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</style><div role="note" class="hatnote navigation-not-searchable">For the Saturn G ring, see <a href="Rings_of_Saturn" title="Rings of Saturn">Rings of Saturn</a>.</div>

<p>In <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a>, a <b>G-ring</b> or <b>Grothendieck ring</b> is a <a href="Noetherian_ring" title="Noetherian ring">Noetherian</a> <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> such that the map of any of its <a href="Local_ring" title="Local ring">local rings</a> to the <a href="Completion_(ring_theory)" class="mw-redirect" title="Completion (ring theory)">completion</a> is regular (defined below). Almost all Noetherian rings that occur naturally in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> or <a href="Number_theory" title="Number theory">number theory</a> are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings. The concept is named after <a href="Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a>.

</p><p>A ring that is both a G-ring and a <a href="J-2_ring" title="J-2 ring">J-2 ring</a> is called a <a href="Quasi-excellent_ring" class="mw-redirect" title="Quasi-excellent ring">quasi-excellent ring</a>, and if in addition it is <a href="Universally_catenary" class="mw-redirect" title="Universally catenary">universally catenary</a> it is called an <a href="Excellent_ring" title="Excellent ring">excellent ring</a>.

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

<ul><li>A (Noetherian) ring <i>R</i> containing a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <i>k</i> is called <b><a href="Geometrically_regular" class="mw-redirect" title="Geometrically regular">geometrically regular</a></b> over <i>k</i> if for any <a href="Finite_extension" class="mw-redirect" title="Finite extension">finite extension</a> <i>K</i> of <i>k</i> the ring <i>R</i> ⊗<sub><i>k</i></sub> <i>K</i> is a <a href="Regular_ring" class="mw-redirect" title="Regular ring">regular ring</a>.</li>

<li>A <a href="Homomorphism" title="Homomorphism">homomorphism</a> of rings from <i>R</i> to <i>S</i> is called <b>regular</b> if it is flat and for every <i>p</i> ∈ Spec(<i>R</i>) the fiber <i>S</i> ⊗<sub><i>R</i></sub> <i>k</i>(<i>p</i>) is geometrically regular over the <a href="Residue_field" title="Residue field">residue field</a> <i>k</i>(<i>p</i>) of <i>p</i>. (see also <a href="Popescu's_theorem" title="Popescu's theorem">Popescu's theorem</a>.)</li>

<li>A ring is called a local G-ring if it is a Noetherian local ring and the map to its completion (with respect to its <a href="Maximal_ideal" title="Maximal ideal">maximal ideal</a>) is regular.</li>

<li>A ring is called a G-ring if it is Noetherian and all its <a href="Localization_(commutative_algebra)" title="Localization (commutative algebra)">localizations</a> at <a href="Prime_ideal" title="Prime ideal">prime ideals</a> are local G-rings. (It is enough to check this just for the maximal ideals, so in particular local G-rings are G-rings.)</li></ul>

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

<ul><li>Every <a href="Field_(mathematics)" title="Field (mathematics)">field</a> is a G-ring</li>

<li>Every complete Noetherian local ring is a G-ring</li>

<li>Every ring of convergent <a href="Power_series" title="Power series">power series</a> in a finite number of variables over <b>R</b> or <b>C</b> is a G-ring.</li>

<li>Every <a href="Dedekind_domain" title="Dedekind domain">Dedekind domain</a> in <a href="Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> 0, and in particular the ring of <a href="Integer#Algebraic_properties" title="Integer">integers</a>, is a G-ring, but in positive characteristic there are Dedekind domains (and even <a href="Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation rings</a>) that are not G-rings.</li>

<li>Every localization of a G-ring is a G-ring</li>

<li>Every finitely generated <a href="Algebra_over_a_ring" class="mw-redirect" title="Algebra over a ring">algebra</a> over a G-ring is a G-ring. This is a <a href="Theorem" title="Theorem">theorem</a> due to Grothendieck.</li></ul>

<p>Here is an example of a discrete valuation ring <i>A</i> of characteristic <i>p</i>>0 which is not a G-ring. If <i>k</i> is any field of characteristic <i>p</i> with [<i>k</i> : <i>k</i><sup><i>p</i></sup>] = ∞ and <i>R</i> = <i>k</i>[[<i>x</i>]] and <i>A</i> is the <a href="Subring" title="Subring">subring</a> of power series Σ<i>a<sub>i</sub>x<sup>i</sup></i> such that [<i>k</i><sup><i>p</i></sup>(<i>a</i><sub>0</sub>,<i>a</i><sub>1</sub>,...) : <i>k</i><sup><i>p</i></sup>] is finite then the formal fiber of <i>A</i> over the generic point is not geometrically regular so <i>A</i> is not a G-ring. Here <i>k</i><sup><i>p</i></sup> denotes the image of <i>k</i> under the <a href="Frobenius_morphism" class="mw-redirect" title="Frobenius morphism">Frobenius morphism</a> <i>a</i>→<i>a</i><sup><i>p</i></sup>.

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

<ul><li>A. Grothendieck, J. Dieudonné, <a rel="nofollow" class="external text" href="http://www.numdam.org/item?id=PMIHES_1965__24__5_0"><i>Eléments de géométrie algébrique IV</i></a> Publ. Math. IHÉS 24 (1965), section 7</li>

<li>H. Matsumura, <i>Commutative algebra</i> <style data-mw-deduplicate="TemplateStyles:r1238218222">

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</style><a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8053-7026-9</bdi>, chapter 13.</li></ul>

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This article is issued from <a class="external text" title="Last edited on 2023-08-12" href="https://en.wikipedia.org/wiki/?title=G-ring&oldid=1170052092">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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