How to apply a double centralizer property on a faithful module of a self-injective Artin algebra?
$begingroup$
Let all considered algebras be Artin algebras and let all considered modules be finitely generated.
Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent:
$bullet$ $Ae$-dom.dim.$(A)geq 2$
$bullet$ There holds a double centralizer property $A=text{End}(Ae_{eAe})$
(This is theorem 2.10 of http://www.sciencedirect.com/science/article/pii/S002186930098726X#).
Corollary 2.11 of the same paper states:
Let $A$ be a self-injectie algebra, let $M$ be a faithful
$A$-module, and let $B:=text{End}_A(M)$. Then there is a double centralizer property $A=text{End}(M_B)$.
The proof of 2.11 shows that dom.dim.($B$)$geq 2$.
Now my question is:
How to apply 2.10 in order to show 2.11?
It seems that the $M$ in 2.11 is the $Ae$ in 2.10. But why is $M$ projective? and must $M$ be a minimal faithful projective module?
Thanks for the help.
abstract-algebra ring-theory representation-theory homological-algebra
asked Aug 7 '15 at 14:29
Stein ChenStein Chen
1157
$endgroup$
add a comment |
$begingroup$
Let all considered algebras be Artin algebras and let all considered modules be finitely generated.
Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent:
$bullet$ $Ae$-dom.dim.$(A)geq 2$
$bullet$ There holds a double centralizer property $A=text{End}(Ae_{eAe})$
(This is theorem 2.10 of http://www.sciencedirect.com/science/article/pii/S002186930098726X#).
Corollary 2.11 of the same paper states:
Let $A$ be a self-injectie algebra, let $M$ be a faithful
$A$-module, and let $B:=text{End}_A(M)$. Then there is a double centralizer property $A=text{End}(M_B)$.
The proof of 2.11 shows that dom.dim.($B$)$geq 2$.
Now my question is:
How to apply 2.10 in order to show 2.11?
It seems that the $M$ in 2.11 is the $Ae$ in 2.10. But why is $M$ projective? and must $M$ be a minimal faithful projective module?
Thanks for the help.
abstract-algebra ring-theory representation-theory homological-algebra
asked Aug 7 '15 at 14:29
Stein ChenStein Chen
1157
$endgroup$
add a comment |
$begingroup$
Let all considered algebras be Artin algebras and let all considered modules be finitely generated.
Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent:
$bullet$ $Ae$-dom.dim.$(A)geq 2$
$bullet$ There holds a double centralizer property $A=text{End}(Ae_{eAe})$
(This is theorem 2.10 of http://www.sciencedirect.com/science/article/pii/S002186930098726X#).
Corollary 2.11 of the same paper states:
Let $A$ be a self-injectie algebra, let $M$ be a faithful
$A$-module, and let $B:=text{End}_A(M)$. Then there is a double centralizer property $A=text{End}(M_B)$.
The proof of 2.11 shows that dom.dim.($B$)$geq 2$.
Now my question is:
How to apply 2.10 in order to show 2.11?
It seems that the $M$ in 2.11 is the $Ae$ in 2.10. But why is $M$ projective? and must $M$ be a minimal faithful projective module?
Thanks for the help.
abstract-algebra ring-theory representation-theory homological-algebra
asked Aug 7 '15 at 14:29
Stein ChenStein Chen
1157
$endgroup$
Let all considered algebras be Artin algebras and let all considered modules be finitely generated.
Let $A$ be left-QF-3 with minimal faithful left ideal $Ae$. Then the following are equivalent:
$bullet$ $Ae$-dom.dim.$(A)geq 2$
$bullet$ There holds a double centralizer property $A=text{End}(Ae_{eAe})$
(This is theorem 2.10 of http://www.sciencedirect.com/science/article/pii/S002186930098726X#).
Corollary 2.11 of the same paper states:
Let $A$ be a self-injectie algebra, let $M$ be a faithful
$A$-module, and let $B:=text{End}_A(M)$. Then there is a double centralizer property $A=text{End}(M_B)$.
The proof of 2.11 shows that dom.dim.($B$)$geq 2$.
Now my question is:
How to apply 2.10 in order to show 2.11?
It seems that the $M$ in 2.11 is the $Ae$ in 2.10. But why is $M$ projective? and must $M$ be a minimal faithful projective module?
Thanks for the help.
abstract-algebra ring-theory representation-theory homological-algebra
abstract-algebra ring-theory representation-theory homological-algebra
asked Aug 7 '15 at 14:29
Stein ChenStein Chen
1157
asked Aug 7 '15 at 14:29
Stein ChenStein Chen
1157
asked Aug 7 '15 at 14:29
Stein ChenStein Chen
1157
asked Aug 7 '15 at 14:29
Stein ChenStein Chen
1157
asked Aug 7 '15 at 14:29
Stein ChenStein Chen
1157
1157
add a comment |
add a comment |
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