Namespaces
Variants
Actions

Difference between revisions of "Serre subcategory"

From Encyclopedia of Mathematics
Jump to: navigation, search
(one references section)
m (→‎References: + ZBL link)
 
Line 51: Line 51:
 
====References====
 
====References====
 
<table>
 
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  J.-.P. Serre,  "Groupes d'homotopie et classes de groupes abéliens"  ''Ann. of Math.'' , '''58''' :  2  (1953)  pp. 258–294</TD></TR>
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  J.-.P. Serre,  "Groupes d'homotopie et classes de groupes abéliens"  ''Ann. of Math.'' , '''58''' :  2  (1953)  pp. 258–294 {{ZBL|0052.19303}}</TD></TR>
 
<TR><TD valign="top">[2]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR>
 
<TR><TD valign="top">[2]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules, and categories" , '''1''' , Springer  (1973)</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top">  N. Popesco,  P. Gabriel,  "Caractérisations des catégories abéliennes avec générateurs et limites inductives exactes"  ''C.R. Acad. Sci. Paris'' , '''258''' :  17  (1964)  pp. 4188–4190</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top">  N. Popesco,  P. Gabriel,  "Caractérisations des catégories abéliennes avec générateurs et limites inductives exactes"  ''C.R. Acad. Sci. Paris'' , '''258''' :  17  (1964)  pp. 4188–4190</TD></TR>
 
<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Popescu,  "Abelian categories with applications to rings and modules" , Acad. Press  (1973)</TD></TR>
 
<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>
 
</table>

Latest revision as of 16:39, 17 March 2023


A locally small full subcategory $ \mathfrak S $ of an Abelian category $ \mathfrak A $ such that for every exact sequence

$$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0 $$

in $ \mathfrak A $ it is the case that $ B \in \mathfrak S $ if and only if $ A \in \mathfrak S $ and $ C \in \mathfrak S $. In this context, local smallness of a category is the condition: A collection of representatives of the isomorphism classes of subobjects of any object forms a set. Serre subcategories can be characterized as kernels of functors defined on $ \mathfrak A $.

Given a Serre subcategory, one can define the quotient category $ \mathfrak A / \mathfrak S $, whose objects are the objects of $ \mathfrak A $ and whose morphisms are defined by

$$ \mathop{\rm Mor} _ {\mathfrak A / \mathfrak S } ( X, Y) = \ \lim\limits _ { {Y ^ \prime , X/X ^ \prime \in \mathfrak S } vec } \ \mathop{\rm Mor} _ {\mathfrak A } ( X ^ \prime , Y/Y ^ \prime ) . $$

The quotient category $ \mathfrak A / \mathfrak S $ is Abelian.

A Serre subcategory is called localizing if the canonical functor $ T: \mathfrak A \rightarrow \mathfrak A / \mathfrak S $ has a right adjoint $ S: \mathfrak A / \mathfrak S \rightarrow \mathfrak A $, called the section functor. If $ \mathfrak A $ is a Grothendieck category with coproducts, then a Serre subcategory is localizing if and only if it is closed under coproducts. Thus one obtains a generalization of the classical theory of localization of modules over a commutative ring. This method embraces many constructions of rings of fractions and torsion theories (radicals) of modules over associative rings.

The concept of a Serre subcategory was introduced by J.-P. Serre [1], who called them classes. By using this concept he obtained a far-reaching generalization of a theorem of Hurewicz (see Homotopy group).

Comments

Serre subcategories are also called thick subcategories or dense subcategories. See also Localization in categories.

References

[1] J.-.P. Serre, "Groupes d'homotopie et classes de groupes abéliens" Ann. of Math. , 58 : 2 (1953) pp. 258–294 Zbl 0052.19303
[2] C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973)
[3] N. Popesco, P. Gabriel, "Caractérisations des catégories abéliennes avec générateurs et limites inductives exactes" C.R. Acad. Sci. Paris , 258 : 17 (1964) pp. 4188–4190
[a1] N. Popescu, "Abelian categories with applications to rings and modules" , Acad. Press (1973)
How to Cite This Entry:
Serre subcategory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Serre_subcategory&oldid=52847
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article