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  "body": "\n## Summary\n\nA Serre subcategory of an [[abelian category|Y9CZ3MTNF1]] is a nonempty [[full subcategory|V3BFGKBJF7]] closed under subobjects, quotient objects and extensions.\nThese are the subcategories that can be \"divided out\".\nFor each Serre subcategory there is a quotient abelian category in which its objects become zero.\nThe objects sent to zero by an [[exact functor|S1G61TDW0K]] always form a Serre subcategory.\n\n## Explanations\n\n### L1 human\n\nThink of a Serre subcategory as a class of objects that one is willing to ignore.\nFor this to be consistent, the class must contain every piece of its objects and everything glued together from two of its objects.\nThe torsion abelian groups form such a class: a subgroup or quotient of a torsion group is torsion, and so is a group built from two torsion groups.\nDividing out the torsion groups leaves a category in which a map of abelian groups counts as an isomorphism when it becomes one after tensoring with $\\Q$.\n",
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  "msc": [
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  "name": "Serre subcategory",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/CSD/CSD6W0JV0P.md",
  "status": null,
  "tag": "CSD6W0JV0P",
  "uri": "m2://CSD6W0JV0P",
  "url": "https://tamnd.github.io/math2/wiki/serre-subcategory/",
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    "tag": "CSD6W0JV0P",
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