Let R ⊂ A be a Frobenius extension of rings. We prove that: (1) for any left A-module M, A M is Gorenstein projective (injective) if and only if the underlying left R-module R M is Gorenstein projective (injective). (2) if G-proj.dim A M < ∞, then G-proj.dim A M = G-proj.dim R M; the dual for Gorenstein injective dimension also holds. (3) if the extension is split, then G-gldim(A) = G-gldim(R).
We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it preserves and reflects Gorenstein projective objects. We give conditions on when a Frobenius functor preserves the stable categories of Gorenstein projective objects, the singularity categories and the Gorenstein defect categories, respectively. In the appendix, we give a direct proof of the following known result: for an abelian category with enough projectives and injectives, its global Gorenstein projective dimension coincides with its global Gorenstein injective dimension.
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