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Tensor product of free and indecomposable modules over $K[t]$
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Let $A$ be a finite dimensional algebra over an algebraically closed field $K$, $M$ an $(A, K[t])$-bimodule that is free as a $K[t]$-module of rank $d$ and $N$ an irreducible left $K[t]$-module.
Why is the left $A$-module $M otimes_K[t]N$ indecomposable?
I know that such $N$ is of the form $K[t]/(t-lambda)$ for some $lambda in K$ and $M=bigoplus_i=1^d m_i . K[t]$ for a basis $(m_1,cdots,m_d)$, so that $M otimes_K[t]N$ is $d$-dimensional as a vectorspace.
Hence it suffices to show that $mathrmEnd_A(M otimes_K[t]N)$ is local.
I'm not getting any further than that and hope you can help me.
abstract-algebra modules representation-theory tensor-products
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up vote
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Let $A$ be a finite dimensional algebra over an algebraically closed field $K$, $M$ an $(A, K[t])$-bimodule that is free as a $K[t]$-module of rank $d$ and $N$ an irreducible left $K[t]$-module.
Why is the left $A$-module $M otimes_K[t]N$ indecomposable?
I know that such $N$ is of the form $K[t]/(t-lambda)$ for some $lambda in K$ and $M=bigoplus_i=1^d m_i . K[t]$ for a basis $(m_1,cdots,m_d)$, so that $M otimes_K[t]N$ is $d$-dimensional as a vectorspace.
Hence it suffices to show that $mathrmEnd_A(M otimes_K[t]N)$ is local.
I'm not getting any further than that and hope you can help me.
abstract-algebra modules representation-theory tensor-products
Quick note: $Ncong K[t]/(t-lambda)^n$ for some $lambdain K$ and $ngeq 0$. Are there any additional assumptions about $M$ as an $A$-module?
– David Hill
Jul 25 at 18:41
Thanks, I meant $N$ irreducible.
– user577451
Jul 26 at 5:20
I think you must be missing some conditions on $M$. For example, take $A=K$ and $M=K[t]^d$. Then $Motimes_K[t]Ncong K^d$, which is not indecomposable as a module for $A=K$.
– Jeremy Rickard
Jul 26 at 9:58
add a comment |Â
up vote
3
down vote
favorite
up vote
3
down vote
favorite
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$, $M$ an $(A, K[t])$-bimodule that is free as a $K[t]$-module of rank $d$ and $N$ an irreducible left $K[t]$-module.
Why is the left $A$-module $M otimes_K[t]N$ indecomposable?
I know that such $N$ is of the form $K[t]/(t-lambda)$ for some $lambda in K$ and $M=bigoplus_i=1^d m_i . K[t]$ for a basis $(m_1,cdots,m_d)$, so that $M otimes_K[t]N$ is $d$-dimensional as a vectorspace.
Hence it suffices to show that $mathrmEnd_A(M otimes_K[t]N)$ is local.
I'm not getting any further than that and hope you can help me.
abstract-algebra modules representation-theory tensor-products
Let $A$ be a finite dimensional algebra over an algebraically closed field $K$, $M$ an $(A, K[t])$-bimodule that is free as a $K[t]$-module of rank $d$ and $N$ an irreducible left $K[t]$-module.
Why is the left $A$-module $M otimes_K[t]N$ indecomposable?
I know that such $N$ is of the form $K[t]/(t-lambda)$ for some $lambda in K$ and $M=bigoplus_i=1^d m_i . K[t]$ for a basis $(m_1,cdots,m_d)$, so that $M otimes_K[t]N$ is $d$-dimensional as a vectorspace.
Hence it suffices to show that $mathrmEnd_A(M otimes_K[t]N)$ is local.
I'm not getting any further than that and hope you can help me.
abstract-algebra modules representation-theory tensor-products
edited Jul 26 at 5:18
asked Jul 25 at 10:32
user577451
Quick note: $Ncong K[t]/(t-lambda)^n$ for some $lambdain K$ and $ngeq 0$. Are there any additional assumptions about $M$ as an $A$-module?
– David Hill
Jul 25 at 18:41
Thanks, I meant $N$ irreducible.
– user577451
Jul 26 at 5:20
I think you must be missing some conditions on $M$. For example, take $A=K$ and $M=K[t]^d$. Then $Motimes_K[t]Ncong K^d$, which is not indecomposable as a module for $A=K$.
– Jeremy Rickard
Jul 26 at 9:58
add a comment |Â
Quick note: $Ncong K[t]/(t-lambda)^n$ for some $lambdain K$ and $ngeq 0$. Are there any additional assumptions about $M$ as an $A$-module?
– David Hill
Jul 25 at 18:41
Thanks, I meant $N$ irreducible.
– user577451
Jul 26 at 5:20
I think you must be missing some conditions on $M$. For example, take $A=K$ and $M=K[t]^d$. Then $Motimes_K[t]Ncong K^d$, which is not indecomposable as a module for $A=K$.
– Jeremy Rickard
Jul 26 at 9:58
Quick note: $Ncong K[t]/(t-lambda)^n$ for some $lambdain K$ and $ngeq 0$. Are there any additional assumptions about $M$ as an $A$-module?
– David Hill
Jul 25 at 18:41
Quick note: $Ncong K[t]/(t-lambda)^n$ for some $lambdain K$ and $ngeq 0$. Are there any additional assumptions about $M$ as an $A$-module?
– David Hill
Jul 25 at 18:41
Thanks, I meant $N$ irreducible.
– user577451
Jul 26 at 5:20
Thanks, I meant $N$ irreducible.
– user577451
Jul 26 at 5:20
I think you must be missing some conditions on $M$. For example, take $A=K$ and $M=K[t]^d$. Then $Motimes_K[t]Ncong K^d$, which is not indecomposable as a module for $A=K$.
– Jeremy Rickard
Jul 26 at 9:58
I think you must be missing some conditions on $M$. For example, take $A=K$ and $M=K[t]^d$. Then $Motimes_K[t]Ncong K^d$, which is not indecomposable as a module for $A=K$.
– Jeremy Rickard
Jul 26 at 9:58
add a comment |Â
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Quick note: $Ncong K[t]/(t-lambda)^n$ for some $lambdain K$ and $ngeq 0$. Are there any additional assumptions about $M$ as an $A$-module?
– David Hill
Jul 25 at 18:41
Thanks, I meant $N$ irreducible.
– user577451
Jul 26 at 5:20
I think you must be missing some conditions on $M$. For example, take $A=K$ and $M=K[t]^d$. Then $Motimes_K[t]Ncong K^d$, which is not indecomposable as a module for $A=K$.
– Jeremy Rickard
Jul 26 at 9:58