(also nonabelian homological algebra)
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A full subcategory $T$ of an abelian category $A$ is a Serre subcategory if it is nonempty and for any exact sequence
$M'$ is in $T$ if $M$ and $M''$ are in $T$.
Alternative definition: a full subcategory $T$ of an abelian category such that for every short exact sequence
where the colimit is over all $X',Y'$ in $A$ such that $Y'$ and $X/X'$ are in $T$. The quotient category $A/T$ is abelian.
Very often the same definition of Serre subcategory is used in an arbitrary abelian category $A$ (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 $A$ one denotes by $\bar{T}$ the full subcategory of $A$ generated by all objects $N$ for which any (nonzero) subquotient of $N$ in $T$ has a (nonzero) subobject from $T$. This becomes an idempotent operation on the class of subcategories of $A$ with $T\subset \bar{T}$ iff $T$ is topologizing. Moreover $\bar{T}$ 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 $T\mapsto\bar{T}$.
Carl Faith, Algebra: Rings, Modules, Categories, vol. 1, Chapter 15, Grundlehren der mathematischen Wissenschaften 190, Springer (1973) [doi:10.1007/978-3-642-80634-6]
Last revised on August 28, 2022 at 17:34:26. See the history of this page for a list of all contributions to it.