Lobb numbers: Difference between revisions

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In mathematics, a Dieudonné module introduced by Template:Harvs, is a module over the non-commutative Dieudonné ring, which is generated over the ring of Witt vectors by two special endomorphisms F and V called the Frobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.

Finite flat commutative group schemes over a perfect field k of positive characteristic p can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting. The basic object is the Dieudonné ring D = W(k){F,V}/(FV − p), which is a quotient of the ring of noncommutative polynomials, with coefficients in Witt vectors of k. F and V are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors. Jean Dieudonné and Pierre Cartier constructed an antiequivalence of categories between finite commutative group schemes over k of order a power of "p" and modules over D with finite W(k)-length. The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf CW of Witt co-vectors. This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by taking a direct limit of finite length Witt vectors under successive Verschiebung maps V: W_n → W_n+1, and then completing. Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected p-group schemes correspond to D-modules for which F is nilpotent, and étale group schemes correspond to modules for which F is an isomorphism.

Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field. Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze p-divisible groups. Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in Wiles's work on the Shimura–Taniyama conjecture.

Dieudonné rings

If k is a field of characteristic p, its ring of Witt vectors consists of sequences (w₁,w₂,w₃,...) of elements of k, and has an endomorphism σ induced by the Frobenius endomorphism of k, so (w₁,w₂,w₃,...)^σ = (wTemplate:Su,wTemplate:Su,wTemplate:Su,...). The Dieudonné ring, often denoted by E_k or D_k, is the non-commutative ring over W(k) generated by 2 elements F and V subject to the relations

FV = VF = p

Fw = w^σF

wV = Vw^σ.

It is a Z-graded ring, where the piece of degree n∈Z is a 1-dimensional free module over W(k), spanned by V⁻ⁿ if n≤0 and by Fⁿ if n≥0.

Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by F and V.

Dieudonné modules and groups

Special sorts of modules over the Dieudonné ring correspond to certain algebraic group schemes. For example, finite length modules over the Dieudonné ring form an abelian category equivalent to the opposite of the category of finite commutative p-group schemes over k.

Examples

If $X$ is the constant group scheme $\mathbb {Z} /p\mathbb {Z}$ over $k$ , then its corresponding Dieudonné module $\mathbf {D} (X)$ is $k$ with $F=\mathrm {Frob} _{k}$ and $V=0$ .
For the scheme of p-th roots of unity $X=\mu _{p}$ , then its corresponding Dieudonné module is $\mathbf {D} (X)=k$ with $F=0$ and $V=\mathrm {Frob} _{k}^{-1}$ .
For $X=\alpha _{p}$ , defined as the kernel of the Frobenius $\mathbb {G} _{a}\to \mathbb {G} _{a}$ , the Dieudonné module is $\mathbf {D} (X)=k$ with $F=V=0$ .
If $X=E[p]$ is the p-torsion of an elliptic curve over k (with p-torsion in k), then the Dieudonné module depends on whether E is supersingular or not.

References

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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.

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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.

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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.

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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.

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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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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.

