Homological invariants of unbounded complexes under Frobenius extensions

Let R subset of S be a Frobenius extension of rings and M an S-complex. It is proven that if the Gorenstein projective (injective) dimension of M as an S-complex is finite, then the Gorenstein projective (injective) dimension of M as an S-complex equals the Gorenstein projective (injective) dimension of M as an R-complex. As a corollary one has that if the projective (injective) dimension of M as an S-complex is finite, then the projective (injective) dimension of M as an S-complex equals the projective (injective) dimension of M as an R-complex. This statement extends a known result of Nakayama and Tsuzuku.