Let $R$ be a commutative ring with unity.
(1) Let $M$ be an $R$-module having a flat resolution of length $1$ . Then for every $R$-module $N$, we have $\operatorname{Tor}_i^R(M,N)=0, \forall i \ge 2$. Hence the functor $\operatorname{Tor}_1^R(M,-):$ Mod-$R \to $ Mod-$R$ is left exact, and moreover it is a covariant, additive functor. So we can consider its right derived functors... do we have a nice description of these right derived functors ?
(2) Let $M$ be an $R$-module having a projective resolution of length $1$ . Then, for every $R$-module $N$, we have $\operatorname{Ext}_R^i(M,N)=0, \forall i \ge 2$. Hence the functor $\operatorname{Ext}^1_R(M,-):$ Mod-$R \to $ Mod-$R$ is right exact , and moreover it is a covariant, additive functor. So we can consider its left derived functors ... do we have a nice description of these left derived functors?