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  "body": "\n## Summary\n\nA Cohen–Macaulay ring is a Noetherian ring whose local rings have [[depth|M2RADBFW99]] equal to their [[Krull dimension|MVGBS3J1P7]].\nThis class contains [[regular local rings|63XXTF6B5M]], [[complete intersection rings|EPM2HKGMZS]] and [[Gorenstein rings|T6HCX959PJ]].\nCohen–Macaulay rings are the setting where dimension theory works best, since systems of parameters are regular sequences.\nThey also have a good duality theory through the [[canonical module|W5ECXPM6FW]].\n\n## Explanations\n\n### L1 human\n\nThe dimension of a ring counts its independent directions, and the depth counts how many independent equations can be imposed one after the other without hitting a zero divisor.\nIn a Cohen–Macaulay ring these two counts agree at every point.\nPolynomial rings over a field are Cohen–Macaulay, and so is the ring of the two coordinate axes $K[x,y]/(xy)$.\nThe ring $K[x,y]/(x^2, xy)$, a line with an extra \"fuzzy\" point at the origin, is not.\n",
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