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Deleted the well, and clarified that the dimension has to be finite.
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Dave Benson
Dave Benson
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Well, theThe definition of tame representation type usually assumes that you're dealing with a finite dimensional algebra $A$ over a field $k$. In this case, the definition of a one parameter family of modules is that there is an $A$-$k[T]$-bimodule $X$ which is finitely generated and free over $k[T]$, and then the modules in the family are all required to be of the form $X \otimes_{k[T]} M$ for some $k[T]$-module $M$. The requirement is then that in any particular (finite) dimension over $k$, all but a finite number of indecomposable $A$-modules belong to a finite set of one parameter families. In other words, $k[T]$ is the model for tame representation type.

For an infinite dimensional algebra, I guess one can use the same definition, and then $k[x]$ would be tame. But you should beware that the finite/tame/wild trichotomy has only been proved for finite dimensional algebras, and it's not clear that one should try to apply the same definitions to infinite dimensional algebras. For example, there are algebras with no finite dimensional representations at all, and then do you want to say that those have finite representation type? It's all a bit unsatisfactory.

Well, the definition of tame representation type usually assumes that you're dealing with a finite dimensional algebra $A$ over a field $k$. In this case, the definition of a one parameter family of modules is that there is an $A$-$k[T]$-bimodule $X$ which is finitely generated and free over $k[T]$, and then the modules in the family are all required to be of the form $X \otimes_{k[T]} M$ for some $k[T]$-module $M$. The requirement is then that in any particular dimension over $k$, all but a finite number of indecomposable $A$-modules belong to a finite set of one parameter families. In other words, $k[T]$ is the model for tame representation type.

For an infinite dimensional algebra, I guess one can use the same definition, and then $k[x]$ would be tame. But you should beware that the finite/tame/wild trichotomy has only been proved for finite dimensional algebras, and it's not clear that one should try to apply the same definitions to infinite dimensional algebras. For example, there are algebras with no finite dimensional representations at all, and then do you want to say that those have finite representation type? It's all a bit unsatisfactory.

The definition of tame representation type usually assumes that you're dealing with a finite dimensional algebra $A$ over a field $k$. In this case, the definition of a one parameter family of modules is that there is an $A$-$k[T]$-bimodule $X$ which is finitely generated and free over $k[T]$, and then the modules in the family are all required to be of the form $X \otimes_{k[T]} M$ for some $k[T]$-module $M$. The requirement is then that in any particular (finite) dimension over $k$, all but a finite number of indecomposable $A$-modules belong to a finite set of one parameter families. In other words, $k[T]$ is the model for tame representation type.

For an infinite dimensional algebra, I guess one can use the same definition, and then $k[x]$ would be tame. But you should beware that the finite/tame/wild trichotomy has only been proved for finite dimensional algebras, and it's not clear that one should try to apply the same definitions to infinite dimensional algebras. For example, there are algebras with no finite dimensional representations at all, and then do you want to say that those have finite representation type? It's all a bit unsatisfactory.

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Dave Benson
Dave Benson
  • 16.2k
  • 2
  • 42
  • 95

Well, the definition of tame representation type usually assumes that you're dealing with a finite dimensional algebra $A$ over a field $k$. In this case, the definition of a one parameter family of modules is that there is an $A$-$k[T]$-bimodule $X$ which is finitely generated and free over $k[T]$, and then the modules in the family are all required to be of the form $X \otimes_{k[T]} M$ for some $k[T]$-module $M$. The requirement is then that in any particular dimension over $k$, all but a finite number of indecomposable $A$-modules belong to a finite set of one parameter families. In other words, $k[T]$ is the model for tame representation type.

For an infinite dimensional algebra, I guess one can use the same definition, and then $k[x]$ would be tame. But you should beware that the finite/tame/wild trichotomy has only been proved for finite dimensional algebras, and it's not clear that one should try to apply the same definitions to infinite dimensional algebras. For example, there are algebras with no finite dimensional representations at all, and then do you want to say that those have finite representation type? It's all a bit unsatisfactory.