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<title>Hilbert–Samuel function</title>

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Hilbert–Samuel function

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a> the Hilbert–Samuel function, named after <a href="David_Hilbert" title="David Hilbert">David Hilbert</a> and <a href="Pierre_Samuel" title="Pierre Samuel">Pierre Samuel</a>,<a href="#cite_note-1">[1]</a> of a nonzero <a href="Finitely_generated_module" title="Finitely generated module">finitely generated</a> <a href="Module_(mathematics)" title="Module (mathematics)">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> over a commutative <a href="Noetherian_ring" title="Noetherian ring">Noetherian</a> <a href="Local_ring" title="Local ring">local ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> and a <a href="Primary_ideal" title="Primary ideal">primary ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi _{M}^{I}:\mathbb {N} \rightarrow \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle \chi _{M}^{I}:\mathbb {N} \rightarrow \mathbb {N} }</annotation>

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</math><img src="./8c5cf0b4ba94f9b3cc2762088861097509ba0eff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.322ex; height:3.176ex;" alt="{\displaystyle \chi _{M}^{I}:\mathbb {N} \rightarrow \mathbb {N} }" loading="lazy"> such that, for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\in \mathbb {N} }">

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<annotation encoding="application/x-tex">{\displaystyle n\in \mathbb {N} }</annotation>

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</math><img src="./d059936e77a2d707e9ee0a1d9575a1d693ce5d0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle n\in \mathbb {N} }" loading="lazy">,

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<annotation encoding="application/x-tex">{\displaystyle \chi _{M}^{I}(n)=\ell (M/I^{n}M)}</annotation>

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</math><img src="./20ee32ac41ec62d9a00c1736010549f04ef4d06c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.977ex; height:3.176ex;" alt="{\displaystyle \chi _{M}^{I}(n)=\ell (M/I^{n}M)}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ell }">

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<annotation encoding="application/x-tex">{\displaystyle \ell }</annotation>

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</math><img src="./f066e981e530bacc07efc6a10fa82deee985929e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }" loading="lazy"> denotes the <a href="Length_of_a_module" title="Length of a module">length</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">. It is related to the <a href="Hilbert_function" class="mw-redirect" title="Hilbert function">Hilbert function</a> of the <a href="Associated_graded_module" class="mw-redirect" title="Associated graded module">associated graded module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {gr} _{I}(M)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {gr} _{I}(M)}</annotation>

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</math><img src="./cea15c68f6e93b7f02bee015ae4779ad9c61c2b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.386ex; height:2.843ex;" alt="{\displaystyle \operatorname {gr} _{I}(M)}" loading="lazy"> by the identity

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<annotation encoding="application/x-tex">{\displaystyle \chi _{M}^{I}(n)=\sum _{i=0}^{n}H(\operatorname {gr} _{I}(M),i).}</annotation>

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</math><img src="./f88dbf1f804244c8b2018cd2d69e42b65e26f988.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.201ex; height:6.843ex;" alt="{\displaystyle \chi _{M}^{I}(n)=\sum _{i=0}^{n}H(\operatorname {gr} _{I}(M),i).}" loading="lazy"></dd></dl>

For sufficiently large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy">, it coincides with a polynomial function of degree equal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dim(\operatorname {gr} _{I}(M))}">

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<annotation encoding="application/x-tex">{\displaystyle \dim(\operatorname {gr} _{I}(M))}</annotation>

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</math><img src="./afe25a4c171615b52d776e81bd0cd6a2636b95dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.071ex; height:2.843ex;" alt="{\displaystyle \dim(\operatorname {gr} _{I}(M))}" loading="lazy">, often called the Hilbert-Samuel polynomial (or <a href="Hilbert_polynomial" class="mw-redirect" title="Hilbert polynomial">Hilbert polynomial</a>).<a href="#cite_note-ica-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

For the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of <a href="Formal_power_series" title="Formal power series">formal power series</a> in two variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[[x,y]]}">

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<annotation encoding="application/x-tex">{\displaystyle k[[x,y]]}</annotation>

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</math><img src="./ac7c20ba30a3d3487d56a835a6a467e95f48106d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.318ex; height:2.843ex;" alt="{\displaystyle k[[x,y]]}" loading="lazy"> taken as a module over itself and the ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

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</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> generated by the monomials x2 and y3 we have

<mtext> and in general </mtext>

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<annotation encoding="application/x-tex">{\displaystyle \chi (1)=6,\quad \chi (2)=18,\quad \chi (3)=36,\quad \chi (4)=60,{\text{ and in general }}\chi (n)=3n(n+1){\text{ for }}n\geq 0.}</annotation>

