Tensor Product Does Not Behave Additively on Delooping Level

The Finitistic Dimension Conjecture is one of the most important homological conjectures in the representation theory of finite-dimensional algebras. Recently, Gélinas arXiv:2004.04828 proposed the notion of ``delooping level’’ to study the Finitistic Dimension Conjecture. This article reports a misconception between the delooping level and the tensor product of algebras. Namely, for finte-dimensional algebras $A$ and $B$, it is not always true that $\mathrm{dell}(A{\otimes }_{\mathbf{k}}B)=\mathrm{dell}(A)+\mathrm{dell}(B)$.