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{{quotebox|align=right|width=33%
|quote=It turns  out that  there  are applications  of our functors
which  make use  of the analogous  transformations      which we
like  to think  of as a change      of basis    for a fixed root-
system  - a tilting  of the axes relative  to the roots which
results  in a different  subset of roots  lying  in the positive  cone.  ...   For  this reason, and because  the word
'tilt'  inflects  easily,  we call  our functors      {{underline|tilting functors}}  or simply  {{underline|tilts}}.
|source={{harvtxt|Brenner|Butler|1980|loc=p.103}}}}
 
In [[mathematics]], specifically [[representation theory]], '''tilting theory''' describes a way to relate the [[module (mathematics)|module categories]] of two algebras using so-called '''tilting modules''' and associated '''tilting functors'''. Here, the second algebra is the [[endomorphism ring|endomorphism algebra]] of a tilting module over the first algebra.
 
Tilting theory was motivated by the introduction of reflection [[functor]]s by {{harvtxt|Bernšteĭn|Gelfand|Ponomarev|1973}}; these functors were used to relate representations of two [[quiver (mathematics)|quiver]]s. These functors were reformulated by {{harvtxt|Auslander|Platzeck|Reiten|1979}}, and generalized by {{harvtxt|Brenner|Butler|1980}} who introduced tilting functors. {{harvtxt|Happel|Ringel|1982}} defined tilted algebras and tilting modules as further generalizations of this.
 
==Definitions==
Suppose that ''A'' is a finite-dimensional [[unital algebra|unital]] [[associative algebra]] over some [[field (mathematics)|field]]. A [[finitely-generated module|finitely-generated]] right ''A''-module ''T'' is called a '''tilting module''' if it has the following three properties:
*''T'' has [[projective dimension]] at most 1, in other words it is a [[quotient module|quotient]] of a [[projective module]] by a projective submodule.
*[[Ext functor|Ext]]{{su|p=1|b=''A''}}(''T'',''T'') = 0.
*The right ''A''-module ''A'' is the [[kernel (algebra)|kernel]] of a [[surjective]] morphism between finite direct sums of direct summands of ''T''.
 
Given such a tilting module, we define the [[endomorphism ring|endomorphism algebra]] ''B''&nbsp;=&nbsp;End<sub>''A''</sub>(''T''). This is another finite-dimensional algebra, and ''T'' is a finitely-generated left ''B''-module.  
The '''tilting functors''' Hom<sub>''A''</sub>(''T'',&minus;), Ext{{su|p=1|b=''A''}}(''T'',&minus;), &minus;⊗<sub>''B''</sub>''T'' and [[tor functor|Tor]]{{su|p=''B''|b=1}}(&minus;,''T'') relate the category mod-''A'' of finitely-generated right ''A''-modules to the category mod-''B'' of finitely-generated right ''B''-modules.
 
In practice one often considers [[hereditary ring|hereditary]] finite dimensional algebras ''A'' because the module categories over such algebras are fairly well understood. The endomorphism algebra of a tilting module over a hereditary finite dimensional algebra is called a '''tilted algebra'''.
 
==Facts==
Suppose ''A'' is a finite-dimensional algebra, ''T'' is a tilting module over ''A'', and ''B''&nbsp;=&nbsp;End<sub>''A''</sub>(''T''). Write ''F''=Hom<sub>''A''</sub>(''T'',&minus;), ''F&prime;''=Ext{{su|p=1|b=''A''}}(''T'',&minus;), ''G''=&minus;⊗<sub>''B''</sub>''T'', and ''G&prime;''=Tor{{su|p=''B''|b=1}}(&minus;,''T''). ''F'' is [[adjoint functor|right adjoint]] to ''G'' and ''F&prime;'' is right adjoint to ''G&prime;''.
 
