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For finitely generated modules $M$ and $N $ over a commutative Noetherianlocal ring $R$, we give various sufficient criteria for detecting freeness of$M$ or $N$ via vanishing of some finitely many Ext modules$\textrm{Ext}^i_R(M,N)$ and finiteness of certain homological dimension of$\textrm{Hom}_R(M,N)$. Some of our results provide partial progress towardsanswering a question of Ghosh-Takahashi and also generalize their main resultsin many ways, for instance, by reducing the number of vanishing. Certainspecial cases of our results allow us to address the Auslander-Reitenconjecture for modules whose (self) dual has finite projective dimension. Amongother things, the notion of residually faithful modules and certain variationsof it play a crucial role in our study. As some applications, we provideaffirmative answers to two questions raised by Tony Se on $n$-semidualizingmodules. Namely, we show that over a local ring of depth $t$, every$(t-1)$-semidualizing module of finite G-dimension is free. Moreover, weestablish that for normal domains which satisfy Serre's condition $(S_3)$ andare locally Gorenstein in co-dimension two, the class of $1$-semidualizingmodules form a subgroup of the divisor class group. These two groups coincidewhen, in addition, the ring is locally regular in co-dimension two.

Finite homological dimension of Hom and vanishing of Ext