On Gorenstein Projective Dimensions of Unbounded Complexes

Abstract

Let R → S be a ring homomorphism and X be a complex of R-modules. Then the complex of S-modules S ⊗ _R ^L X in the derived category D(S) is constructed in the natural way. This paper is devoted to dealing with the relationships of the Gorenstein projective dimension of an R-complex X (possibly unbounded) with those of the S-complex S ⊗ _R ^L X. It is shown that if R is a Noetherian ring of finite Krull dimension and ϕ: R → S is a faithfully flat ring homomorphism, then for any homologically degree-wise finite complex X, there is an equality Gpd _R X = Gpd _S(S ⊗ _R ^L X). Similar result is obtained for Ding projective dimension of the S-complex S ⊗ _R ^L X.

Keywords

Gorenstein projective dimension / Ding projective dimension / Faithfully flat ring homomorphism