A note on Grothendieck groups of periodic derived categories

Abstract

The m-periodic derived category of an abelian category is a natural Z/mZ-periodic analogue of the usual derived category. We determine the Grothendieck group of the periodic derived category of a skeletally small abelian category with enough projectives. 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 Z-module if m is even, and is an F2-vector space if m is odd. Moreover, in both cases of parity of m, the rank of the Grothendieck group 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 by the author in [6] as a periodic analogue of tilting objects, is independent of the choice of T.