<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Noncommutative projective geometry</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Noncommutative_projective_geometry"> <link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Noncommutative_projective_geometry rootpage-Noncommutative_projective_geometry skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Noncommutative projective geometry</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<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 mathematics, <b>noncommutative projective geometry</b> is a noncommutative analog of <a href="Projective_geometry" title="Projective geometry">projective geometry</a> in the setting of <a href="Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">noncommutative algebraic geometry</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>
<ul><li>The quantum plane, the most basic example, is the <a href="Quotient_ring" title="Quotient ring">quotient ring</a> of the free ring:</li></ul>
<dl><dd><dl><dd><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\langle x,y\rangle /(yx-qxy)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>x</mi>
<mo>,</mo>
<mi>y</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<mi>y</mi>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>q</mi>
<mi>x</mi>
<mi>y</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k\langle x,y\rangle /(yx-qxy)}</annotation>
</semantics>
</math></span><img src="./904f98a4194606ff08bedf0608774bb3176d0dfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.392ex; height:2.843ex;" alt="{\displaystyle k\langle x,y\rangle /(yx-qxy)}" loading="lazy"></span></dd></dl></dd></dl>
<ul><li>More generally, the <b>quantum polynomial ring</b> is the quotient ring:</li></ul>
<dl><dd><dl><dd><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\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>k</mi>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mo stretchy="false">(</mo>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<mo>−<!-- − --></mo>
<msub>
<mi>q</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msub>
<msub>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle k\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}</annotation>
</semantics>
</math></span><img src="./ea900de348c74a0a2813dec8372adfbc9f13dcdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.195ex; height:3.009ex;" alt="{\displaystyle k\langle x_{1},\dots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}" loading="lazy"></span></dd></dl></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Proj_construction">Proj construction</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1236090951">
/* start https://en.wikipedia.org/ */
.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}
/* end https://en.wikipedia.org/ */
</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Proj_construction" title="Proj construction">Proj construction</a></div>
<p>By definition, the Proj of a graded ring <i>R</i> is the quotient category of the category of finitely generated graded modules over <i>R</i> by the subcategory of torsion modules. If <i>R</i> is a commutative Noetherian graded ring generated by degree-one elements, then the Proj of <i>R</i> in this sense is equivalent to the category of <a href="Coherent_sheaf" title="Coherent sheaf">coherent sheaves</a> on the usual Proj of <i>R</i>. Hence, the construction can be thought of as a generalization of the Proj construction for a commutative graded ring.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Elliptic_algebra" title="Elliptic algebra">Elliptic algebra</a></li>
<li><a href="Calabi%E2%80%93Yau_algebra" class="mw-redirect" title="Calabi–Yau algebra">Calabi–Yau algebra</a></li>
<li><a href="Sklyanin_algebra" title="Sklyanin algebra">Sklyanin algebra</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */
.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}
/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFAjitabh1994" class="citation cs2">Ajitabh, Kaushal (1994), <a rel="nofollow" class="external text" href="http://dspace.mit.edu/bitstream/handle/1721.1/28088/31369741.pdf?sequence=1"><i>Modules over regular algebras and quantum planes</i></a> <span class="cs1-format">(PDF)</span> (Ph.D. thesis)</cite></li>
<li><cite id="CITEREFArtin1992" class="citation cs2"><a href="Michael_Artin" title="Michael Artin">Artin, Michael</a> (1992), "Geometry of quantum planes", <i>Contemporary Mathematics</i>, <b>124</b>: <span class="nowrap">1–</span>15, <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=1144023">1144023</a></cite></li>
<li><cite id="CITEREFRogalski2014" class="citation arxiv cs1">Rogalski, D (2014). "An introduction to Noncommutative Projective Geometry". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1403.3065">1403.3065</a></span> [<a rel="nofollow" class="external text" href="https://arxiv.org/archive/math.RA">math.RA</a>].</cite></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2021-08-28" href="https://en.wikipedia.org/wiki/?title=Noncommutative_projective_geometry&oldid=1041050783">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
</body></html>