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  "body": "\n## Summary\n\nAn algebra $B$ over a commutative ring $A$ is finitely presented if it can be described by finitely many generators and finitely many relations.\nThat is, $B \\cong A[x_1, \\ldots, x_n]/I$ with $I$ a finitely generated ideal.\nOver a Noetherian ring every [[finite type algebra|AC5TFFQT9M]] is finitely presented, so the distinction matters only over non-Noetherian bases.\nFinite presentation is the right finiteness condition for limit arguments and for étale and smooth morphisms in algebraic geometry.\n\n## Explanations\n\n### L1 human\n\nTo describe a ring by a finite amount of data, one lists some generators and some polynomial equations between them.\nFor example, $\\Z[i]$ is $\\Z[x]/(x^2 + 1)$: one generator and one relation.\nFinite type asks only for finitely many generators, while finite presentation also asks that finitely many equations imply all the others.\nOver a ring like $\\Z$ or a field, the second demand is automatic.\n",
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