Relative cohomology of complexes

Let R be an arbitrary ring and C a complex with finite Gorenstein projective dimension (that is, the supremum of Gorenstein projective dimension of all R-modules in C is finite). For each complex D, we define the nth relative cohomology group Ext(gp)(n) (C, D) by the equality Ext(gp)(n) (C, D) = H(n)Hom(G, D), where G -> C is a strict Gorenstein projective precover of C. If D is a complex with finite Gorenstein injective dimension (that is, the supremum of Gorenstein injective dimension of all R-modules in D is finite), then one can use a dual argument to define a notion of relative cohomology group Ext(gI)(n)(C, D). We show that if C is a complex with finite Gorenstein projective dimension and D a complex with finite Gorenstein injective dimension, then for each n is an element of Z there exists an isomorphism Ext(gp)(n) (C, D) congruent to Ext(gI)(n).(C, D). This shows that the relative cohomology functor of complexes is balanced. Some induced exact sequences concerning relative cohomology groups are considered. (C) 2018 Elsevier Inc. All rights reserved.