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  "body": "\n## Summary\n\nThe depth of a module $M$ along an ideal $I$ is the common length of the maximal [[regular sequences|69R5R2FCS5]] on $M$ inside $I$.\nFor a Noetherian local ring it is bounded by the [[Krull dimension|MVGBS3J1P7]], and the rings where equality holds are the [[Cohen–Macaulay rings|N1QJQEGBFC]].\nDepth can be computed with the [[Ext functor|4P7GBWJ4RR]], with [[Koszul complexes|YZY11W08SD]] or with [[local cohomology|6JB30BXDA2]].\n\n## Explanations\n\n### L1 human\n\nDepth counts how many independent equations one can impose, one after the other, without any of them being a zero divisor.\nDimension counts how many independent directions a space has.\nFor nice rings, like polynomial rings, the two numbers agree.\nFor the ring $K[x,y]/(x^2, xy)$ near the origin, the dimension is $1$ but the depth is $0$, because every element vanishing at the origin is killed by $x$.\n",
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