# Tensor products of n-complete algebras

## TL;DR

This paper investigates how tensor products of certain algebraic structures called n-complete algebras behave, showing that under specific conditions, their tensor product maintains a property called (n+m)-completeness, extending understanding of algebraic tensor products.

## Contribution

The paper proves that the tensor product of acyclic n- and m-complete algebras is (n+m)-complete, providing new insights into the preservation of algebraic properties under tensor operations.

## Key findings

- Tensor products of acyclic n- and m-complete algebras are (n+m)-complete.
- Higher Auslander algebra does not preserve d-representation finiteness but preserves d-completeness.
- Provides a necessary condition for tensor products to be (n+m)-representation finite.

## Abstract

If $A$ and $B$ are $n$- and $m$-representation finite $k$-algebras, then their tensor product $\Lambda = A\otimes_k B$ is not in general $(n+m)$-representation finite. However, we prove that if $A$ and $B$ are acyclic and satisfy the weaker assumption of $n$- and $m$-completeness, then $\Lambda$ is $(n+m)$-complete. This mirrors the fact that taking higher Auslander algebra does not preserve $d$-representation finiteness in general, but it does preserve $d$-completeness. As a corollary, we get the necessary condition for $\Lambda$ to be $(n+m)$-representation finite which was found by Herschend and Iyama by using a certain twisted fractionally Calabi-Yau property.