Basic solution (linear programming) - Revision history

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2014-05-30T06:25:40Z

Reverted edits by 111.93.185.90 (talk) to last version by Yobot

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	In [[algebra]], a commutative ''k''-algebra ''A'' is said to be '''0-smooth''' if it satisfies the following lifting property: given a ''k''-algebra ''C'', an ideal ''N'' of ''C'' whose square is zero and a ''k''-algebra map <math>u: A \to C/N</math>, there exists a ''k''-algebra map <math>v: A \to C</math> such that ''u'' is ''v'' followed by the canonical map. If there exists at most one such a lifting ''v'', then ''A'' is said to be '''0-unramified''' (or '''0-neat'''). ''A'' is said to be '''0-étale''' if it is '''0-smooth''' and '''0-unramified'''.

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	~~== ''I''-smooth ==~~
	~~Let ''B''~~ be ~~an ''A''-algebra~~ and ~~suppose ''B'' is given~~ the ~~''I''-adic topology~~, ~~''I'' an ideal~~ of ~~''B''~~. ~~We say~~ '~~'B'' is '''''I''-smooth over ''A''''' if it satisfies~~ the ~~lifting property: given an ''A''-algebra ''C'~~', ~~an ideal ''N'' of ''C'' whose square is zero and an ''A''-algebra map~~ <~~math~~>~~u: B \to C/N~~<~~/math~~> ~~that is continuous when~~ <~~math~~>~~C/N~~<~~/math~~> ~~is given~~ the ~~discrete topology, there exists an ''A''-algebra map~~ <~~math~~>~~v: B \to C~~<~~/math~~> ~~such that ''u'' is ''v'' followed by~~ the ~~canonical map~~. ~~As before, if there exists at most one such lift ''v''~~, ~~then ''B'' is said~~ to ~~be '''''I''-unramified over ''A''''' (or '''''I''-neat''')~~. ~~''B'' is said~~ to ~~be '''''I''-étale''' if it is '''''I''-smooth''' and '''''I''-unramified'''. If ''I'' is~~ the ~~zero ideal and ''A'' is a field~~, ~~these notions coincide~~ with ~~0-smooth etc~~. as ~~defined above.~~

	~~A standard example is~~ this: ~~let ''A'' be a ring,~~ <~~math~~>~~B = A[\![t_1~~, ~~\ldots, t_n]\!]~~<~~/math~~> ~~and~~ <~~math~~>~~I = (t_1~~, ~~\ldots~~, ~~t_n)~~.<~~/math~~> ~~Then ''B'' is ''I''~~-~~smooth over ''A''~~.

	~~Let ''A'' be~~ a ~~noetherian local ring~~ with ~~maximal ideal~~ <~~math~~>~~\mathfrak{m}~~<~~/math~~> ~~that is~~ a ~~''k''~~-~~algebra~~. ~~Then ''A'~~' is <~~math~~>~~\mathfrak{m}~~<~~/math~~>~~-smooth over~~ '~~'k'' if~~ and ~~only if~~ <~~math~~>A ~~\otimes_k k'~~<~~/math~~> ~~is a regular local ring for any finite field extension~~ <~~math~~>k'~~</math>~~ of '~~'k''~~.<~~ref~~>~~{{harvnb\|Matsumura\|1986\|loc=Theorem 28~~.~~7}}<~~/~~ref>~~

	== ~~See also~~ ==
	*[[étale morphism]]
	*[[formally smooth morphism]]

	~~== References ==~~
	~~{{reflist}}~~
	* H. ~~Matsumura~~ '~~'Commutative ring theory~~.~~'' Translated from the Japanese by M~~. ~~Reid~~. ~~Second edition~~. ~~Cambridge Studies in Advanced Mathematics~~, 8.

	[[~~Category~~:~~Algebra~~]]

	~~{{algebra~~-~~stub}}~~

en>Jaanizri: Removed internal link - didn't notice it was a disambig page.

2014-01-24T12:05:57Z

Removed internal link - didn't notice it was a disambig page.

