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| {{for|regular Cauchy sequence|Cauchy sequence#In constructive mathematics}}
| | Jewelry are some things everyone loves wearing. Allow be stones, gold, silver or pewter, there is a sort of knickknack that everyone will enjoy. Now, to selected that all pets aren't far behind in this race for jewelry, plenty of people manufacture pet jewelry, yes jewelry for pets! Jewelry that includes chains, lockets, collars, and anything fancy one can envisage.<br><br>Most belonging to the times all of us talk about conflicts, we mostly for you to the conflict around the world; but, what to your conflict within your inner diy? There is too many conflicts going on within us and the sterling peace symbols remind us to strive for peace within ourselves.<br><br>Cufflinks, custom steak branding irons, and swiss army knives is also another popular gift options for groomsmen. Besides these, you can get engraved cufflinks for any father, your brother-in-law, some other important people present at the wedding.<br><br><br><br>It isn't always simple choose your favorite jewelry that will go collectively with your dress. All this depends from which you shop, you might find a particular necklace made for you and also the dress may have, but sadly you can't find matching earrings. Yes, it is often a tiring activity, all the combo 'n' match, but simply because said you have to be wise here. You'll always get a wedding jewelry sets which can complete in them.<br><br>Don't ignore your girlfriend's age and personality. A younger woman in her 20s is lively and bubbly; thus, a pink or sky blue pendant is brilliant. For older women on the other hand, black Pendants or red, will really make them feel appreciated and loved. Jewelry is crucial in every girl's lifestyle mainly because they compliment a dress-up costume. Regardless within the price, accessories need to well along with a skirt, a blouse or perhaps with coloring of your sweetheart's eyes.<br><br>It makes idea to shine your silver wear using a clean and soft polishing cloth after wearing them. You should have dirt and oils dealt with as a consequence of this is what.<br><br>In past years, costume jewelry attempt to look like real boulders. Now, it's popular just to embrace the sparkling accent of faux diamonds and uric acid. A lot of these stones are oversized or have rough cuts. As an alternative to trying to mimic fine jewelry, just together with over guidelines pieces.<br><br>Once a designer creates enough pieces, it's only like any other product. The jewelry needs being marketed and sold! This may start solely as as designers wearing pretty own pieces, but in order to run that product, designers need exposure. Jewelry parties among friends and friends of friends is usually good technique sell dresses. However, the internet is significant marketplace presently there are some very well-known ecommerce sites that sell jewelry.<br><br>In case you loved this short article and you would want to receive more info concerning [https://www.youtube.com/watch?v=pns2W-mSPU8 Quantum Pendants] i implore you to visit the web site. |
| In [[commutative algebra]], a regular sequence is a sequence of elements of a [[commutative ring]] which are as independent as possible, in a precise sense. This is the algebraic analogue of the geometric notion of a [[complete intersection]].
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| ==Definitions==
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| For a commutative ring ''R'' and an ''R''-[[Module (mathematics)|module]] ''M'', an element ''r'' in ''R'' is called a '''non-zero-divisor on ''M'' ''' if ''r m'' = 0 implies ''m'' = 0 for ''m'' in ''M''. An ''' ''M''-regular sequence''' is a sequence
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| :''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> in ''R''
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| such that ''r''<sub>''i''</sub> is a non-zero-divisor on ''M''/(''r''<sub>1</sub>, ..., ''r''<sub>''i''-1</sub>)''M'' for ''i'' = 1, ..., ''d''.<ref>N. Bourbaki. ''Algèbre. Chapitre 10. Algèbre Homologique.'' Springer-Verlag (2006). X.9.6.</ref> Some authors also require that ''M''/(''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub>)''M'' is not zero. Intuitively, to say that
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| ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> is an ''M''-regular sequence means that these elements "cut ''M'' down" as much as possible, when we pass successively from ''M'' to ''M''/(''r''<sub>1</sub>)''M'', to ''M''/(''r''<sub>1</sub>, ''r''<sub>2</sub>)''M'', and so on.
