Relative cohomology of complexes

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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 n th relative cohomology group Ext GP n ( C , D ) by the equality Ext GP n ( C , D ) = H n H om ( 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 ∈ Z there exists an isomorphism Ext GP n ( C , D ) ≅ 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.