Abstract. Let R be a commutative ring and S be an R-algebra. It is wellknown that if N is an injective R-module, then Hom R (S, N ) is an injective S-module. The converse is not true, not even if R is a commutative noetherian local ring and S is its completion, but it is close: It is a special case of our main theorem that in this setting, an R-module N with Ext >0 R (S, N ) = 0 is injective if Hom R (S, N ) is an injective S-module.

For a commutative ring R and a faithfully flat R-algebra S we prove, under mild extra assumptions, that an R-module M is Gorenstein flat if and only if the left S-module S ⊗ R M is Gorenstein flat, and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Hom R (S, N ) is Gorenstein injective. We apply these results to the study of Gorenstein homological dimensions of unbounded complexes. In particular, we prove two theorems on stability of these dimensions under faithfully flat (co-)base change. Date: 12 May 2016. 2010 Mathematics Subject Classification. 13D05; 13D02. Key words and phrases. Gorenstein injective dimension, faithfully flat co-base change, Gorenstein flat dimension, faithfully flat base change.This result compares to the statement about modules in Theorem 1.2.By significantly relaxing the conditions on the rings, the results of this paper improve results obtained by Christensen and Holm [7], by Christensen and Sather-Wagstaff [11], and by Liu and Ren [22]. Details pertaining to Theorems 1.1 and 1.2 are given in Remarks 4.13 and 6.8; the trend is that the rings in [11,22] are assumed to be commutative noetherian and, more often than not, of finite Krull dimension.The paper is organized as follows: In section 2 we set the notation and recall some background material. Sections 3-4 focus on the Gorenstein injective dimension, and Sections 5-6 deal with the Gorenstein flat dimension. Section 7 has some closing remarks and, finally, an appendix by Bennis answers a question raised in an earlier version of this paper. ComplexesLet R be a ring with identity. We consider only unitary R-modules, and we employ the convention that R acts on the left. That is, an R-module is a left R-module, and right R-modules are treated as modules over the opposite ring, denoted R • .2.1 Complexes. Complexes of R-modules, R-complexes for short, is our object of study. Let M be an R-complex. With homological grading, M has the formone switches to cohomological grading by settingFor n ∈ Z the symbol M n denotes the quotient complex of M with (M n ) i = M i for i n and (M n ) i = 0 for i < n.The subcomplexes B(M ) and Z(M ) of boundaries and cycles, the quotient complex C(M ) of cokernels, and the subquotient complex H(M ) of homology all have