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| An ''R''-regular sequence is called simply a '''regular sequence'''. That is, ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> is a regular sequence if ''r''<sub>1</sub> is a non-zero-divisor in ''R'', ''r''<sub>2</sub> is a non-zero-divisor in the ring ''R''/(''r''<sub>1</sub>), and so on. In geometric language, if ''X'' is an [[Spectrum of a ring|affine scheme]] and ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> is a regular sequence in the ring of regular functions on ''X'', then we say that the closed subscheme {''r''<sub>1</sub>=0, ..., ''r''<sub>''d''</sub>=0} ⊂ ''X'' is a '''[[complete intersection]]''' subscheme of ''X''.
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| For example, ''x'', ''y''(1-''x''), ''z''(1-''x'') is a regular sequence in the polynomial ring '''C'''[''x'', ''y'', ''z''], while ''y''(1-''x''), ''z''(1-''x''), ''x'' is not a regular sequence. But if ''R'' is a [[Noetherian ring|Noetherian]] [[local ring]] and the elements ''r''<sub>''i''</sub> are in the maximal ideal, or if ''R'' is a [[Graded algebra|graded ring]] and the ''r''<sub>''i''</sub> are homogeneous of positive degree, then any permutation of a regular sequence is a regular sequence.
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| Let ''R'' be a Noetherian ring, ''I'' an ideal in ''R'', and ''M'' a finitely generated ''R''-module. The '''[[depth (ring theory)|depth]]''' of ''I'' on ''M'', written depth<sub>''R''</sub>(''I'', ''M'') or just depth(''I'', ''M''), is the supremum of the lengths of all ''M''-regular sequences of elements of ''I''. When ''R'' is a Noetherian local ring and ''M'' is a finitely generated ''R''-module, the '''depth''' of ''M'', written depth<sub>''R''</sub>(''M'') or just depth(''M''), means depth<sub>''R''</sub>(''m'', ''M''); that is, it is the supremum of the lengths of all ''M''-regular sequences in the maximal ideal ''m'' of ''R''. In particular, the '''depth''' of a Noetherian local ring ''R'' means the depth of ''R'' as a ''R''-module. That is, the depth of ''R'' is the maximum length of a regular sequence in the maximal ideal.
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| For a Noetherian local ring ''R'', the depth of the zero module is ∞,<ref>A. Grothendieck. EGA IV, Part 1. Publications Mathématiques de l'IHÉS 20 (1964), 259 pp. 0.16.4.5.</ref> whereas the depth of a nonzero finitely generated ''R''-module ''M'' is at most the [[Krull dimension#Krull dimension of a module|Krull dimension]] of ''M'' (also called the dimension of the support of ''M'').<ref>N. Bourbaki. ''Algèbre Commutative. Chapitre 10.'' Springer-Verlag (2007). Th. X.4.2.</ref>
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| ==Examples==
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| *For a prime number ''p'', the local ring '''Z'''<sub>(''p'')</sub> is the subring of the rational numbers consisting of fractions whose denominator is not a multiple of ''p''. The element ''p'' is a non-zero-divisor in '''Z'''<sub>(''p'')</sub>, and the quotient ring of '''Z'''<sub>(''p'')</sub> by the ideal generated by ''p'' is the field '''Z'''/(''p''). Therefore ''p'' cannot be extended to a longer regular sequence in the maximal ideal (''p''), and in fact the local ring '''Z'''<sub>(''p'')</sub> has depth 1.
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| *For any field ''k'', the elements ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> in the polynomial ring ''A'' = ''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>] form a regular sequence. It follows that the [[Localization of a ring|localization]] ''R'' of ''A'' at the maximal ideal ''m'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) has depth at least ''n''. In fact, ''R'' has depth equal to ''n''; that is, there is no regular sequence in the maximal ideal of length greater than ''n''.
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| *More generally, let ''R'' be a [[regular local ring]] with maximal ideal ''m''. Then any elements ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> of ''m'' which map to a basis for ''m''/''m''<sup>2</sub> as an ''R''/''m''-vector space form a regular sequence.