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</math><img src="./b0633f28b642ccfc8d688d81fd9b9d9956f82b3c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:92.363ex; height:2.843ex;" alt="{\displaystyle \chi (1)=6,\quad \chi (2)=18,\quad \chi (3)=36,\quad \chi (4)=60,{\text{ and in general }}\chi (n)=3n(n+1){\text{ for }}n\geq 0.}" loading="lazy"><a href="#cite_note-ica-2">[2]</a></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Degree_bounds">Degree bounds</h2></div>

Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the <a href="Artin%E2%80%93Rees_lemma" title="Artin–Rees lemma">Artin–Rees lemma</a>. We denote by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{I,M}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{I,M}}</annotation>

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</math><img src="./1f0ed171ae4676ee8b83b1e5cded8e966c6b994d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.737ex; height:2.843ex;" alt="{\displaystyle P_{I,M}}" loading="lazy"> the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.

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Theorem—Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R,m)}">

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<annotation encoding="application/x-tex">{\displaystyle (R,m)}</annotation>

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</math><img src="./b34d74345d9bbb0bf3615696ba322f6d74e5bbb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.648ex; height:2.843ex;" alt="{\displaystyle (R,m)}" loading="lazy"> be a Noetherian local ring and I an m-<a href="Primary_ideal" title="Primary ideal">primary ideal</a>. If

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<annotation encoding="application/x-tex">{\displaystyle 0\to M'\to M\to M''\to 0}</annotation>

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</math><img src="./1a12da2cdc62593c2b3d9e1a796698a052f8ccab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.043ex; height:2.509ex;" alt="{\displaystyle 0\to M'\to M\to M''\to 0}" loading="lazy"></dd></dl>

is an exact sequence of finitely generated R-modules and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M/IM}">

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<annotation encoding="application/x-tex">{\displaystyle M/IM}</annotation>

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</math><img src="./0765180478a3f248c4ec9b270df59ab95c18b0a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.219ex; height:2.843ex;" alt="{\displaystyle M/IM}" loading="lazy"> has finite length,<a href="#cite_note-3">[3]</a> then we have:<a href="#cite_note-4">[4]</a>

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<annotation encoding="application/x-tex">{\displaystyle P_{I,M}=P_{I,M'}+P_{I,M''}-F}</annotation>

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</math><img src="./aeb95085b51199deb89f995b80c500359a628c34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:26.243ex; height:3.009ex;" alt="{\displaystyle P_{I,M}=P_{I,M'}+P_{I,M''}-F}" loading="lazy"></dd></dl>

where F is a polynomial of degree strictly less than that of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{I,M'}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{I,M'}}</annotation>

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</math><img src="./e2bbb6ba1d0219cb952b45137c36fc19165450d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.309ex; height:3.009ex;" alt="{\displaystyle P_{I,M'}}" loading="lazy"> and having positive leading coefficient. In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M'\simeq M}">

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<annotation encoding="application/x-tex">{\displaystyle M'\simeq M}</annotation>

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</math><img src="./7ad64ce88515666e6318b1464e603b5da8fc4c60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.724ex; height:2.509ex;" alt="{\displaystyle M'\simeq M}" loading="lazy">, then the degree of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{I,M''}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{I,M''}}</annotation>

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</math><img src="./5ba3e72d2c7be16f190124f847f55a1ad5b6b99a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.676ex; height:3.009ex;" alt="{\displaystyle P_{I,M''}}" loading="lazy"> is strictly less than that of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{I,M}=P_{I,M'}}">

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<annotation encoding="application/x-tex">{\displaystyle P_{I,M}=P_{I,M'}}</annotation>

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</math><img src="./1c6ab187e1d2715488d02fdb46fb45173e6c4d23.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.145ex; height:3.009ex;" alt="{\displaystyle P_{I,M}=P_{I,M'}}" loading="lazy">.

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Proof: Tensoring the given exact sequence with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/I^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle R/I^{n}}</annotation>

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</math><img src="./450dc301658614a35b0cc329114812e4055cf50f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.361ex; height:2.843ex;" alt="{\displaystyle R/I^{n}}" loading="lazy"> and computing the kernel we get the exact sequence:

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<annotation encoding="application/x-tex">{\displaystyle 0\to (I^{n}M\cap M')/I^{n}M'\to M'/I^{n}M'\to M/I^{n}M\to M''/I^{n}M''\to 0,}</annotation>

