Wrapped distribution - Revision history

en>Jodosma: typo: coefficents→coefficients

2014-08-17T20:34:24Z

typo: coefficents→coefficients

← Older revision		Revision as of 22:34, 17 August 2014
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	~~In [[commutative algebra]]~~ the ~~'''Hilbert–Samuel function''', named after [[David Hilbert]] and [[Pierre Samuel]],<ref>H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero:~~ I~~. Ann.~~ of ~~Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203~~.</ref> of a nonzero finitely generated [[module (mathematics)\|module]] <math>M</math> over a commutative [[Noetherian]] [[local ring]] <math>A</math> and a [[primary ideal]] <math>I</math> of <math>A</math> is ~~the map <math>\chi_{M}^{~~I~~}:\mathbb{N}\rightarrow\mathbb{N}</math> such that, for all <math>n\~~in~~\mathbb{N}</math>,~~		Darrick may be the name I prefer to be called with it really is not one of the most masculine concept. My house is now in Georgia but I will have to push in a full year or dual. I work as an invoicing officer and I'll be promoted soon. To play country music may be the thing he loves quite a few. She's been working on her website for a certain period now. Go here here: http://www.fundacionenergia.net/[http://www.Google.Co.uk/search?hl=en&gl=us&tbm=nws&q=canada-goose-outlet.asp&gs_l=news canada-goose-outlet.asp]<br><br>Also visit my web-site: [http://www.fundacionenergia.net/canada-goose-outlet.asp Canada Goose Baratos]

	~~:<math>\chi_{M}^{I}(n)=\ell(M/I^{n+1}M)</math>~~

	~~where <math>\ell</math> denotes the [[length of~~ a ~~module\|length]] over <math>A</math>~~. ~~It is related to the [[Hilbert function]] of the [[associated graded module]] <math>\operatorname{gr}_I(M)</math> by the identity~~

	~~: <math>\chi_M^~~I ~~(n)=\sum_{i=0}^n H(\operatorname{gr}_I(M),i),</math>~~

	~~For sufficiently large <math>n</math>, it coincides with a polynomial function of degree equal to <math>\dim(\operatorname{gr}_I(M))-1</math>.<ref>Atiyah, M. F.~~ and ~~MacDonald,~~ I~~. G. ''Introduction to Commutative Algebra'~~'. ~~Reading, MA: Addison–Wesley, 1969.</ref>~~

	~~==Examples==~~

	~~For~~ the ~~[[ring (mathematics)\|ring]] of [[formal power series]] in two variables <math>k[[x,y]]</math> taken as~~ a ~~module over itself and graded by the order and the ideal generated by the monomials~~ '~~'x''<sup>2</sup> and ''y''<sup>3</sup> we have~~

	~~: <math>\chi(1)=1,\quad \chi(2)=3,\quad \chi(3)=5,\quad \chi(4)=6\text{ and } \chi(k)=6\text{~~ for ~~}k > 4~~.</~~math><ref>Ibidem<~~/~~ref>~~

	~~== Degree bounds ==~~
	~~Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence~~. ~~However, it is still reasonably close to being additive, as a consequence of the [[Artin-Rees lemma]]. We denote by <math>P_{I, M}</math> the Hilbert-Samuel polynomial; i~~.~~e., it coincides with the Hilbert–Samuel function for large integers.~~

	~~Let <math>(R, m)<~~/~~math> be a Noethrian local ring and ''I'' a m-~~[~~[primary ideal]]. If~~
	:~~<math>0 \to M' \to M \to M'' \to 0</math>~~
	~~is an exact sequence of finitely generated ''R''-modules and if <math>M~~/~~I M<~~/~~math> has finite length,<ref>This implies that <math>M'/IM'</math> and <math>M''/IM''</math> also have finite length~~.~~</ref> then we have:<ref>{{harvnb\|Eisenbud\|loc=Lemma 12~~.~~3}}<~~/~~ref>~~
	~~:<math>P_{I, M}~~ = ~~P_{I, M'} + P_{I, M''} - F</math>~~
	where ''F'' is a polynomial of degree strictly less than that of <math>P_{I, M'}</math> and having positive leading coefficient. In particular, if <math>M' \simeq M</math>, then the degree of <math>P_{I, M''}</math> is strictly less than that of <math>P_{I, M} = ~~P_{I, M'}</math>.~~

