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For a ring $R$ define the category of complexes of length $n \geq 2$ as the category $C_n(R)$ with objects the complexes of the form $0 \rightarrow X_n \rightarrow \cdots X_1 \rightarrow 0$ with the morphisms being the usual morphisms of complexes.

Question: For a given $n$, for which $R$ is $C_n(R)$ of finite representation type, meaning that it has only finitely many indecomposable objects?

We can restrict to $R$ being a finite dimensional algebra for now, where we should have that $C_n(R)$ is equivalent to the module category of the algebra $R \otimes A_n$ with $A_n$ being the upper triangular matrix algebra with non-zero entries only on the main diagonal and the diagonal right next to it (so $A_n$ is the Nakayama algebra with Kupisch series [2,2,...,2,1]).

Note that this interpretation easily shows that for example in the case when $R$ is a field, $C_n(R)$ is representation-finite.

Some computer tests suggest that $R$ has to have Loewy length at most 3, or else $C_n(R)$ is representation-infinite for $n \geq 3$. Is this correct?

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