A note on Grothendieck groups of periodic derived categories

Abstract

We determine Grothendieck groups of periodic derived categories. In particular, we prove that the Grothendieck group of the $m$-periodic derived category of finitely generated modules over an Artin algebra is a free $\mathbb{Z}$-module if $m$ is even but an $\mathbb{F}_2$-vector space if $m$ is odd. Its rank is equal to the number of isomorphism classes of simple modules in both cases. As an application, we prove that the number of non-isomorphic summands of a strict periodic tilting object $T$, which was introduced in [S21] as a periodic analogue of tilting objects, is independent of the choice of $T$.