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Tensor Product Of Free Abelian Groups Pdf

 
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MessagePosté le: Mar 28 Mar - 18:04 (2017)    Sujet du message: Tensor Product Of Free Abelian Groups Pdf Répondre en citant




Tensor Product Of Free Abelian Groups Pdf > bit.ly/2o7MjU6









Tensor Product Of Free Abelian Groups Pdf

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Array,,programming,,languages[edit]The,,tensor,,product,,as,,defined,,above,,satisfies,,a,,universal,,propertyThe,,tensor,,product,,of,,fields,,is,,closely,,related,,to,,Galois,,theory:,,if,,,say,,,A,,=,,R[x],,/,,f(x),,,where,,f,,is,,some,,irreducible,,polynomial,,with,,coefficients,,in,,R,,,the,,tensor,,product,,can,,be,,calculated,,asExample:,,If,,G,,is,,an,,abelian,,group,,,G,,⊗,,Z,,Z,,/,,n,,=,,G,,/,,n,,G,,{displaystyle,,Gotimes,,{mathbb,,Transclusion,,expansion,,time,,report,,(%,ms,calls,template),,100.00%,,211.689,,1,,-total,,20.43%,,43.255,,3,,Template:Citation,,19.54%,,41.371,,3,,Template:Navbox,,17.18%,,36.363,,1,,Template:Cn,,15.22%,,32.214,,1,,Template:Fix,,13.90%,,29.433,,1,,Template:Tensors,,10.20%,,21.586,,6,,Template:Sectionlink,,8.64%,,18.293,,2,,Template:Navboxsubgroup,,8.53%,,18.056,,2,,Template:Categoryhandler,,8.24%,,17.447,,10,,Template:Harvnb,,8,,}mathbb,,Transclusion,,expansion,,time,,report,,(%,ms,calls,template),,100.00%,,211.689,,1,,-total,,20.43%,,43.255,,3,,Template:Citation,,19.54%,,41.371,,3,,Template:Navbox,,17.18%,,36.363,,1,,Template:Cn,,15.22%,,32.214,,1,,Template:Fix,,13.90%,,29.433,,1,,Template:Tensors,,10.20%,,21.586,,6,,Template:Sectionlink,,8.64%,,18.293,,2,,Template:Navboxsubgroup,,8.53%,,18.056,,2,,Template:Categoryhandler,,8.24%,,17.447,,10,,Template:Harvnb,,7,,/n=G/nG},,;,,this,,follows,,from,,1Hom,,Z,,⁡,,(,,M,,⊗,,R,,N,,,,,G,,),,≃,,Hom,,R,,⁡,,(,,M,,,,,Hom,,Z,,⁡,,(,,N,,,,,G,,),,),,{displaystyle,,operatorname,,,,2,,{mathbb,,,,1,,}(Motimes,,,,,,0N,G)simeq,,operatorname,,,,9,,,,,,8(M,operatorname,,,,7,,{mathbb,,,,6,,}(N,G))},,ExamplesThe,,,tensor,,,product,,,of,,,two,,,modules,,,A,,,and,,,B,,,over,,,a,,,commutative,,,ring,,,R,,,is,,,defined,,,in,,,exactly,,,the,,,same,,,way,,,as,,,the,,,tensor,,,product,,,of,,,vector,,,spaces,,,over,,,a,,,field:⟨,,r,,⋅,,g,,,,,h,,⋅,,s,,⟩,,=,,r,,⋅,,⟨,,g,,,,,h,,⟩,,⋅,,s,,,,,r,,,,,s,,∈,,R,,It,,,should,,,be,,,mentioned,,,that,,,,though,,,called,,,"tensor,,,product",,,,this,,,is,,,not,,,a,,,tensor,,,product,,,of,,,graphs,,,in,,,the,,,above,,,sense;,,,actually,,,it,,,is,,,the,,,category-theoretic,,,product,,,in,,,the,,,category,,,of,,,graphs,,,and,,,graph,,,homomorphismsThe,,,pairing,,,is,,,left,,,R-linear,,,in,,,its,,,left,,,argument,,,,and,,,right,,,R-linear,,,in,,,its,,,right,,,argument:

Tensor,,product,,of,,algebras[edit]If,S,is,a,subring,of,a,ring,R,,then,M,⊗,R,N,{displaystyle,Motimes,{R}N},is,the,quotient,group,of,M,⊗,S,N,{displaystyle,Motimes,{S}N},by,the,subgroup,generated,by,x,r,⊗,S,y,−,x,⊗,S,r,y,,,r,∈,R,,,x,∈,M,,,y,∈,N,{displaystyle,xrotimes,{S}y-xotimes,{S}ry,,rin,R,xin,M,yin,N},,,where,x,⊗,S,y,{displaystyle,xotimes,{S}y},is,the,image,of,(,x,,,y,),{displaystyle,(x,y)},under,⊗,:,M,,N,→,M,⊗,S,N,Tensor,,,product,,,of,,,vector,,,spaces[edit]References[edit]Example:,,,We,,,propose,,,to,,,compare,,,R,,,⊗,,,Z,,,R,,,{displaystyle,,,mathbb,,,{R},,,otimes,,,{mathbb,,,{Z},,,}mathbb,,,{R},,,},,,and,,,R,,,⊗,,,R,,,R,,,{displaystyle,,,mathbb,,,{R},,,otimes,,,{mathbb,,,{R},,,}mathbb,,,{R},,,},,,with,,,the,,,differential,,,given,,,by:,,,for,,,x,,,in,,,Xi,,,and,,,y,,,in,,,Yj,Given,linear,maps,f,:,M,→,M,′,{displaystyle,f:Mto,M'},of,right,modules,over,a,ring,R,and,g,:,N,→,N,′,{displaystyle,g:Nto,N'},of,left,modules,,there,is,a,unique,group,homomorphismIn,,,the,,,general,,,case,,,,each,,,element,,,of,,,the,,,tensor,,,product,,,of,,,modules,,,gives,,,rise,,,to,,,a,,,left,,,R-linear,,,map,,,,to,,,a,,,right,,,R-linear,,,map,,,,and,,,to,,,an,,,R-bilinear,,,formThe,,construction,,has,,a,,consequence,,that,,tensoring,,is,,a,,functor:,,each,,right,,R-module,,M,,determines,,the,,functorOther,examples,of,tensor,products[edit]

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