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Topic: Regular local ring

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	Regular local ring - Wikipedia, the free encyclopedia
		In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is exactly the same as its Krull dimension.
		Regular local rings were originally defined by Wolfgang Krull, but they first became prominent in the work of Oscar Zariski, who showed that geometrically, a regular local ring corresponds to a smooth point on an algebraic variety.
		Any discrete valuation ring is a regular local ring of dimension 1 and the regular local rings of dimension 1 are exactly the discrete valuation rings.
	en.wikipedia.org /wiki/Regular_local_ring (708 words)

	Springer Online Reference Works
		For a local ring, this is the determination of a regular local ring birationally equivalent to it.
		The existence of a resolving system (the local uniformization theorem) was proved for arbitrary varieties over a field of characteristic zero (see [1]), and also for two-dimensional varieties over any field and three-dimensional varieties over an algebraically closed field of characteristic other than 2, 3 or 5 (see [2]).
		For (local) uniformization in analytic geometry and in the theory of functions of a complex variable (Riemann surfaces) cf.
	eom.springer.de /L/l060230.htm (295 words)

	PlanetMath: regular local ring
		elements, so the maximal ideals of regular local rings have a minimal number of generators.
		This is version 3 of regular local ring, born on 2002-12-27, modified 2004-04-23.
		(Commutative rings and algebras :: Local rings and semilocal rings :: Regular local rings)
	planetmath.org /encyclopedia/RegularLocalRing.html (116 words)

	Cohen-Macaulay ring - Wikipedia, the free encyclopedia
		In mathematics, a Cohen-Macaulay ring is a particular type of commutative ring possessing some of the algebraic-geometric properties of a collection of nonsingular points, such as local equidimensionality.
		A local Cohen-Macaulay ring is defined as a commutative noetherian local ring with Krull dimension equal to its depth.
		Non-singularity (regularity) is still stronger— it corresponds to the notion of smoothness of a geometric object at a particular point.
	en.wikipedia.org /wiki/Cohen-Macaulay_ring (446 words)

	Abstract (Site not responding. Last check: 2007-11-03)
		Abstract: Lyubeznik proved that for F-modules over a regular ring that is a finitely generated algebra over a regular local ring of characteristic p > 0, F-finite implies DCC, i.e., finite length as an F-module.
		We discuss Lyubeznik's results, and then strengthen them by showing that whenever DCC holds for an F-module over a regular ring, there are only finitely many F-submodules.
		We also discuss consequences of this result for Artinian modules M over an arbitrary local ring with an action of Frobenius F such that F(rm) = F(r)F(m) for all r in the ring and and m in the module.
	www.nd.edu /~magic05/abstracts/hochster.html (110 words)

	[No title]
		Abstract: Let K be a noetherian ring and S a commutative algebra, which is essentially of finite type over K and is projective as a K-module.
		Abstract: Let R be a Noetherian local ring with infinite residue field k and I an R-ideal.
		Abstract: We study the analogy between the global theory of liaison of varieties in projective space and the local theory of liaison of ideals in a local ring.
	www.ms.uky.edu /~corso/NotreDame/talks.html (1617 words)

	OUOSU ring theory seminar speakers for 2004-5
		November 5, 2004: Warrem McGovern, Bowling Green StateUniversity, Rings of quotients of C(X) ABSTRACT: Q(X) and q(X) denote the maximal and classical rings of quotients of C(X) (resp.) It is known that, since C(X) is a semiprime ring, Q(X) is always von-Neumann regular.
		Inspired by a classic treatment of Ore [1932], a consideration of the arithmetic of such a ring \(R \) on an element-by-element basis (rather than by an ideal-theoretic analysis) leads to a very nice description of the internal structure of the indecomposables.
		The work is not complete: there is a detailed analysis of the structure of the endomorphism rings of the indecomposables underway; and some general questions about the nature of localizations intermediate between the first Weyl algebra and the Weyl division algebra.
	www.math.ohiou.edu /~lopez/log0405.html (826 words)

