A note on odd periodic derived Hall algebras

Let [Formula: see text] be an odd positive integer and [Formula: see text] be the [Formula: see text]-periodic derived category of a finitary hereditary Abelian category [Formula: see text]. In this note, we prove that there is an embedding of algebras from the derived Hall algebra of [Formula: see text] defined by Xu–Chen [Hall algebras of odd periodic triangulated categories, Algebr. Represent. Theory 16(3) (2013) 673–687] to the extended derived Hall algebra of [Formula: see text] 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.