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  "body": "\n## Summary\n\nA complete intersection ring is a local ring that is, after completion, a [[regular local ring|63XXTF6B5M]] divided by an ideal generated by a [[regular sequence|69R5R2FCS5]].\nIt is the algebraic version of a space cut out by exactly as many equations as its codimension.\nEvery complete intersection ring is [[Gorenstein|T6HCX959PJ]], and hence [[Cohen–Macaulay|N1QJQEGBFC]].\n\n## Explanations\n\n### L1 human\n\nIn three-dimensional space, a surface is usually cut out by one equation and a curve by two.\nWhen a space of codimension $r$ is cut out by exactly $r$ equations, it is a complete intersection.\nThe plane curve $xy = 0$, the union of two axes, is a complete intersection: one equation in the plane.\nThe three coordinate axes in space form a curve that needs three equations, $xy = yz = zx = 0$, so they are not a complete intersection.\n",
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