	AMCA: The Pierce-Birkhoff Conjecture by James J. Madden (Site not responding. Last check: 2007-11-03)
		The classical theory of Enriques and Zariski on complete ideals in 2-dimensional regular local rings provides the tools needed to extend Mahé's 1983 result to arbitrary 2-dimensional regular real algebras.
		We shall describe the generalization needed and present some interesting examples related to sequences of point blow-ups along a valuation centered on a 3-dimensional regular local ring which show some of the obstacles to proving the needed generalization.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Mathematical Conference Abstracts.
	at.yorku.ca /c/a/c/v/66.htm (239 words)

	Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, Vol. 47, No. 1, pp. 121-135, 2006 (Site not responding. Last check: 2007-11-03)
		Abstract: We study the effect of a quadratic transformation on the degree function of a $0$-dimensional ideal with only one Rees valuation in a $2$-dimensional regular local ring with algebraically closed residue field.
		A number of important results of Zariski and Lipman about complete ideals in a $2$-dimensional regular local ring follow as quick corollaries.
		Necessary and sufficient conditions for the regularity of a $2$-dimensional normal local domain are proved.
	www.emis.de /journals/BAG/vol.47/no.1/8.html (132 words)

	Atlas: Jumping numbers of a simple complete ideal in a two dimensional regular local ring by Tarmo Järvilehto (Site not responding. Last check: 2007-11-03)
		Atlas: Jumping numbers of a simple complete ideal in a two dimensional regular local ring by Tarmo Järvilehto
		Jumping numbers of a simple complete ideal turn out to divide into subsets corresponding to the stars in the associated dual graph.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # carv-10.
	atlas-conferences.com /cgi-bin/abstract/carv-10 (151 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Mark Hovey New papers appearing on hopf between 8/7/04 and 9/2/04 1.
		http://hopf.math.purdue.edu/cgi-bin/generate?/Devinatz/recog Title: Recognizing Hopf algebroids defined by a group action Author: Ethan Devinatz e-mail: devinatz@math.washington.edu Abstract: Let A be a complete noetherian regular local ring, and suppose that S is a profinite group acting continuously on A via ring homomorphisms.
		Let T be the algebra of continuous functions from S to A. Then (A,T) has a canonical structure of a complete Hopf algebroid, determined by the action of S on A. We give necessary and sufficient conditions for a general Hopf algebroid to be of this form.
	claude.math.wesleyan.edu /~mhovey/archive/letter161 (411 words)

	Regular Local Ring -- from Wolfram MathWorld
		A regular local ring is a local ring
		Weisstein, Eric W. "Regular Local Ring." From MathWorld--A Wolfram Web Resource.
		Show your math savvy with a MathWorld T-shirt.
	mathworld.wolfram.com /RegularLocalRing.html (57 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		Commutative algebra (Pandharipande) - compute Tor_1^Z(Z/p,Z/q) - classify finite dimensional commutative algebras over C with (vector space) dimension d - for finite modules over a (noetherian) local ring, prove free = projective = flat; give counterexamples over non-local rings - what's Hilbert's syzygy theorem?
		how does this relate to Serre's theorem on regular local rings?
		give two different definitions of K_0 of a regular local ring (either use all f.g modules, or just the projective ones) and prove their equivalence Algebraic number theory (Wiles) - "prove" that Q has no unramified extensions - compute the class group of K = Q(sqrt(-31)) - what's a Hilbert class field?
	www.math.princeton.edu /graduate/generals/bhatt_bhargav (495 words)

	Hexapedia - Regular local ring (via CobWeb/3.1 planetlab2.cs.virginia.edu) (Site not responding. Last check: 2007-11-03)
		Geometrically, a regular local ring corresponds to a smooth point on an algebraic variety.
		Thus if one forms the ring of germs of holomorphic functions at a smooth point on an algebraic variety defined in several complex variables, that will be a regular local ring.
		If p is an ordinary prime number, the ring of p-adic integers is an example of a discrete valuation ring which does not contain a field.
	www.hexafind.com.cob-web.org:8888 /encyclopedia/Regular_local_ring (286 words)

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