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  "body": "\n## Definition\n\nLet $(R, \\mathfrak m)$ be a [[Noetherian|0670T59HGE]] [[local ring|MZ02BKPHJA]], let $M$ be a [[finitely generated|05QY1VEBYD]] $R$-module, and let $I$ be an [[$\\mathfrak m$-primary ideal|TMTPWVETQK]] of $R$.\nThe *Hilbert–Samuel function* of $M$ with respect to $I$ is the function $\\chi_M^I \\colon \\N \\to \\N$ given by\n$$\\chi_M^I(n) = \\ell(M/I^n M),$$\nwhere $\\ell$ denotes the [[length|CZSCNZH8VY]] of an $R$-module.\n\n## Remarks\n\nThe length is finite because $M/I^n M$ is a finitely generated module over the Artinian ring $R/I^n$.\nThere is a polynomial $P_M^I$ with rational coefficients, the Hilbert–Samuel polynomial, such that $\\chi_M^I(n) = P_M^I(n)$ for all large $n$.\nIf $M \\neq 0$, its degree is the Krull dimension of $M$, so it does not depend on $I$.\nExample: for $R = k[x, y]_{(x, y)}$, with $k$ a field, $M = R$ and $I = \\mathfrak m$, the module $R/\\mathfrak m^n$ has the $k$-basis of monomials $x^a y^b$ with $a + b \u003c n$, so $\\chi(n) = n(n+1)/2$.\nNon-example: for the same ring and $I = (x)$, which is not $\\mathfrak m$-primary, the module $R/(x)^n$ has infinite length, so the formula does not define a function to $\\N$.\nSome authors use $\\ell(M/I^{n+1} M)$ instead, which shifts the variable by one.\n",
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