Noetherian if the set of its subobjects satisfies the ascending chain condition, i.e. if every sequence ${\displaystyle X_{1}\subseteq X_{2}\subseteq \cdots }$ of subobjects of ${\displaystyle X}$ eventually becomes stationary. This is the case if and only if every subobject of X is finitely generated. (In the case ${\displaystyle {\cal {A}}=\operatorname {Mod} (R)}$, this notion is equivalent to the familiar notion of Noetherian modules.) 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.

Notes

References

• Popescu, Nicolae (1973). Abelian categories with applications to rings and modules. Academic Press.
• Stenström, Bo T. (1975). Rings of Quotients: An Introduction to Methods of Ring Theory. Springer-Verlag. ISBN 978-0-387-07117-6.

External links

• Tsalenko, M.Sh. (2001) [1994], "Grothendieck category", Encyclopedia of Mathematics, EMS Press
• Abelian Categories, notes by Daniel Murfet. Section 2.3 covers Grothendieck categories.