Frobenius functors, stable equivalences and K-theory of Gorenstein projective modules

Owing to the difference in K-theory, an example by Dugger and Shipley implies that equivalence of stable categories Gorenstein projective modules should not be a Quillen equivalence. We give sufficient necessary condition for Frobenius pair faithful functors between two abelian equivalence, which is also equivalent induce mutually inverse equivalences objects. show category objects Waldhausen category, then K-groups are introduced characterized. As applications, we Morita type preserve K-groups, CM-finiteness CM-freeness. Two specific examples path algebras presented illustrate results, K0 K1-groups calculated.