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For finitely generated modules and over a commutative Noetherian local ring , we give various sufficient criteria for detecting freeness of or via vanishing of some Ext modules and finiteness of certain homological dimension of . Some of our results provide partial progress towards answering a question of Ghosh-Takahashi and also generalize their main results in many ways, for instance, by reducing the number of vanishing. As some applications, we provide affirmative answers to two questions raised by Tony Se on -semidualizing modules. In particular, we establish that for normal domains which satisfy Serre's condition and are locally Gorenstein in co-dimension two, the class of -semidualizing modules form a subgroup of the divisor class group. These two groups coincide when, in addition, the ring is locally regular in co-dimension two.