# When is the category of complexes of finite type?

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?