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</math><img src="./06ef71929b71e73d44391e670ff156f3cd14089f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.591ex; height:3.009ex;" alt="{\displaystyle 0\to (I^{n}M\cap M')/I^{n}M'\to M'/I^{n}M'\to M/I^{n}M\to M''/I^{n}M''\to 0,}" loading="lazy"></dd></dl>

which gives us:

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<annotation encoding="application/x-tex">{\displaystyle \chi _{M}^{I}(n-1)=\chi _{M'}^{I}(n-1)+\chi _{M''}^{I}(n-1)-\ell ((I^{n}M\cap M')/I^{n}M')}</annotation>

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</math><img src="./c1540d86be5089b7ac2f1fd109f7e3f7fed318bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:64.165ex; height:3.343ex;" alt="{\displaystyle \chi _{M}^{I}(n-1)=\chi _{M'}^{I}(n-1)+\chi _{M''}^{I}(n-1)-\ell ((I^{n}M\cap M')/I^{n}M')}" loading="lazy">.</dd></dl>

The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large n and some k,

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<annotation encoding="application/x-tex">{\displaystyle I^{n}M\cap M'=I^{n-k}((I^{k}M)\cap M')\subset I^{n-k}M'.}</annotation>

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</math><img src="./2243c601e8482a93277741904194c15a9f347151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.942ex; height:3.176ex;" alt="{\displaystyle I^{n}M\cap M'=I^{n-k}((I^{k}M)\cap M')\subset I^{n-k}M'.}" loading="lazy"></dd></dl>

Thus,

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<annotation encoding="application/x-tex">{\displaystyle \ell ((I^{n}M\cap M')/I^{n}M')\leq \chi _{M'}^{I}(n-1)-\chi _{M'}^{I}(n-k-1)}</annotation>

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</math><img src="./947fd58b1fbdfcccb69b4601b3ef8023eb14a7dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:54.388ex; height:3.343ex;" alt="{\displaystyle \ell ((I^{n}M\cap M')/I^{n}M')\leq \chi _{M'}^{I}(n-1)-\chi _{M'}^{I}(n-k-1)}" loading="lazy">.</dd></dl>

This gives the desired degree bound.

<div class="mw-heading mw-heading2"><h2 id="Multiplicity">Multiplicity</h2></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

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<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

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</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a local ring of Krull dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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<annotation encoding="application/x-tex">{\displaystyle d}</annotation>

</semantics>

</math><img src="./e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">, with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy">-primary ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy">, its Hilbert polynomial has leading term of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {e}{d!}}\cdot n^{d}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {e}{d!}}\cdot n^{d}}</annotation>

</semantics>

</math><img src="./b57f92b7c19f61e360c26cd000afad0ea79932b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:6.865ex; height:4.843ex;" alt="{\displaystyle {\frac {e}{d!}}\cdot n^{d}}" loading="lazy"> for some integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy">. This integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> is called the multiplicity of the ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I}</annotation>

</semantics>

</math><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy">. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I=m}</annotation>

</semantics>

</math><img src="./25a6fa5860e46d039b3271cad82876aba8597aae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.311ex; height:2.176ex;" alt="{\displaystyle I=m}" loading="lazy"> is the maximal ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">, one also says <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> is the multiplicity of the local ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy">.

The multiplicity of a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> of a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is defined to be the multiplicity of the corresponding local ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X,x}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X,x}}</annotation>

</semantics>

</math><img src="./0617c163a03d1584ad15e192f2d488ffaaef7d0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.88ex; height:2.843ex;" alt="{\displaystyle {\mathcal {O}}_{X,x}}" loading="lazy">.

<ul><li><a href="J-multiplicity" title="J-multiplicity">j-multiplicity</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203.

</li>

<li id="cite_note-ica-2">^ <a href="#cite_ref-ica_2-0">a</a> <a href="#cite_ref-ica_2-1">b</a> Atiyah, M. F. and MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison–Wesley, 1969.

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> This implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M'/IM'}">

<msup>

</msup>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M'/IM'}</annotation>

</semantics>

</math><img src="./9b0d481962004375c6c5421b796e86408a5452e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.701ex; height:3.009ex;" alt="{\displaystyle M'/IM'}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M''/IM''}">

<msup>

</msup>

</mrow>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M''/IM''}</annotation>

</semantics>

</math><img src="./fc5cfa4aadeed0fcb4454223b03d64b28e1ffe03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.606ex; height:3.009ex;" alt="{\displaystyle M''/IM''}" loading="lazy"> also have finite length.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="David_Eisenbud" title="David Eisenbud">Eisenbud, David</a>, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-94268-8</bdi>. Lemma 12.3.

</li>

</ol></div></div><div><div class="zim-footer">

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