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  "body": "\n## Statement\n\nLet $(R, \\mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]].\nFor $n \\ge 0$ let $\\chi(n) = \\ell(R/\\mathfrak m^n)$ be the [[length|CZSCNZH8VY]] of $R/\\mathfrak m^n$; this is the [[Hilbert–Samuel function|BNQVS83GZV]] of $R$.\nThen $\\chi(n)$ agrees with a polynomial in $n$ for all large $n$, and the following three integers are equal:\n\n1. the [[Krull dimension|MVGBS3J1P7]] $\\dim R$;\n2. the least number $\\delta(R)$ of elements of $\\mathfrak m$ that generate an [[$\\mathfrak m$-primary ideal|TMTPWVETQK]];\n3. the degree $d(R)$ of the polynomial that agrees with $\\chi(n)$ for large $n$.\n\n## Remarks\n\nThe usual proof shows $\\delta(R) \\ge d(R) \\ge \\dim R \\ge \\delta(R)$.\nThe first inequality holds because the Hilbert–Samuel polynomial of an ideal generated by $s$ elements has degree at most $s$, and its degree does not depend on the $\\mathfrak m$-primary ideal.\nThe second goes by induction on $d(R)$.\nTake a chain of primes $\\mathfrak p_0 \\subsetneq \\cdots \\subsetneq \\mathfrak p_r$ and an element $x \\in \\mathfrak p_1 \\setminus \\mathfrak p_0$.\nThen the ring $R/(\\mathfrak p_0 + (x))$ has a smaller $d$ and still contains a chain of length $r - 1$.\nThe third goes by building, with the [[prime avoidance lemma|FMP5AWG6PR]], elements $x_1, \\dots, x_{\\dim R}$ such that each $x_i$ lies outside every prime of height $i - 1$ that contains $x_1, \\dots, x_{i-1}$.\nFor example, for $R = k[x, y]_{(x, y)}$ with $k$ a field, $\\chi(n) = n(n+1)/2$, of degree $2$, the ideal $\\mathfrak m = (x, y)$ needs two generators, and $\\dim R = 2$.\nConsequences: $\\dim R \\le \\dim_k \\mathfrak m/\\mathfrak m^2$ with $k = R/\\mathfrak m$, so $\\dim R$ is finite, and [[Krull's height theorem|TBRTZMHFTV]] follows.\n\n## Explanations\n\n### L1 human\n\nThere are three natural ways to measure the size of a local ring.\nOne counts the longest chain of prime ideals.\nOne counts how many equations are needed to cut out the point.\nOne measures how fast the ring grows modulo higher and higher powers of the maximal ideal.\nThe theorem says that the three counts always give the same number.\n\n### L2 human\n\nThe theorem turns the order-theoretic Krull dimension into something computable.\nThe number $\\delta(R)$ gives upper bounds, since any $\\mathfrak m$-primary ideal with $s$ generators shows $\\dim R \\le s$.\nThe degree $d(R)$ connects dimension with [[Hilbert polynomials|RMH4FBCFDA]] and with the [[multiplicity|JX20MNDDXT]] of the ring.\nA list of $\\dim R$ elements that generates an $\\mathfrak m$-primary ideal is a [[system of parameters|95QTEPMDA4]].\n",
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