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{{for|regular Cauchy sequence|Cauchy sequence#In constructive mathematics}} | |||
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]]. | |||
==Definitions== | |||
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 | |||
:''r''<sub>1</sub>, ..., ''r''<sub>''d''</sub> in ''R'' | |||
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 | |||
''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. | |||
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''. | |||
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. | |||
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. | |||
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> | |||
==Examples== | |||
*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. | |||
*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''. | |||
*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. | |||
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. | |||
==Applications== | |||
*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: | |||
:<math>0\rightarrow R^{\binom{d}{d}} \rightarrow\cdots \rightarrow | |||
R^{\binom{d}{1}} \rightarrow R \rightarrow R/(r_1,\ldots,r_d) | |||
\rightarrow 0</math> | |||
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. | |||
*If ''I'' is an ideal generated by a regular sequence in a ring ''R'', then the associated graded ring | |||
:<math>\oplus_{j\geq 0} I^j/I^{j+1}</math> | |||
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. | |||
==See also== | |||
*[[Complete intersection ring]] | |||
*[[Koszul complex]] | |||
*[[Depth (ring theory)]] | |||
*[[Cohen-Macaulay ring]] | |||
==Notes== | |||
{{reflist}} | |||
== References == | |||
* {{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 }} | |||
* {{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 }} | |||
* 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 | |||
* [[David Eisenbud]], ''Commutative Algebra with a View Toward Algebraic Geometry''. Springer Graduate Texts in Mathematics, no. 150. ISBN 0-387-94268-8 | |||
*{{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}} | |||
[[Category:Commutative algebra]] | |||
[[Category:Dimension]] | |||
Revision as of 10:46, 20 January 2014
28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. 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.
Definitions
For a commutative ring R and an R-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
- r1, ..., rd in R
such that ri is a non-zero-divisor on M/(r1, ..., ri-1)M for i = 1, ..., d.[1] Some authors also require that M/(r1, ..., rd)M is not zero. Intuitively, to say that r1, ..., rd is an M-regular sequence means that these elements "cut M down" as much as possible, when we pass successively from M to M/(r1)M, to M/(r1, r2)M, and so on.
An R-regular sequence is called simply a regular sequence. That is, r1, ..., rd is a regular sequence if r1 is a non-zero-divisor in R, r2 is a non-zero-divisor in the ring R/(r1), and so on. In geometric language, if X is an affine scheme and r1, ..., rd is a regular sequence in the ring of regular functions on X, then we say that the closed subscheme {r1=0, ..., rd=0} ⊂ X is a complete intersection subscheme of X.
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 local ring and the elements ri are in the maximal ideal, or if R is a graded ring and the ri are homogeneous of positive degree, then any permutation of a regular sequence is a regular sequence.
Let R be a Noetherian ring, I an ideal in R, and M a finitely generated R-module. The depth of I on M, written depthR(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 depthR(M) or just depth(M), means depthR(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.
For a Noetherian local ring R, the depth of the zero module is ∞,[2] whereas the depth of a nonzero finitely generated R-module M is at most the Krull dimension of M (also called the dimension of the support of M).[3]
Examples
- For a prime number p, the local ring Z(p) 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(p), and the quotient ring of Z(p) 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(p) has depth 1.
- For any field k, the elements x1, ..., xn in the polynomial ring A = k[x1, ..., xn] form a regular sequence. It follows that the localization R of A at the maximal ideal m = (x1, ..., xn) 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.
- More generally, let R be a regular local ring with maximal ideal m. Then any elements r1, ..., rd of m which map to a basis for m/m2 as an R/m-vector space form a regular sequence.
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. 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.
Applications
- If r1, ..., rd is a regular sequence in a ring R, then the Koszul complex is an explicit free resolution of R/(r1, ..., rd) as an R-module, of the form:
In the special case where R is the polynomial ring k[r1, ..., rd], this gives a resolution of k as an R-module.
- If I is an ideal generated by a regular sequence in a ring R, then the associated graded ring
is isomorphic to the polynomial ring (R/I)[x1, ..., xd]. In geometric terms, it follows that a local complete intersection subscheme Y of any scheme X has a normal bundle which is a vector bundle, even though Y may be singular.
See also
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 - 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
- David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Springer Graduate Texts in Mathematics, no. 150. ISBN 0-387-94268-8
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010