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Let $k$ be a field with infinite cardinality and $A$ a finite dimensional $k$-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers $n_1<n_2< \dots$ such that for every $n_i$ there are infinitely many indecomposable $A$-modules with length $n_i$. This does not hold if the cardinality of $k$ is finite. My question is, wether there is a version of this conjecture for an arbitrary Artin algebra $A$, not necessarily over a field? Of course we have to restrict $A$ since it includes $k$-Algebras for a finite field $k$. I could imagine something like: The cardinality of $A$ is infinite.

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    $\begingroup$ $A$ infinite is not enough, as you could take $A=B\times K$, where $B$ is a finite dimensional algebra over a finite field, and $K$ is an infinite field. $\endgroup$ Jul 19, 2021 at 15:17
  • $\begingroup$ New suggestion: $A$ infinite and connected. $\endgroup$
    – kevkev1695
    Jul 19, 2021 at 22:44

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