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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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+In mathematics, a '''Dieudonné module''' introduced by {{harvs|txt|last=Dieudonné|authorlink=Jean Dieudonné|year1=1954|year2=1957b}},  is a [[module (mathematics)|module]] over the non-commutative '''Dieudonné ring''', which is generated over the ring of [[Witt vector]]s by two special  endomorphisms ''F'' and ''V'' called the Frobenius and Verschiebung operators. They are used for studying finite flat commutative group schemes.
+Finite flat commutative group schemes over a perfect [[field (mathematics)|field]] ''k'' of positive characteristic ''p'' can be studied by transferring their geometric structure to a (semi-)linear-algebraic setting.  The basic object is the Dieudonné ring ''D'' = ''W''(''k''){''F'',''V''}/(''FV''&nbsp;&minus;&nbsp;''p''), which is a quotient of the ring of noncommutative polynomials, with coefficients in [[Witt vectors]] of ''k''.  ''F'' and ''V'' are the Frobenius and Verschiebung operators, and they may act nontrivially on the Witt vectors.  [[Jean Dieudonné]] and [[Pierre Cartier (mathematician)|Pierre Cartier]] constructed an [[antiequivalence of categories]] between finite commutative group schemes over ''k'' of order a power of "p" and modules over ''D'' with finite ''W''(''k'')-length.  The Dieudonné module functor in one direction is given by homomorphisms into the abelian sheaf ''CW'' of Witt co-vectors.  This sheaf is more or less dual to the sheaf of Witt vectors (which is in fact representable by a group scheme), since it is constructed by  taking a direct limit of finite length Witt vectors under successive Verschiebung maps ''V'': ''W''<sub>n</sub> → ''W''<sub>n+1</sub>, and then completing.  Many properties of commutative group schemes can be seen by examining the corresponding Dieudonné modules, e.g., connected ''p''-group schemes correspond to ''D''-modules for which ''F'' is nilpotent, and étale group schemes correspond to modules for which ''F'' is an isomorphism.
+Dieudonné theory exists in a somewhat more general setting than finite flat groups over a field.  Oda's 1967 thesis gave a connection between Dieudonné modules and the first de Rham cohomology of abelian varieties, and at about the same time, Grothendieck suggested that there should be a crystalline version of the theory that could be used to analyze ''p''-divisible groups.  Galois actions on the group schemes transfer through the equivalences of categories, and the associated deformation theory of Galois representations was used in [[Andrew Wiles|Wiles]]'s work on the [[Shimura–Taniyama conjecture]].
+==Dieudonné rings==
+If ''k'' is a field of characteristic ''p'', its ring of [[Witt vector]]s consists of sequences (''w''<sub>1</sub>,''w''<sub>2</sub>,''w''<sub>3</sub>,...) of elements of ''k'', and has an endomorphism σ induced by the Frobenius endomorphism of ''k'', so (''w''<sub>1</sub>,''w''<sub>2</sub>,''w''<sub>3</sub>,...)<sup>σ</sup> = (''w''{{su|b=1|p=''p''}},''w''{{su|b=2|p=''p''}},''w''{{su|b=3|p=''p''}},...). The '''Dieudonné ring''', often denoted by ''E''<sub>''k''</sub> or D<sub>''k''</sub>, is the non-commutative ring over ''W''(''k'') generated by 2 elements ''F'' and ''V'' subject to the relations
+:''FV'' = ''VF'' = ''p''
+:''Fw'' = ''w''<sup>&sigma;</sup>''F''
+:''wV'' = ''Vw''<sup>&sigma;</sup>.
+It is a ''Z''-graded ring, where the piece of  degree ''n''∈''Z'' is a 1-dimensional free module over ''W''(''k''), spanned by ''V''<sup>&minus;''n''</sup> if ''n''≤0 and by ''F''<sup>''n''</sup> if ''n''≥0.
+Some authors define the Dieudonné ring to be the completion of the ring above for the ideal generated by ''F'' and ''V''.
+==Dieudonné modules and groups==
+Special sorts of modules over the Dieudonné ring correspond to certain algebraic group schemes. For example, finite length modules over the Dieudonné ring form an abelian category equivalent to the opposite of the category of finite commutative ''p''-group schemes over ''k''.
+==Examples==
+* If <math>X</math> is the constant group scheme <math>\mathbb{Z}/p\mathbb{Z}</math> over <math>k</math>, then its corresponding Dieudonné module <math> \mathbf{D}(X) </math> is <math>k</math> with <math> F = \mathrm{Frob}_k</math> and <math> V = 0 </math>.
+* For the scheme of ''p''-th roots of unity <math>X = \mu_p</math>, then its corresponding Dieudonné module is <math> \mathbf{D}(X) = k</math> with <math> F = 0 </math> and <math> V = \mathrm{Frob}_k^{-1}</math>.
+* For <math> X = \alpha_p</math>, defined as the kernel of the Frobenius <math> \mathbb{G}_{a} \to \mathbb{G}_{a}</math>, the Dieudonné module is <math> \mathbf{D}(X) = k</math> with <math> F = V = 0 </math>.
+* If <math> X = E[p] </math> is the ''p''-torsion of an elliptic curve over ''k'' (with ''p''-torsion in ''k''), then the Dieudonné module depends on whether ''E'' is [[supersingular elliptic curve|supersingular]] or not.
+==References==
+*{{Citation | last1=Cartier | first1=Pierre | title=Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) | url=http://www.famaf.unc.edu.ar/andrus/papers/cartier1.pdf | publisher=Librairie Universitaire, Louvain | mr=0148665 | year=1962 | chapter=Groupes algébriques et groupes formels | pages=87–111}}
+*{{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Lie groups and Lie hyperalgebras over a field of characteristic p>0. IV | jstor=2372633 | mr=0071718 | year=1955 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=77 | pages=429–452}}
+*{{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Lie groups and Lie hyperalgebras over a field of characteristic p>0. VI | jstor=2372686 | mr=0094413 | year=1957 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=79 | pages=331–388}}
+*{{Citation | last1=Dieudonné | first1=Jean | author1-link=Jean Dieudonné | title=Groupes de Lie et hyperalgèbres de Lie sur un corps de caractéristique p>0. VII | doi=10.1007/BF01342790 | mr=0098146 | year=1957b | journal=[[Mathematische Annalen]] | issn=0025-5831 | volume=134 | pages=114–133}}
+*{{eom|id=D/d031640|first=I.V.|last= Dolgachev}}
+*{{Citation | last1=Manin | first1=Ju. I. | title=Theory of commutative formal groups over fields of finite characteristic | doi=10.1070/RM1963v018n06ABEH001142 | mr=0157972 | year=1963 | journal=Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk | issn=0042-1316 | volume=18 | issue=6 | pages=3–90}}
+{{DEFAULTSORT:Dieudonne module}}
+[[Category:Algebraic groups]]

Lobb numbers: Difference between revisions

Latest revision as of 12:43, 25 January 2014

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Dieudonné rings

Dieudonné modules and groups

Examples

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Lobb numbers: Difference between revisions

Latest revision as of 12:43, 25 January 2014

Dieudonné rings

Dieudonné modules and groups

Examples

References

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