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An [[Abelian category|Abelian category]] with a set of generators (cf. [[Generator of a category|Generator of a category]]) and satisfying the following axiom: There exist coproducts (sums) of arbitrary families of objects, and for each directed family of subobjects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451501.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451502.png" />, of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451503.png" />, and any subobject <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451504.png" />, the following equality is valid:
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An [[Abelian category]] with a set of generators (cf. [[Generator of a category]]) and satisfying the following axiom: There exist [[coproduct]]s (sums) of arbitrary families of objects, and for each directed family of subobjects $U_i$, $i \in I$, of an object $A$, and any subobject $V$, the following equality holds:
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$$
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\left({ \bigcup_{i \in I} U_i }\right) \cap V = \bigcup_{i \in I} \left({ U_i  \cap V }\right)
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451505.png" /></td> </tr></table>
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The [[Modules, category of|category of modules]] (left or right) over an arbitrary [[Associative rings and algebras|associative ring]] $R$ with an [[Ring with identity|identity element]] and the category of sheaves of $R$-modules over an arbitrary topological space (cf. [[Sheaf theory]]) are Grothendieck categories. A full subcategory $\mathfrak{S}$ of the category ${}_R \mathfrak{M}$ of left $R$-modules is known as a ''localizing subcategory'' if it is closed with respect to [[colimit]]s and if, in an exact sequence
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$$
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0 \rightarrow A' \rightarrow A \rightarrow A''' \rightarrow 0
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$$
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the object $A$ belongs to $\mathfrak{S}$ if and only if both $A'$ and $A''$ belong to $\mathfrak{S}$. Each localizing subcategory makes it possible to construct the quotient category ${}_R \mathfrak{M} / \mathfrak{S}$. An Abelian category is a Grothendieck category if and only if it is equivalent to some quotient category of the type ${}_R \mathfrak{M} / \mathfrak{S}$.
  
The category of left (right) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451506.png" />-modules over an arbitrary associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451507.png" /> with an identity element and the category of sheaves of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451508.png" />-modules over an arbitrary topological space are Grothendieck categories. A full subcategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g0451509.png" /> of the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515010.png" /> of left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515011.png" />-modules is known as a localizing subcategory if it is closed with respect to colimits and if, in an exact sequence
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In a Grothendieck category each object has an [[injective envelope]], and for this reason Grothendieck categories are well suited for use in homological applications.
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515012.png" /></td> </tr></table>
 
 
 
the object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515013.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515014.png" /> if and only if both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515016.png" /> belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515017.png" />. Each localizing subcategory makes it possible to construct the quotient category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515018.png" />. An Abelian category is a Grothendieck category if and only if it is equivalent to some quotient category of the type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045150/g04515019.png" />.
 
 
 
In a Grothendieck category each object has an injective envelope, and for this reason Grothendieck categories are well suited for use in homological applications.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "Sur quelques points d'algèbre homologique"  ''Tôhoku Math. J. (2)'' , '''9'''  (1957)  pp. 119–221</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I. Bucur,  A. Deleanu,  "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. [N. Popescu] Popesco,  P. Gabriel,  "Charactérisation des catégories abéliennes avec générateurs et limites inductives exactes"  ''C.R. Acad. Sci.'' , '''258'''  (1964)  pp. 4188–4190</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Grothendieck,  "Sur quelques points d'algèbre homologique"  ''Tôhoku Math. J. (2)'' , '''9'''  (1957)  pp. 119–221</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top">  I. Bucur,  A. Deleanu,  "Introduction to the theory of categories and functors" , Wiley  (1968)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  N. [N. Popescu] Popesco,  P. Gabriel,  "Charactérisation des catégories abéliennes avec générateurs et limites inductives exactes"  ''C.R. Acad. Sci.'' , '''258'''  (1964)  pp. 4188–4190</TD></TR>
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</table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Popescu,  "Abelian categories with applications to rings and modules" , Acad. Press  (1973)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Popescu,  "Abelian categories with applications to rings and modules" , Acad. Press  (1973)</TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 19:42, 30 October 2016

An Abelian category with a set of generators (cf. Generator of a category) and satisfying the following axiom: There exist coproducts (sums) of arbitrary families of objects, and for each directed family of subobjects $U_i$, $i \in I$, of an object $A$, and any subobject $V$, the following equality holds: $$ \left({ \bigcup_{i \in I} U_i }\right) \cap V = \bigcup_{i \in I} \left({ U_i \cap V }\right) $$

The category of modules (left or right) over an arbitrary associative ring $R$ with an identity element and the category of sheaves of $R$-modules over an arbitrary topological space (cf. Sheaf theory) are Grothendieck categories. A full subcategory $\mathfrak{S}$ of the category ${}_R \mathfrak{M}$ of left $R$-modules is known as a localizing subcategory if it is closed with respect to colimits and if, in an exact sequence $$ 0 \rightarrow A' \rightarrow A \rightarrow A''' \rightarrow 0 $$ the object $A$ belongs to $\mathfrak{S}$ if and only if both $A'$ and $A''$ belong to $\mathfrak{S}$. Each localizing subcategory makes it possible to construct the quotient category ${}_R \mathfrak{M} / \mathfrak{S}$. An Abelian category is a Grothendieck category if and only if it is equivalent to some quotient category of the type ${}_R \mathfrak{M} / \mathfrak{S}$.

In a Grothendieck category each object has an injective envelope, and for this reason Grothendieck categories are well suited for use in homological applications.

References

[1] A. Grothendieck, "Sur quelques points d'algèbre homologique" Tôhoku Math. J. (2) , 9 (1957) pp. 119–221
[2] I. Bucur, A. Deleanu, "Introduction to the theory of categories and functors" , Wiley (1968)
[3] N. [N. Popescu] Popesco, P. Gabriel, "Charactérisation des catégories abéliennes avec générateurs et limites inductives exactes" C.R. Acad. Sci. , 258 (1964) pp. 4188–4190


Comments

References

[a1] N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973)
How to Cite This Entry:
Grothendieck category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Grothendieck_category&oldid=14714
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article