Tor functors

Let \(R\) be a commutative ring, let \(B\) be an \(R\)-module, and let \(n\geq 0\). The functor \(\Tor_n^R(\wait, B)\) is the \(n\)th left derived functor of \(\wait\tensor_R B\): \[ \Tor_n^R(A, B) \coloneqq L_n(\wait\tensor_R B)(A) = H_n(P_\bullet\tensor_R B) ,\] where \(P_\bullet \to A\) is a projective resolution of the \(R\)-module \(A\).