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| An important case is when the depth of a local ring ''R'' is equal to its [[Krull dimension]]: ''R'' is then said to be '''[[Cohen-Macaulay ring|Cohen-Macaulay]]'''. The three examples shown are all Cohen-Macaulay rings. Similarly, a finitely generated ''R''-module ''M'' is said to be '''Cohen-Macaulay''' if its depth equals its dimension.
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| ==Applications==
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| *If ''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> is a regular sequence in a ring ''R'', then the [[Koszul complex]] is an explicit [[Resolution (algebra)|free resolution]] of ''R''/(''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub>) as an ''R''-module, of the form:
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| :<math>0\rightarrow R^{\binom{d}{d}} \rightarrow\cdots \rightarrow
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| R^{\binom{d}{1}} \rightarrow R \rightarrow R/(r_1,\ldots,r_d)
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| \rightarrow 0</math>
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| In the special case where ''R'' is the polynomial ring ''k''[''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub>], this gives a resolution of ''k'' as an ''R''-module.
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| *If ''I'' is an ideal generated by a regular sequence in a ring ''R'', then the associated graded ring
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| :<math>\oplus_{j\geq 0} I^j/I^{j+1}</math>
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| is isomorphic to the polynomial ring (''R''/''I'')[''x''<sub>1</sub>, ..., ''x''<sub>''d''</sub>]. In geometric terms, it follows that a [[Complete intersection ring|local complete intersection]] subscheme ''Y'' of any scheme ''X'' has a [[normal bundle]] which is a vector bundle, even though ''Y'' may be singular. | |
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| ==See also==
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| *[[Complete intersection ring]]
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| *[[Koszul complex]]
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| *[[Depth (ring theory)]]
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| *[[Cohen-Macaulay ring]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Algèbre. Chapitre 10. Algèbre Homologique | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-34492-6 | doi=10.1007/978-3-540-34493-3 | mr=2327161 | year=2006 }}
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| * {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Algèbre Commutative. Chapitre 10 | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-34394-3 | doi=10.1007/978-3-540-34395-0 | mr=2333539 | year=2007 }}
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| * Winfried Bruns; Jürgen Herzog, ''Cohen-Macaulay rings''. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge, 1993. xii+403 pp. ISBN 0-521-41068-1
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| * [[David Eisenbud]], ''Commutative Algebra with a View Toward Algebraic Geometry''. Springer Graduate Texts in Mathematics, no. 150. ISBN 0-387-94268-8
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| *{{Citation | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Éléments de géometrie algébrique IV. Première partie | url=http://www.numdam.org/numdam-bin/fitem?id=PMIHES_1964__20__5_0 | mr=0173675 | year=1964 | journal=Publications Mathématiques de l'Institut des Hautes Études Scientifiques | volume=20 | pages=1–259}}
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| [[Category:Commutative algebra]]
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| [[Category:Dimension]]
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Most belonging to the times all of us talk about conflicts, we mostly for you to the conflict around the world; but, what to your conflict within your inner diy? There is too many conflicts going on within us and the sterling peace symbols remind us to strive for peace within ourselves.
Cufflinks, custom steak branding irons, and swiss army knives is also another popular gift options for groomsmen. Besides these, you can get engraved cufflinks for any father, your brother-in-law, some other important people present at the wedding.
It isn't always simple choose your favorite jewelry that will go collectively with your dress. All this depends from which you shop, you might find a particular necklace made for you and also the dress may have, but sadly you can't find matching earrings. Yes, it is often a tiring activity, all the combo 'n' match, but simply because said you have to be wise here. You'll always get a wedding jewelry sets which can complete in them.
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It makes idea to shine your silver wear using a clean and soft polishing cloth after wearing them. You should have dirt and oils dealt with as a consequence of this is what.
In past years, costume jewelry attempt to look like real boulders. Now, it's popular just to embrace the sparkling accent of faux diamonds and uric acid. A lot of these stones are oversized or have rough cuts. As an alternative to trying to mimic fine jewelry, just together with over guidelines pieces.
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