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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]].\nA *Serre subcategory* of $\\mathcal{A}$ is a nonempty [[full subcategory|V3BFGKBJF7]] $\\mathcal{C}$ of $\\mathcal{A}$ with the following property.\nFor every [[short exact sequence|KJKNJMGKZZ]] $0 \\to A' \\to A \\to A'' \\to 0$ in $\\mathcal{A}$, the object $A$ lies in $\\mathcal{C}$ if and only if $A'$ and $A''$ lie in $\\mathcal{C}$.\n\n## Remarks\n\nEquivalently, $\\mathcal{C}$ is nonempty and closed under subobjects, quotient objects and extensions.\nIt then contains a zero object and is closed under isomorphisms, and it is itself an abelian category with exact inclusion.\nIn a Grothendieck category, a Serre subcategory closed under arbitrary direct sums is called a *localizing subcategory*.\n\nExample: the torsion abelian groups form a Serre subcategory of the category of abelian groups.\nSubgroups and quotients of torsion groups are torsion.\nIf $0 \\to A' \\to A \\to A'' \\to 0$ is exact with $A'$ and $A''$ torsion and $a \\in A$, then $na$ maps to $0$ in $A''$ for some $n \\ge 1$, so $na \\in A'$ and $mna = 0$ for some $m \\ge 1$.\n\nNon-example: the free abelian groups do not form a Serre subcategory.\nThe sequence $0 \\to \\Z \\xrightarrow{2} \\Z \\to \\Z/2\\Z \\to 0$ is exact with $\\Z$ free, but the quotient $\\Z/2\\Z$ is not free.\n",
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