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  "body": "\n## Summary\n\nA Noetherian scheme is a [[scheme|5A35TYZD02]] that is covered by finitely many affine pieces, each the spectrum of a [[Noetherian ring|0670T59HGE]].\nIt is the finiteness condition under which most of the standard theory of schemes works well.\nAll varieties over a field and all schemes of finite type over $\\Z$ are Noetherian.\n\n## Explanations\n\n### L1 human\n\nA Noetherian ring is one where every ideal has finitely many generators, so every system of polynomial equations is equivalent to a finite one.\nA Noetherian scheme is built from finitely many pieces of this kind.\nThe finiteness has geometric consequences, for example the scheme splits into finitely many irreducible components.\nThe spectrum of a polynomial ring in infinitely many variables is a typical scheme that is not Noetherian.\n",
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