{{harvtxt|Brenner|Butler|1980}} showed that tilting functors give equivalences between certain subcategories of mod-''A'' and mod-''B''. Specifically, if we define the two subcategories <math>\mathcal{F}=\ker(F)</math> and <math>\mathcal{T}=\ker(F')</math> of ''A''-mod, and the two subcategories <math>\mathcal{X}=\ker(G)</math> and <math>\mathcal{Y}=\ker(G')</math> of ''B''-mod, then <math>(\mathcal{T},\mathcal{F})</math> is a [[torsion pair]] in ''A''-mod (i.e. <math>\mathcal{T}</math> and <math>\mathcal{F}</math> are maximal subcategories with the property <math>\operatorname{Hom}(\mathcal{T},\mathcal{F})=0</math>; this implies that every ''M'' in ''A''-mod admits a natural short exact sequence <math>0\to U\to M\to V\to 0</math> with ''U'' in <math>\mathcal{T}</math> and ''V'' in <math>\mathcal{F}</math>) and <math>(\mathcal{X},\mathcal{Y})</math> is a torsion pair in ''B''-mod. Further, the restrictions of the functors ''F'' and ''G'' yield inverse [[equivalence (category theory)|equivalence]]s between <math>\mathcal{T}</math> and <math>\mathcal{Y}</math>, while the restrictions of ''F&prime;'' and ''G&prime;'' yield inverse equivalences between <math>\mathcal{F}</math> and <math>\mathcal{X}</math>. (Note that these equivalences switch the order of the torsion pairs <math>(\mathcal{T},\mathcal{F})</math> and <math>(\mathcal{X},\mathcal{Y})</math>.)
 
Tilting theory may be seen as a generalization of [[Morita equivalence]] which is recovered if ''T'' is a [[injective cogenerator|projective generator]]; in that case <math>\mathcal{T}=\operatorname{mod}-A</math> and <math>\mathcal{Y}=\operatorname{mod}-B</math>.
 
If ''A'' has finite [[global dimension]], then ''B'' also has finite global dimension, and the difference of ''F'' and ''F''' induces an isometry between the [[Grothendieck group]]s K<sub>0</sub>(''A'') and K<sub>0</sub>(''B'').
 
In case ''A'' is hereditary (i.e. ''B'' is a tilted algebra), the global dimension of ''B'' is at most 2, and the torsion pair <math>(\mathcal{X},\mathcal{Y})</math> splits, i.e. every indecomposable object of ''B''-mod is either in <math>\mathcal{X}</math> or in <math>\mathcal{Y}</math>.
 
{{harvtxt|Happel|1988}} and {{harvtxt|Cline, Parshall, Scott|1986}} showed that in general ''A'' and ''B'' are [[derived equivalence|derived equivalent]] (i.e. the [[derived category|derived categories]] D<sup>b</sub>(''A''-mod) and D<sup>b</sub>(''B''-mod) are equivalent as [[triangulated category|triangulated categories]]).
 
==Generalizations and extensions==
A '''generalized tilting module''' over the finite-dimensional algebra ''A'' is a right ''A''-module ''T'' with the following three properties:
*''T'' has finite projective dimension.
*[[Ext functor|Ext]]{{su|p=''i''|b=''A''}}(''T'',''T'') = 0 for all ''i''>0.
*There is an exact sequence <math>0\to A\to T_1\to \dots\to T_n\to 0</math> where the ''T<sub>i</sub>'' are finite direct sums of direct summands of ''T''.
These generalized tilting modules also yield derived equivalences between ''A'' and ''B'', where ''B''=End<sub>''A''</sub>(''T'').
 
{{harvtxt|Rickard|1989}} extended the results on derived equivalence by proving that two finite-dimensional algebras ''R'' and ''S'' are derived equivalent if and only if ''S'' is the endomorphism algebra of a "tilting complex" over ''R''. Tilting complexes are generalizations of generalized tilting modules. A version of this theorem is valid for arbitrary rings ''R'' and ''S''.
 
{{harvtxt|Happel, Reiten, Smalø|1996}} defined tilting objects in hereditary abelian categories in which all Hom- and Ext-spaces are finite-dimensional over some [[algebraically closed field]] ''k''. The endomorphism algebras of these tilting objects are the '''quasi-tilted algebras''', a generalization of tilted algebras. The quasi-tilted algebras over ''k'' are precisely the finite-dimensional algebras over ''k'' of global dimension &le; 2 such that every indecomposable module either has projective dimension &le; 1 or injective dimension &le; 1. {{harvtxt|Happel|2001}} classified the hereditary abelian categories that can appear in the above construction.
 
{{harvtxt|Colpi & Fuller|2007}} defined tilting objects ''T'' in an arbitrary [[abelian category]] ''C''; their definition requires that ''C'' contain the direct sums of arbitrary (possibly infinite) numbers of copies of ''T'', so this is not a direct generalization of the finite-dimensional situation considered above. Given such a tilting object with endomorphism ring ''R'', they establish tilting functors that provide equivalences between a torsion pair in ''C'' and a torsion pair in ''R''-Mod, the category of ''all'' ''R''-modules.
 
From the theory of [[cluster algebra]]s came the definition of '''cluster category''' and '''cluster tilted algebra''' associated to a hereditary algebra ''A''. A cluster tilted algebra arises from a tilted algebra as a certain [[semidirect product]], and the cluster category of ''A'' summarizes all the module categories of cluster tilted algebras arising from ''A''.
 
