A note on odd periodic derived Hall algebras

Let $m$ be an odd positive integer and $D_m({𝒜})$ be the $m$-periodic derived category of a finitary hereditary Abelian category ${𝒜}$. In this note, we prove that there is an embedding of algebras from the derived Hall algebra of $D_m({𝒜})$ defined by Xu–Chen [Hall algebras of odd periodic triangulated categories, Algebr. Represent. Theory16(3) (2013) 673–687] to the extended derived Hall algebra of $D_m({𝒜})$ defined in [H. Zhang, Periodic derived Hall algebras of hereditary Abelian categories, preprint (2023), arXiv:2303.02912v2]. This homomorphism is given on basis elements, rather than just on generating elements.