Hom and Ext, Revisited

Let $R$ be a commutative Noetherian local ring and $M,N$ be finitely generated $R$-modules. We prove a number of results of the form: if $Ho{m}_R(M,N)$ has some nice properties and $\mbox{Ext}^{1 \leq i \leq n}_R(M,N)=0$forsome$n$,then$M$(andsometimes$N$) must be be close to free. Our methods are quite elementary, yet they suffice to give a unified treatment, simplify, and sometimes extend a number of results in the literature.