==References==
*{{Citation | editor1-last=Angeleri Hügel | editor1-first=Lidia | editor2-last=Happel | editor2-first=Dieter | editor3-last=Krause | editor3-first=Henning | title=Handbook of tilting theory | url=http://www.maths.ed.ac.uk/~aar/papers/tilting.pdf | publisher=[[Cambridge University Press]] | series=London Mathematical Society Lecture Note Series | isbn=978-0-521-68045-5; 978-0-521-68045-5 | doi=10.1017/CBO9780511735134 | mr=2385175 | year=2007 | volume=332}}
*{{Citation | last1=Assem | first1=Ibrahim | editor1-last=Balcerzyk | editor1-first=Stanisław | editor2-last=Józefiak | editor2-first=Tadeusz | editor3-last=Krempa | editor3-first=Jan | editor4-last=Simson | editor4-first=Daniel | editor5-last=Vogel | editor5-first=Wolfgang | title=Topics in algebra, Part 1 (Warsaw, 1988) | url=http://journals.impan.gov.pl/BC/oldindex.html | publisher=PWN | location=Warszawa | series=Banach Center Publ. | mr=1171230 | year=1990 | journal=Banach Center Publications | volume=26 | chapter=Tilting theory---an introduction | pages=127–180}}
*{{Citation | last1=Auslander | first1=Maurice | last2=Platzeck | first2=María Inés | last3=Reiten | first3=Idun | title=Coxeter functors without diagrams | doi=10.2307/1998978 | mr=530043 | year=1979 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=250 | pages=1–46}}
*{{Citation | last1=Bernšteĭn | first1=I. N. | last2=Gelfand | first2=I. M. | last3=Ponomarev | first3=V. A. | title=Coxeter functors, and Gabriel's theorem | doi=10.1070/RM1973v028n02ABEH001526 | mr=0393065 | year=1973 | journal=Russian mathematical surveys | issn=0042-1316 | volume=28 | issue=2 | pages=17–32}}
*{{Citation | last1=Brenner | first1=Sheila | last2=Butler | first2=M. C. R. | title=Representation theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979) | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Math. | doi=10.1007/BFb0088461 | mr=607151 | year=1980 | volume=832 | chapter=Generalizations of the Bernstein-Gel'fand-Ponomarev reflection functors | pages=103–169}}
*{{Citation| last1=Cline | first1= E.| last2=Parshall| first2=B.|last3=Scott|first3=L.|title=Derived categories and Morita theory|journal=Algebra|volume=104|year=1986|pages=397–409|url=http://www.math.virginia.edu/~lls2l/derived_categories_and_morita_theory.pdf}}
*{{Citation|journal=Transactions of the American Mathematical Society|volume=359|number=2|date=February 2007|pages=741–765|url=http://www.ams.org/journals/tran/2007-359-02/S0002-9947-06-03909-2/S0002-9947-06-03909-2.pdf|title=Tilting Objects in Abelian Categories and Quasitilted Rings|first1=Riccardo|last1=Colpi|first2=Kent R.|last2=Fuller}}
*{{Citation| first1=Dieter|last1=Happel|first2=Idun|last2=Reiten|first3=S.O.|last3=Smalø|title=Tilting in abelian categories and quasitilted algebras|journal=Memoirs American Mathematical Society|volume=575|year=1996}}
*{{Citation | last1=Happel | first1=Dieter | last2=Ringel | first2=Claus Michael | title=Tilted algebras | doi=10.2307/1999116 | mr=675063 | year=1982 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=274 | issue=2 | pages=399–443}}
*{{Citation | last1=Happel | first1=Dieter | title=Triangulated categories in the representation theory of finite-dimensional algebras|year=1988|series=London Mathematical Society Lecture Notes|volume=119|publisher=Cambridge University Press|url=http://staff.ustc.edu.cn/~yhbao/WRTA2012/download/III-3-triangulated-categories-in-the-representation-theory-of-finite-dimensional-algebras.pdf}}
*{{Citation|first1=Dieter|last1=Happel|title=A characterization of hereditary categories with tilting object|journal=Invent. Math.|volume=144|year=2001|number=2|pages=381–398}}
*{{Citation | last1=Rickard| first1=Jeremy|title=Morita theory for derived categories|journal=Journal London Mathematical Society|number=2|volume=39|year=1989|pages=436–456|url=http://www1.ahu.edu.cn/math/mathweb2/wrta2012/download/III-4-Morita-theory-for-derived-categories.pdf}}
*{{eom|id=T/t130130|title=Tilting theory|first=L.|last= Unger}}
 
[[Category:Representation theory]]

Revision as of 12:51, 15 March 2013

Template:Quotebox

In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra.

