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  "body": "\n## Summary\n\nA finitely presented module is a module given by finitely many generators and finitely many relations among them.\nIt is the cokernel of a linear map between two finitely generated [[free modules|0DZ9DKAKZR]], so it is described by a finite matrix.\nEvery finitely presented module is [[finitely generated|05QY1VEBYD]], and over a [[Noetherian ring|0670T59HGE]] the two notions agree.\nMany statements about modules, such as the behaviour of $\\Hom$ under localization, need finite presentation rather than finite generation.\n\n## Explanations\n\n### L1 human\n\nA module is finitely presented when you can describe it completely with a finite list of generators and a finite list of rules.\nFor example, $\\Z/6\\Z$ has one generator $e$ and one rule $6e = 0$.\nA presentation is like a finite recipe for building the module from a free module.\nOver familiar rings, such as $\\Z$ or polynomial rings in finitely many variables over a field, every finitely generated module has such a recipe.\n",
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