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<title>Complete intersection ring</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Complete intersection ring</span></span>

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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"><p>In <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a>, a <b>complete intersection ring</b> is a <a href="Commutative_ring" title="Commutative ring">commutative ring</a> similar to the <a href="Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate rings</a> of varieties that are <a href="Complete_intersection" title="Complete intersection">complete intersections</a>. Informally, they can be thought of roughly as the <a href="Local_ring" title="Local ring">local rings</a> that can be defined using the "minimum possible" number of relations.

</p><p>For Noetherian local rings, there is the following chain of inclusions:

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<dl><dd><b><a href="Universally_catenary_ring" class="mw-redirect" title="Universally catenary ring">Universally catenary rings</a></b> ⊃ <b><a href="Cohen%E2%80%93Macaulay_ring" title="Cohen–Macaulay ring">Cohen–Macaulay rings</a></b> ⊃ <b><a href="Gorenstein_ring" title="Gorenstein ring">Gorenstein rings</a></b> ⊃ <b></b> ⊃ <b><a href="Regular_local_ring" title="Regular local ring">regular local rings</a></b></dd></dl>

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

<p>A local complete intersection ring is a <a href="Noetherian_ring" title="Noetherian ring">Noetherian</a> <a href="Local_ring" title="Local ring">local ring</a> whose <a href="Completion_(ring_theory)" class="mw-redirect" title="Completion (ring theory)">completion</a> is the quotient of a <a href="Regular_local_ring" title="Regular local ring">regular local ring</a> by an ideal generated by a <a href="Regular_sequence" title="Regular sequence">regular sequence</a>. Taking the completion is a minor technical complication caused by the fact that not all local rings are quotients of regular ones. For rings that are quotients of regular local rings, which covers most local rings that occur in algebraic geometry, it is not necessary to take completions in the definition.

</p><p>There is an alternative intrinsic definition that does not depend on embedding the ring in a regular local ring.

If <i>R</i> is a Noetherian local ring with maximal ideal <i>m</i>, then the dimension of <i>m</i>/<i>m</i><sup>2</sup> is called the <b>embedding dimension</b> emb dim (<i>R</i>) of <i>R</i>. Define a graded algebra <i>H</i>(<i>R</i>) as the homology of the <a href="Koszul_complex" title="Koszul complex">Koszul complex</a> with respect to a minimal system of generators of <i>m</i>/<i>m</i><sup>2</sup>; up to isomorphism this only depends on <i>R</i> and not on the choice of the generators of <i>m</i>. The dimension of <i>H</i><sub>1</sub>(<i>R</i>) is denoted by ε<sub>1</sub> and is called the <a href="First_deviation" class="mw-redirect" title="First deviation">first deviation</a> of <i>R</i>; it vanishes if and only if <i>R</i> is regular.

A Noetherian local ring is called a <b>complete intersection ring</b> if its

embedding dimension is the sum of the dimension and the first deviation:

</p>

<p>There is also a recursive characterization of local complete intersection rings that can be used as a definition, as follows. Suppose that <i>R</i> is a complete Noetherian local ring. If <i>R</i> has dimension greater than 0 and <i>x</i> is an element in the maximal ideal that is not a zero divisor then <i>R</i> is a complete intersection ring if and only if <i>R</i>/(<i>x</i>) is. (If the maximal ideal consists entirely of zero divisors then <i>R</i> is not a complete intersection ring.) If <i>R</i> has dimension 0, then <a href="#CITEREFWiebe1969">Wiebe (1969)</a> showed that it is a complete intersection ring if and only if the <a href="Fitting_ideal" title="Fitting ideal">Fitting ideal</a> of its maximal ideal is non-zero.

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

<div class="mw-heading mw-heading3"><h3 id="Regular_local_rings">Regular local rings</h3></div>

<p><a href="Regular_local_ring" title="Regular local ring">Regular local rings</a> are complete intersection rings, but the converse is not true: the ring <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x]/(x^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle k[x]/(x^{2})}</annotation>

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</math></span><img src="./152b7c94aa6b8a03eb284cc318ae1cfc2164b8cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.19ex; height:3.176ex;" alt="{\displaystyle k[x]/(x^{2})}" loading="lazy"></span> is a 0-dimensional complete intersection ring that is not regular.

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<div class="mw-heading mw-heading3"><h3 id="Not_a_complete_intersection">Not a complete intersection</h3></div>

<p>An example of a locally complete intersection ring which is not a complete intersection ring is given by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x,y]/(y-x^{2},x^{3})}">

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<annotation encoding="application/x-tex">{\displaystyle k[x,y]/(y-x^{2},x^{3})}</annotation>

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</math></span><img src="./ac119c9b7d89d85997eafa27ce80ec802afaa9b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.793ex; height:3.176ex;" alt="{\displaystyle k[x,y]/(y-x^{2},x^{3})}" loading="lazy"></span>which has length 3 since it is isomorphic as a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> vector space to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\oplus k\cdot x\oplus k\cdot x^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle k\oplus k\cdot x\oplus k\cdot x^{2}}</annotation>

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</math></span><img src="./adda8b669ff7b9d2bc767368ceba211b05523835.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.386ex; height:2.843ex;" alt="{\displaystyle k\oplus k\cdot x\oplus k\cdot x^{2}}" loading="lazy"></span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading3"><h3 id="Counterexample">Counterexample</h3></div>