	~~Proof: Tensoring the given exact sequence with <math>R/I^n</math> and computing the kernel we get the exact sequence:~~
	~~:<math>0 \to (I^n M \cap M')/I^n M' \to M'/I^n M' \to M/I^n M \to M''/I^n M'' \to 0,</math>~~
	~~which gives~~ us:
	~~:<math>\chi_M^I(n-1)~~ = ~~\chi_{M'}^I(n~~-~~1) + \chi_{M''}^I(n-1) - \ell((I^n M \cap M')/I^n M')</math>.~~
	~~The third term on the right can be estimated by Artin~~-~~Rees~~. ~~Indeed, by the lemma, for large ''n'' and some ''k'',~~
	~~:<math>I^n M \cap M'~~ = ~~I^{n~~-~~k} ((I^k M) \cap M') \subset I^{n~~-~~k} M'~~.<~~/math~~>
	~~Thus,~~
	:<~~math~~>~~\ell((I^n M \cap M')~~ / ~~I^n M') \le \chi^I_{M'}(n-1) - \chi^I_{M'}(n-k-1)<~~/~~math>~~.
	~~This gives the desired degree bound~~.

	~~==References==~~
	~~<references~~/>

	* [[David Eisenbud\|Eisenbud, David]], ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-~~Verlag, 1995, ISBN 0~~-~~387-94268-8~~.

	~~{{DEFAULTSORT:Hilbert-Samuel function}}~~
	~~[[Category:Commutative algebra]~~]

en>John of Reading: Typo/general fixing, replaced: Dircetional → Directional using AWB

2014-01-09T15:35:40Z

Typo/general fixing, replaced: Dircetional → Directional using AWB

← Older revision		Revision as of 17:35, 9 January 2014
Line 1:		Line 1:
	~~From slicing aloin to dicing~~ a ~~pineapple~~, ~~being aware~~ of ~~how to function with knives~~ is ~~crucial to good results~~ in the ~~kitchen~~. ~~With a deer hanging beneath the back deck, this~~ is ~~my favorite tool~~ to ~~rid~~ the ~~carcass~~ of ~~its hide Possibly~~ the ~~sharpest knife~~ I ~~personal~~, ~~the sweeping belly gets into each and every nook and back-~~of-the~~-knee cranny~~, and ~~each~~ and ~~every now~~ and ~~then I just take a break~~ and ~~appear at~~ the ~~issue~~, ~~marveling that such a utilitarian object can be so dang attractive~~. ~~I personally really feel like any knife with a gut hook~~ is a ~~waste~~ of ~~steal~~.<br><br>~~In a expert knife set you will see knife handles made from Micarda which is synthetic material~~. ~~When hunting for a skilled knife set you will come across many varieties of knives out there in the regional retailer~~. ~~But when acquiring make positive you are comfy~~ with the ~~knife that you have bought~~. ~~The life of~~ a ~~superior high~~-~~quality knife typically lasts longer than that~~ of ~~other folks. It~~'~~s a Chef Knife Set In Case should~~ that ~~everybody need to~~ have ~~a bread knife no matter whether you cook or not~~.<br><br>~~Deciding upon a knife manage is like deciding upon a car~~, ~~do you want a sports car or truck for speed or a 4 wheel drive for off road terrain~~, ~~or maybe just a classy car or truck to show off with. As with automobiles there~~ is ~~no all in~~ a ~~single remedy~~, ~~pick your knife deal with wisely~~. ~~Fascinating how knife fashions have changed in 60 years - my father~~'~~s old Case hunting knife dressed deer~~, ~~bear~~, ~~cut wood, but appears like a toy next to modern day "Rambo" poseur knives.~~<br><br>~~A knife must be solidly constructed~~, ~~light weight~~, ~~and have a 440HC blade - why that~~'~~s Kershaw! Could be~~ the ~~Finest compact knife obtainable, you never will need massive hands to operate it perfectly,~~ and ~~it really is made in~~ the ~~USA! Simply because this knife is so lightweight, this is a cause it tends to make it to be~~ the ~~very Chefs Choice Knife Sharpener 120 Best Price best little knife~~ to ~~carry. If you cherished this posting and you would like~~ to ~~obtain much more details with regards~~ to ~~[http:~~//~~www.thebestkitchenknivesreviews.com~~/~~best~~-~~chef~~-~~knives~~-~~reviews~~/ ~~Best Chefs Knife] kindly visit our site~~. ~~Now you have removed~~ the ~~Black Knife from your pocket~~ and ~~your holding it in your hand~~.<br><br>~~The finest way to choose the most effective sports knife for yourself is to 1st have a clear understanding of the kind of knife you want~~. ~~The purpose for which you demand~~ the ~~sports knife need to be clear in your thoughts~~. ~~You can also use a sports knife for various purposes like cutting a paper~~, ~~slicing~~ a ~~fruit~~, ~~cutting clothes and~~ in ~~some situations even for self defense~~. ~~The knife is either a folder or fixed blade knife. Carbon Fiber~~ : ~~Thin strands of carbon that are set in resin.~~		In [[commutative algebra]] the '''Hilbert–Samuel function''', named after [[David Hilbert]] and [[Pierre Samuel]],<ref>H. Hironaka, Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I. Ann. of Math. 2nd Ser., Vol. 79, No. 1. (Jan., 1964), pp. 109-203.</ref> of a nonzero finitely generated [[module (mathematics)\|module]] <math>M</math> over a commutative [[Noetherian]] [[local ring]] <math>A</math> and a [[primary ideal]] <math>I</math> of <math>A</math> is the map <math>\chi_{M}^{I}:\mathbb{N}\rightarrow\mathbb{N}</math> such that, for all <math>n\in\mathbb{N}</math>,

