A Note on Odd Periodic derived Hall algebras

Let $m$ be an odd positive integer and $D_m(\mathcal{A})$ be the $m$-periodic derived category of a finitary hereditary abelian category $\mathcal{A}$. In this note, we prove that there is an embedding of algebras from the derived Hall algebra of $D_m(\mathcal{A})$ defined by Xu-Chen [13] to the extended derived Hall algebra of $D_m(\mathcal{A})$ defined in [16]. This homomorphism is given on basis elements, rather than just on generating elements.