Tilting theory was motivated by the introduction of reflection functors by Template:Harvtxt; these functors were used to relate representations of two quivers. These functors were reformulated by Template:Harvtxt, and generalized by Template:Harvtxt who introduced tilting functors. Template:Harvtxt defined tilted algebras and tilting modules as further generalizations of this.

Definitions

Suppose that A is a finite-dimensional unital associative algebra over some field. A finitely-generated right A-module T is called a tilting module if it has the following three properties:

Given such a tilting module, we define the endomorphism algebra B = EndA(T). This is another finite-dimensional algebra, and T is a finitely-generated left B-module. The tilting functors HomA(T,−), ExtTemplate:Su(T,−), −⊗BT and TorTemplate:Su(−,T) relate the category mod-A of finitely-generated right A-modules to the category mod-B of finitely-generated right B-modules.

In practice one often considers hereditary finite dimensional algebras A because the module categories over such algebras are fairly well understood. The endomorphism algebra of a tilting module over a hereditary finite dimensional algebra is called a tilted algebra.

Facts

Suppose A is a finite-dimensional algebra, T is a tilting module over A, and B = EndA(T). Write F=HomA(T,−), F′=ExtTemplate:Su(T,−), G=−⊗BT, and G′=TorTemplate:Su(−,T). F is right adjoint to G and F′ is right adjoint to G′.

Template:Harvtxt showed that tilting functors give equivalences between certain subcategories of mod-A and mod-B. Specifically, if we define the two subcategories and of A-mod, and the two subcategories and of B-mod, then is a torsion pair in A-mod (i.e. and are maximal subcategories with the property ; this implies that every M in A-mod admits a natural short exact sequence with U in and V in ) and is a torsion pair in B-mod. Further, the restrictions of the functors F and G yield inverse equivalences between and , while the restrictions of F′ and G′ yield inverse equivalences between and . (Note that these equivalences switch the order of the torsion pairs and .)

Tilting theory may be seen as a generalization of Morita equivalence which is recovered if T is a projective generator; in that case and .

If A has finite global dimension, then B also has finite global dimension, and the difference of F and F' induces an isometry between the Grothendieck groups K0(A) and K0(B).

In case A is hereditary (i.e. B is a tilted algebra), the global dimension of B is at most 2, and the torsion pair splits, i.e. every indecomposable object of B-mod is either in or in .

Template:Harvtxt and Template:Harvtxt showed that in general A and B are derived equivalent (i.e. the derived categories Db(A-mod) and Db(B-mod) are equivalent as triangulated categories).

Generalizations and extensions

A generalized tilting module over the finite-dimensional algebra A is a right A-module T with the following three properties:

These generalized tilting modules also yield derived equivalences between A and B, where B=EndA(T).

Template:Harvtxt extended the results on derived equivalence by proving that two finite-dimensional algebras R and S are derived equivalent if and only if S is the endomorphism algebra of a "tilting complex" over R. Tilting complexes are generalizations of generalized tilting modules. A version of this theorem is valid for arbitrary rings R and S.

Template:Harvtxt defined tilting objects in hereditary abelian categories in which all Hom- and Ext-spaces are finite-dimensional over some algebraically closed field k. The endomorphism algebras of these tilting objects are the quasi-tilted algebras, a generalization of tilted algebras. The quasi-tilted algebras over k are precisely the finite-dimensional algebras over k of global dimension ≤ 2 such that every indecomposable module either has projective dimension ≤ 1 or injective dimension ≤ 1. Template:Harvtxt classified the hereditary abelian categories that can appear in the above construction.

Template:Harvtxt defined tilting objects T in an arbitrary abelian category C; their definition requires that C contain the direct sums of arbitrary (possibly infinite) numbers of copies of T, so this is not a direct generalization of the finite-dimensional situation considered above. Given such a tilting object with endomorphism ring R, they establish tilting functors that provide equivalences between a torsion pair in C and a torsion pair in R-Mod, the category of all R-modules.

From the theory of cluster algebras came the definition of cluster category and cluster tilted algebra associated to a hereditary algebra A. A cluster tilted algebra arises from a tilted algebra as a certain semidirect product, and the cluster category of A summarizes all the module categories of cluster tilted algebras arising from A.

References

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    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
  • 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
  • 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
  • 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
  • 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
  • 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
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