Abstract
We derive in the paper the tensor product functor -<TEX>${\otimes}_R$</TEX>- by using proper <TEX>$\mathcal{GP}_C$</TEX>-resolutions, where C is a semidualizing module. After giving several cases in which different relative homologies agree, we use the Pontryagin duals of <TEX>$\mathcal{G}_C$</TEX>-projective modules to establish a balance result for such relative homology over a Cohen-Macaulay ring with a dualizing module D.
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