Let $\Lambda$ be a commutative ring with $1$. Let $M$ be a $\Lambda$-module. Is it true that $\mathrm{Ext}^i_\Lambda(M,\Lambda)=0$ for all $i\ge1$ if and only if $M$ is free over $\Lambda$?
The 'if' part of the question is clear: compute the Ext groups by a projective resolution of $M$. Since $M$ is free, we may choose the trivial resolution, which makes all Ext groups vanish.
The question is, does the converse hold?
The converse is false; if $M$ is any projective module, then all higher Ext groups vanish, for exactly the same reason (compute the Ext groups by the trivial projective resolution). In general, there are plenty of examples of projective modules that are not free. For example, take a non-principal ideal of a Dedekind domain.