			:<math>\chi_{M}^{I}(n)=\ell(M/I^{n+1}M)</math>

			where <math>\ell</math> denotes the [[length of a module\|length]] over <math>A</math>. It is related to the [[Hilbert function]] of the [[associated graded module]] <math>\operatorname{gr}_I(M)</math> by the identity

			: <math>\chi_M^I (n)=\sum_{i=0}^n H(\operatorname{gr}_I(M),i),</math>

			For sufficiently large <math>n</math>, it coincides with a polynomial function of degree equal to <math>\dim(\operatorname{gr}_I(M))-1</math>.<ref>Atiyah, M. F. and MacDonald, I. G. ''Introduction to Commutative Algebra''. Reading, MA: Addison–Wesley, 1969.</ref>

			==Examples==

			For the [[ring (mathematics)\|ring]] of [[formal power series]] in two variables <math>k[[x,y]]</math> taken as a module over itself and graded by the order and the ideal generated by the monomials ''x''<sup>2</sup> and ''y''<sup>3</sup> we have

			: <math>\chi(1)=1,\quad \chi(2)=3,\quad \chi(3)=5,\quad \chi(4)=6\text{ and } \chi(k)=6\text{ for }k > 4.</math><ref>Ibidem</ref>

			== Degree bounds ==
			Unlike the Hilbert function, the Hilbert–Samuel function is not additive on an exact sequence. However, it is still reasonably close to being additive, as a consequence of the [[Artin-Rees lemma]]. We denote by <math>P_{I, M}</math> the Hilbert-Samuel polynomial; i.e., it coincides with the Hilbert–Samuel function for large integers.

			Let <math>(R, m)</math> be a Noethrian local ring and ''I'' a m-[[primary ideal]]. If
			:<math>0 \to M' \to M \to M'' \to 0</math>
			is an exact sequence of finitely generated ''R''-modules and if <math>M/I M</math> has finite length,<ref>This implies that <math>M'/IM'</math> and <math>M''/IM''</math> also have finite length.</ref> then we have:<ref>{{harvnb\|Eisenbud\|loc=Lemma 12.3}}</ref>
			:<math>P_{I, M} = P_{I, M'} + P_{I, M''} - F</math>
			where ''F'' is a polynomial of degree strictly less than that of <math>P_{I, M'}</math> and having positive leading coefficient. In particular, if <math>M' \simeq M</math>, then the degree of <math>P_{I, M''}</math> is strictly less than that of <math>P_{I, M} = P_{I, M'}</math>.

			Proof: Tensoring the given exact sequence with <math>R/I^n</math> and computing the kernel we get the exact sequence:
			:<math>0 \to (I^n M \cap M')/I^n M' \to M'/I^n M' \to M/I^n M \to M''/I^n M'' \to 0,</math>
			which gives us:
			:<math>\chi_M^I(n-1) = \chi_{M'}^I(n-1) + \chi_{M''}^I(n-1) - \ell((I^n M \cap M')/I^n M')</math>.
			The third term on the right can be estimated by Artin-Rees. Indeed, by the lemma, for large ''n'' and some ''k'',
			:<math>I^n M \cap M' = I^{n-k} ((I^k M) \cap M') \subset I^{n-k} M'.</math>
			Thus,
			:<math>\ell((I^n M \cap M') / I^n M') \le \chi^I_{M'}(n-1) - \chi^I_{M'}(n-k-1)</math>.
			This gives the desired degree bound.

			==References==
			<references/>

			* [[David Eisenbud\|Eisenbud, David]], ''Commutative Algebra with a View Toward Algebraic Geometry'', Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN 0-387-94268-8.

			{{DEFAULTSORT:Hilbert-Samuel function}}
			[[Category:Commutative algebra]]

en>Helpful Pixie Bot: ISBNs (Build KH)

2012-05-12T01:14:16Z

ISBNs (Build KH)

New page

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