Relative Gorenstein global dimension

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Abstract
We study the relative Gorenstein projective global dimension of a ring with respect to a weakly Wakamatsu tilting module [Formula: see text]. We prove that this relative global dimension is finite if and only if the injective dimension of every module in Add[Formula: see text] and the [Formula: see text]-projective dimension of every injective module are both finite (indeed these three dimensions have a common upper bound). When RC satisfies some extra conditions we prove that the relative Gorenstein projective global dimension of [Formula: see text] is always bounded above by the [Formula: see text]-projective global dimension of [Formula: see text], these two dimensions being equal when the class of all [Formula: see text]-Gorenstein projective [Formula: see text]-modules is contained in the Bass class of [Formula: see text] relative to [Formula: see text]. Of course we also give the dual results concerning the relative Gorenstein injective global dimension.

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