<p>Complete intersection local rings are <a href="Gorenstein_ring" title="Gorenstein ring">Gorenstein rings</a>, but the converse is not true: the ring <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x,y,z]/(x^{2},y^{2},xz,yz,z^{2}-xy)=R/I}">

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<annotation encoding="application/x-tex">{\displaystyle k[x,y,z]/(x^{2},y^{2},xz,yz,z^{2}-xy)=R/I}</annotation>

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</math></span><img src="./f82da7195df38ebe3716551475ae004199d14c18.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.18ex; height:3.176ex;" alt="{\displaystyle k[x,y,z]/(x^{2},y^{2},xz,yz,z^{2}-xy)=R/I}" loading="lazy"></span> is a 0-dimensional Gorenstein ring that is not a complete intersection ring. As a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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

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</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span>-vector space this ring is isomorphic to

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<annotation encoding="application/x-tex">{\displaystyle {\frac {k[x,y,z]}{(x^{2},y^{2},xz,yz,z^{2}-xy)}}\cong R_{0}\oplus R_{1}\oplus R_{2}}</annotation>

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</math></span><img src="./809f08395b3784eccac59b30cc4f4c83ae1d383f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.745ex; height:6.509ex;" alt="{\displaystyle {\frac {k[x,y,z]}{(x^{2},y^{2},xz,yz,z^{2}-xy)}}\cong R_{0}\oplus R_{1}\oplus R_{2}}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}=k\cdot 1,R_{1}=k\cdot x\oplus k\cdot y\oplus k\cdot z}">

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<annotation encoding="application/x-tex">{\displaystyle R_{0}=k\cdot 1,R_{1}=k\cdot x\oplus k\cdot y\oplus k\cdot z}</annotation>

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</math></span><img src="./ce8f78b50eecbb466538f5ee471782434d44d04b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.845ex; height:2.509ex;" alt="{\displaystyle R_{0}=k\cdot 1,R_{1}=k\cdot x\oplus k\cdot y\oplus k\cdot z}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{2}=k\cdot z^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{2}=k\cdot z^{2}}</annotation>

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</math></span><img src="./3c9e08fd222d9dbd70d83b8d76f1a09ce7b0e67a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.952ex; height:3.009ex;" alt="{\displaystyle R_{2}=k\cdot z^{2}}" loading="lazy"></span></dd></dl>

<p>showing it is Gorenstein since the top-degree component is dimension <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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<annotation encoding="application/x-tex">{\displaystyle 1}</annotation>

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</math></span><img src="./92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span> and it satisfies the Poincare property. It is not a local complete intersection ring because the ideal <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\subset R}">

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

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</math></span><img src="./cf82feb3f84a530f44b000a8e003e00170e10ed5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.176ex;" alt="{\displaystyle I\subset R}" loading="lazy"></span> is not <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>-regular. For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy}">

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

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</math></span><img src="./c72eb345e496513fb8b2fa4aa8c4d89b855f9a01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.485ex; height:2.009ex;" alt="{\displaystyle xy}" loading="lazy"></span> is a zero-divisor to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math></span><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"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/(x^{2},y^{2})}">

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<annotation encoding="application/x-tex">{\displaystyle R/(x^{2},y^{2})}</annotation>

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</math></span><img src="./4906ebb00b527ff86cd671cd2e468a53835f81c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.368ex; height:3.176ex;" alt="{\displaystyle R/(x^{2},y^{2})}" loading="lazy"></span>.

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

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathoverflow.net/q/122874">"Example of locally complete intersection varieties which are not smooth and not complete intersection"</a>. <i><a href="MathOverflow" title="MathOverflow">MathOverflow</a></i><span class="reference-accessdate">. Retrieved <span class="nowrap">2017-01-04</span></span>.</cite></span>

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

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<ul><li><cite id="CITEREFBrunsHerzog1993" class="citation cs2">Bruns, Winfried; Herzog, Jürgen (1993), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LF6CbQk9uScC"><i>Cohen–Macaulay rings</i></a>, Cambridge Studies in Advanced Mathematics, vol. 39, <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-41068-7</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1251956">1251956</a></cite></li>

<li><cite id="CITEREFMajadasRodicio2010" class="citation book cs1">Majadas, Javier; Rodicio, Antonio G. (2010). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/smoothness-regularity-and-complete-intersection/912909D15CF5892103F256D1A9737BA9"><i>Smoothness, Regularity and Complete Intersection</i></a>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781139107181</bdi>.</cite></li>

<li><cite id="CITEREFTate1957" class="citation cs2"><a href="John_Tate_(mathematician)" title="John Tate (mathematician)">Tate, John</a> (1957), <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.ijm/1255378502">"Homology of Noetherian rings and local rings"</a>, <i>Illinois Journal of Mathematics</i>, <b>1</b>: <span class="nowrap">14–</span>27, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0019-2082">0019-2082</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0086072">0086072</a></cite></li>

<li><cite id="CITEREFWiebe1969" class="citation cs2">Wiebe, Hartmut (1969), "Über homologische Invarianten lokaler Ringe", <i><a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a></i>, <b>179</b>: <span class="nowrap">257–</span>274, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01350771">10.1007/BF01350771</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0025-5831">0025-5831</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0255531">0255531</a></cite></li></ul>

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