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  "body": "\n## Summary\n\nA regular sequence is a list of ring elements in which each element is a non-zero-divisor modulo the earlier ones.\nIt is the algebraic version of a set of independent equations, like the coordinates $x_1, \\ldots, x_n$ of a polynomial ring.\nRegular sequences define the [[depth|M2RADBFW99]] of a module, and the [[Koszul complex|YZY11W08SD]] of a regular sequence is a free resolution.\n\n## Explanations\n\n### L1 human\n\nCut a space by one equation, then cut the result by a second equation, and so on.\nThe sequence of equations is regular when each new equation genuinely cuts down what is left, and never vanishes on a whole piece of it.\nIn $K[x, y]$ the sequence $x, y$ is regular: $x$ is a non-zero-divisor in $K[x,y]$, and $y$ is a non-zero-divisor in $K[x,y]/(x) \\cong K[y]$.\nThe sequence $x, xy$ is not regular, because $xy$ is already zero modulo $x$.\n",
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