A note on Grothendieck groups of periodic derived categories

TL;DR

This paper characterizes the Grothendieck groups of periodic derived categories over Artin algebras, revealing their structure varies with the parity of the period and establishing invariance properties of periodic tilting objects.

Contribution

It determines the structure of Grothendieck groups for periodic derived categories and proves invariance of summand counts of periodic tilting objects.

Findings

Grothendieck group is free over Z for even periods

Grothendieck group is an F_2-vector space for odd periods

Number of summands of periodic tilting objects is invariant

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 $T$.