Localization of self dual algebra over a ring
$begingroup$
Let $R$ be a commutative ring with identity and $A$ an $R$-algebra that is finitely generated, projective $R$-module and has identity. Furthermore, assume that $A$ and $Hom_R(A,R)$ are isomorphic as $A$-modules.
For $varphiin Hom_R(A,R)$ and $ain A$, $avarphi(x)=varphi(ax)$.
If for an element $ain A$ one takes the localization $A_a$, we have that
$A_acong Hom_R(A,R)otimes_A A_a$. My question is, how can I think of the elements in $Hom_R(A,R)otimes_A A_a$?
If $Hom_R(A,R)$ is generated by $eta:Arightarrow R$ as an $A$-module and $ain kereta$, can I define $a^{-1}eta$ as a homomorphism?
commutative-algebra
asked Jan 11 at 12:03
BajoucaBajouca
11
$endgroup$
add a comment |
$begingroup$
Let $R$ be a commutative ring with identity and $A$ an $R$-algebra that is finitely generated, projective $R$-module and has identity. Furthermore, assume that $A$ and $Hom_R(A,R)$ are isomorphic as $A$-modules.
For $varphiin Hom_R(A,R)$ and $ain A$, $avarphi(x)=varphi(ax)$.
If for an element $ain A$ one takes the localization $A_a$, we have that
$A_acong Hom_R(A,R)otimes_A A_a$. My question is, how can I think of the elements in $Hom_R(A,R)otimes_A A_a$?
If $Hom_R(A,R)$ is generated by $eta:Arightarrow R$ as an $A$-module and $ain kereta$, can I define $a^{-1}eta$ as a homomorphism?
commutative-algebra
asked Jan 11 at 12:03
BajoucaBajouca
11
$endgroup$
add a comment |
$begingroup$
Let $R$ be a commutative ring with identity and $A$ an $R$-algebra that is finitely generated, projective $R$-module and has identity. Furthermore, assume that $A$ and $Hom_R(A,R)$ are isomorphic as $A$-modules.
For $varphiin Hom_R(A,R)$ and $ain A$, $avarphi(x)=varphi(ax)$.
If for an element $ain A$ one takes the localization $A_a$, we have that
$A_acong Hom_R(A,R)otimes_A A_a$. My question is, how can I think of the elements in $Hom_R(A,R)otimes_A A_a$?
If $Hom_R(A,R)$ is generated by $eta:Arightarrow R$ as an $A$-module and $ain kereta$, can I define $a^{-1}eta$ as a homomorphism?
commutative-algebra
asked Jan 11 at 12:03
BajoucaBajouca
11
$endgroup$
Let $R$ be a commutative ring with identity and $A$ an $R$-algebra that is finitely generated, projective $R$-module and has identity. Furthermore, assume that $A$ and $Hom_R(A,R)$ are isomorphic as $A$-modules.
For $varphiin Hom_R(A,R)$ and $ain A$, $avarphi(x)=varphi(ax)$.
If for an element $ain A$ one takes the localization $A_a$, we have that
$A_acong Hom_R(A,R)otimes_A A_a$. My question is, how can I think of the elements in $Hom_R(A,R)otimes_A A_a$?
If $Hom_R(A,R)$ is generated by $eta:Arightarrow R$ as an $A$-module and $ain kereta$, can I define $a^{-1}eta$ as a homomorphism?
commutative-algebra
commutative-algebra
asked Jan 11 at 12:03
BajoucaBajouca
11
asked Jan 11 at 12:03
BajoucaBajouca
11
asked Jan 11 at 12:03
BajoucaBajouca
11
asked Jan 11 at 12:03
BajoucaBajouca
11
asked Jan 11 at 12:03
BajoucaBajouca
11
11
add a comment |
add a comment |
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