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Serre subcategory

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homological algebra

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nonabelian homological algebra

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Definition

A (nonempty) full subcategory TT of an abelian category AMod_{A}Mod of (say left) modules over a ring AA is a Serre subcategory if for any exact sequence

0MMM0 0\to M\to M'\to M''\to 0

MM' is in TT iff MM and MM'' are in TT.

Following Serre, one then defines the category A/TA/T whose objects are the objects of AA and where the morphisms in A/TA/T are defined by

Hom A/T(X,Y):=colimHom A(X,Y/Y),\mathrm{Hom}_{A/T}(X,Y) := \mathrm{colim}\, \mathrm{Hom}_A(X',Y/Y'),

where the colimit is over all X,YX',Y' in AA such that YY' and X/XX/X' are in TT. The quotient category A/TA/T is abelian.

Very often the same definition of Serre subcategory is used in an arbitrary abelian category AA (we will say in that case weakly Serre subcategory); but in fact, at least when the abelian category is not a Grothendieck category, it is more appropriate to ask for an additional condition in the definition of Serre subcategory, so that the standard theorems on correspondences with other canonical data in localization theory remain valid.

To this aim, for any subcategory of an arbitrary abelian category AA one denotes by T¯\bar{T} the full subcategory of AA generated by all objects NN for which any (nonzero) subquotient of NN in TT has a (nonzero) subobject from TT. This becomes an idempotent operation on the class of subcategories of AA with TT¯T\subset \bar{T} iff TT is topologizing. Moreover T¯\bar{T} is always thick in the stronger sense (that is, thick and topologizing).

Serre subcategories in the strong sense are those nonempty full subcategories which are stable under the operation TT¯T\mapsto\bar{T}.

Revised on November 19, 2011 15:13:45 by Urs Schreiber (82.113.99.53)