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  "body": "\n## Summary\n\nThe associated graded ring of a [[commutative ring|0310REJJY8]] $R$ with respect to an [[ideal|096Z9MP2ME]] $I$ is the [[graded ring|8KB4SQGC0C]] $\\operatorname{gr}_I(R) = \\bigoplus_{n \\ge 0} I^n/I^{n+1}$.\nIt keeps the leading terms of elements and forgets the rest.\nFor a local ring and its maximal ideal it describes the tangent cone of the point.\nSince it is graded, tools for graded rings such as the [[Hilbert polynomial|RMH4FBCFDA]] apply to local rings through it.\n\n## Explanations\n\n### L1 human\n\nNear a point, a curve looks to first approximation like its tangent lines.\nThe associated graded ring records only these first approximations, by keeping the lowest-degree part of each function.\nFor the integers and a prime $p$, the piece $p^n\\Z/p^{n+1}\\Z$ records one digit in base $p$, and these pieces together form the polynomial ring $\\F_p[t]$.\nThe [[definition|XBZ99KH3RP]] makes this precise.\n",
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