Consider the ideal $(x_1,\dots ,x_n{)}^d\subseteq k[x_1,\dots ,x_n]$, where $k$ is any field. This ideal can be resolved by both the $L$-complexes of Buchsbaum and Eisenbud, and the Eliahou-Kervaire resolution. Both of these complexes admit the structure of an associative DG algebra, and it is a question of Peeva as to whether these DG structures coincide in general. In this paper, we construct an isomorphism of complexes between the aforementioned complexes that is also an isomorphism of algebras with their respective products, thus giving an affirmative answer to Peeva’s question.