← Older revision		Revision as of 14:05, 24 January 2014
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	As a ~~parent~~, ~~your young ones are your greatest treasure~~ and ~~you want all interesting for them including in their dental remedy. In this article~~, ~~I have~~ to ~~anyone with~~ a ~~associated with helpful for how you~~'ll be ~~able~~ to ~~find~~ the ~~suitable Dallas Pediatric Dentist to all your kids.~~<br><br>~~This could possibly not have almost anything to do together with personal dentist~~ and ~~she or she may be very experienced~~. ~~However~~, a ~~dentist for children has been trained to work with children and he or she will often have developed more skills that may match little one better~~.<br><br>~~Abrasives: Hydrated silicas~~ and ~~calcium carbonates are~~ the ~~rationale ingredients~~ - ~~the degree~~ of ~~abrasiveness varies from paste to paste~~. ~~Some dentists recommend toothpastes having a lower Relative Dentin Abrasivity (RDA) score (~~the ~~scale runs from 8~~-~~200) based on the text Consumer Reports~~, ~~especially if you go with~~ an ~~electric toothbrush, most likely~~ to ~~over-brush your teeth, or have receding gums.~~<br><br>~~Dentistry~~ is ~~wide, and~~ the ~~aspect that handle oral look after children usually that of pediatric dentist s. These are dentists that treat infants~~, ~~young children as well as young kids. Generally, pediatric dentists love offspring. For your family, you will find~~ an ~~experienced pediatric dentist in Alexandria VA, one more available Dentist Springfield The state of virginia.~~<br><br>~~After a reason your child~~ is ~~certain have ideas~~. ~~They may want~~ to ~~know something specific like the name for the kids dentist they~~'~~re going~~ to ~~look at~~. ~~They could just want recognize whether or not~~ the ~~procedure will cause pain to~~. ~~Take your time in responding~~. ~~Be honest with your child and do the best can certainly to answer their some questions. When you loved~~ this ~~short article and you would want to receive more information about~~ [~~https:~~/~~/www~~.~~youtube.com~~/~~watch?v=0b_~~-~~CCpP6Pg tampa emergency dentist] please visit our own internet site~~. ~~You can~~ be ~~honest in the event~~ that ~~don~~'~~t are aware of the answer a new question~~. It'~~s also possible to encourage these ask questions of the experts at the appointment.~~<br><br>~~Children should start watching a dentist at the time they have teeth. If there~~ is be a ~~problem with them coming in in early place, it may possibly be cognizant~~ of to take them in with regard to checkup. Those early teeth can tell a dentist a lot about what children tooth they could need in the foreseeable future. Though not all can be known from several teeth coming up, may do be surprised at what could be learned. After that, dental professional should tell you how often they decide to assist you to for dental visits.<br><br>~~So, a verbal specialist does not need to feel helpless while confronting frightened youths~~. ~~There are several things that can be done drugs~~ the ~~situation easier even better enjoyable record their lives involved~~.
			In [[algebra]], a commutative ''k''-algebra ''A'' is said to be '''0-smooth''' if it satisfies the following lifting property: given a ''k''-algebra ''C'', an ideal ''N'' of ''C'' whose square is zero and a ''k''-algebra map <math>u: A \to C/N</math>, there exists a ''k''-algebra map <math>v: A \to C</math> such that ''u'' is ''v'' followed by the canonical map. If there exists at most one such a lifting ''v'', then ''A'' is said to be '''0-unramified''' (or '''0-neat'''). ''A'' is said to be '''0-étale''' if it is '''0-smooth''' and '''0-unramified'''.

