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<title>Geometrically regular ring</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Geometrically regular 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="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, a <b>geometrically regular ring</b> is a <a href="Noetherian_ring" title="Noetherian ring">Noetherian ring</a> over a <a href="Field_(algebra)" class="mw-redirect" title="Field (algebra)">field</a> that remains a <a href="Regular_ring" class="mw-redirect" title="Regular ring">regular ring</a> after any <a href="Finite_extension" class="mw-redirect" title="Finite extension">finite extension</a> of the base field. Geometrically regular <a href="Scheme_(mathematics)" title="Scheme (mathematics)">schemes</a> are defined in a similar way. In older terminology, points with regular <a href="Local_ring" title="Local ring">local rings</a> were called <b>simple points</b>, and points with geometrically regular local rings were called <b>absolutely simple points</b>. Over fields that are of characteristic 0, or algebraically closed, or more generally <a href="Perfect_field" title="Perfect field">perfect</a>, geometrically regular rings are the same as regular rings. Geometric regularity originated when <a href="Claude_Chevalley" title="Claude Chevalley">Claude Chevalley</a> and <a href="Andr%C3%A9_Weil" title="André Weil">André Weil</a> pointed out to <a href="Oscar_Zariski" title="Oscar Zariski">Oscar Zariski</a> (<a href="#CITEREFZariski1947">1947</a>) that, over non-perfect fields, the Jacobian criterion for a simple point of an algebraic variety is not equivalent to the condition that the local ring is regular.

</p><p>A Noetherian local ring containing a field <i>k</i> is geometrically regular over <i>k</i> if and only if it is <a href="Formally_smooth" class="mw-redirect" title="Formally smooth">formally smooth</a> over <i>k</i>.

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

<p><a href="#CITEREFZariski1947">Zariski (1947)</a> gave the following two examples of local rings that are regular but not geometrically regular.

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<ol><li>Suppose that <i>k</i> is a field of characteristic <i>p</i> > 0 and <i>a</i> is an element of <i>k</i> that is not a <i>p</i>th power. Then every point of the curve <i>x</i><sup><i>p</i></sup> + <i>y</i><sup><i>p</i></sup> = <i>a</i> is regular. However over the field <i>k</i>[<i>a</i><sup>1/<i>p</i></sup>], every point of the curve is singular. So the points of this curve are regular but not geometrically regular.</li>

<li>In the previous example, the equation defining the curve becomes reducible over a finite extension of the base field. This is not the real cause of the phenomenon: Chevalley pointed out to Zariski that the curve <i>x</i><sup><i>p</i></sup> + <i>y</i><sup>2</sup> = <i>a</i> (with the notation of the previous example) is absolutely irreducible but still has a point that is regular but not geometrically regular.</li></ol>

<ul><li><a href="Regular_scheme" title="Regular scheme">Regular scheme</a></li></ul>

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

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</style><cite id="CITEREFGrothendieckDieudonné1965" class="citation journal cs1"><a href="Alexander_Grothendieck" title="Alexander Grothendieck">Grothendieck, Alexandre</a>; <a href="Jean_Dieudonn%C3%A9" title="Jean Dieudonné">Dieudonné, Jean</a> (1965). <a rel="nofollow" class="external text" href="http://www.numdam.org/articles/PMIHES_1965__24__5_0">"Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Seconde partie"</a>. <i><a href="Publications_Math%C3%A9matiques_de_l'IH%C3%89S" title="Publications Mathématiques de l'IHÉS">Publications Mathématiques de l'IHÉS</a></i>. <b>24</b>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf02684322">10.1007/bf02684322</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=0199181">0199181</a>.</cite></li>

<li><cite id="CITEREFZariski1947" class="citation cs2"><a href="Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a> (1947), "The concept of a simple point of an abstract algebraic variety.", <i><a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>, <b>62</b> (1): <span class="nowrap">1–</span>52, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fs0002-9947-1947-0021694-1">10.1090/s0002-9947-1947-0021694-1</a></span>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990628">1990628</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=0021694">0021694</a></cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-24" href="https://en.wikipedia.org/wiki/?title=Geometrically_regular_ring&oldid=1302304646">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.

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