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 the equivalence of stable categories of Gorenstein projective modules should not be a Quillen equivalence. We give a sufficient and necessary condition for the Frobenius pair of faithful functors between two abelian categories to be a Quillen equivalence, which is also equivalent to that the Frobenius functors induce mutually inverse equivalences between stable categories of Gorenstein projective objects. We show that the category of Gorenstein projective objects is a Waldhausen category, then Gorenstein $K$-groups are introduced and characterized. As applications, we show that stable equivalences of Morita type preserve Gorenstein $K$-groups, CM-finiteness and CM-freeness. Two specific examples of path algebras are presented to illustrate the results, for which the Gorenstein $K_0$ and $K_1$-groups are calculated.