			A separable algebraic field extension ''L'' of ''k'' is 0-étale over ''k''.<ref>{{harvnb\|Matsumura\|1986\|loc=Theorem 25.3}}</ref> the formal power series ring <math>k[\![t_1, \ldots, t_n]\!]</math> is 0-smooth only when <math>\operatorname{char}k = p</math> and <math>[k: k^p] < \infty</math> (i.e., ''k'' has a finite [[p-basis\|''p''-basis]].)<ref>{{harvnb\|Matsumura\|1986\|loc=pg. 215}}</ref>

			== ''I''-smooth ==
			Let ''B'' be an ''A''-algebra and suppose ''B'' is given the ''I''-adic topology, ''I'' an ideal of ''B''. We say ''B'' is '''''I''-smooth over ''A''''' if it satisfies the lifting property: given an ''A''-algebra ''C'', an ideal ''N'' of ''C'' whose square is zero and an ''A''-algebra map <math>u: B \to C/N</math> that is continuous when <math>C/N</math> is given the discrete topology, there exists an ''A''-algebra map <math>v: B \to C</math> such that ''u'' is ''v'' followed by the canonical map. As before, if there exists at most one such lift ''v'', then ''B'' is said to be '''''I''-unramified over ''A''''' (or '''''I''-neat'''). ''B'' is said to be '''''I''-étale''' if it is '''''I''-smooth''' and '''''I''-unramified'''. If ''I'' is the zero ideal and ''A'' is a field, these notions coincide with 0-smooth etc. as defined above.

			A standard example is this: let ''A'' be a ring, <math>B = A[\![t_1, \ldots, t_n]\!]</math> and <math>I = (t_1, \ldots, t_n).</math> Then ''B'' is ''I''-smooth over ''A''.

			Let ''A'' be a noetherian local ring with maximal ideal <math>\mathfrak{m}</math> that is a ''k''-algebra. Then ''A'' is <math>\mathfrak{m}</math>-smooth over ''k'' if and only if <math>A \otimes_k k'</math> is a regular local ring for any finite field extension <math>k'</math> of ''k''.<ref>{{harvnb\|Matsumura\|1986\|loc=Theorem 28.7}}</ref>

			== See also ==
			*[[étale morphism]]
			*[[formally smooth morphism]]

			== References ==
			{{reflist}}
			* H. Matsumura ''Commutative ring theory.'' Translated from the Japanese by M. Reid. Second edition. Cambridge Studies in Advanced Mathematics, 8.

			[[Category:Algebra]]

			{{algebra-stub}}

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2012-06-11T02:19:39Z

Dating maintenance tags: {{Orphan}}

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As a parent, your young ones are your greatest treasure and you want all interesting for them including in their dental remedy. In this article, I have to anyone with a associated with helpful for how you'll be able to find the suitable Dallas Pediatric Dentist to all your kids. This could possibly not have almost anything to do together with personal dentist and she or she may be very experienced. However, a dentist for children has been trained to work with children and he or she will often have developed more skills that may match little one better. Abrasives: Hydrated silicas and calcium carbonates are the rationale ingredients - the degree of abrasiveness varies from paste to paste. Some dentists recommend toothpastes having a lower Relative Dentin Abrasivity (RDA) score (the scale runs from 8-200) based on the text Consumer Reports, especially if you go with an electric toothbrush, most likely to over-brush your teeth, or have receding gums. Dentistry is wide, and the aspect that handle oral look after children usually that of pediatric dentist s. These are dentists that treat infants, young children as well as young kids. Generally, pediatric dentists love offspring. For your family, you will find an experienced pediatric dentist in Alexandria VA, one more available Dentist Springfield The state of virginia. After a reason your child is certain have ideas. They may want to know something specific like the name for the kids dentist they're going to look at. They could just want recognize whether or not the procedure will cause pain to. Take your time in responding. Be honest with your child and do the best can certainly to answer their some questions. When you loved this short article and you would want to receive more information about [https://www.youtube.com/watch?v=0b_-CCpP6Pg tampa emergency dentist] please visit our own internet site. You can be honest in the event that don't are aware of the answer a new question. It's also possible to encourage these ask questions of the experts at the appointment. Children should start watching a dentist at the time they have teeth. If there is be a problem with them coming in in early place, it may possibly be cognizant of to take them in with regard to checkup. Those early teeth can tell a dentist a lot about what children tooth they could need in the foreseeable future. Though not all can be known from several teeth coming up, may do be surprised at what could be learned. After that, dental professional should tell you how often they decide to assist you to for dental visits. So, a verbal specialist does not need to feel helpless while confronting frightened youths. There are several things that can be done drugs the situation easier even better enjoyable record their lives involved.