Basic properties of Tor for modules

Observation 8.5.9. Let \(R\) be a connective associative ring spectrum. By Observation 8.2.7, the relative tensor product of a connective right \(R\)-module \(M\) and a connective left \(R\)-module \(N\) is connective, and \[ \Tor ^R_0(M,N) = \pi _0(M \otimes _R N) \cong \pi _0(M) \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N). \] In particular, \(\Tor ^R_n(M,N) = 0\) for \(n < 0\) whenever \(M\) and \(N\) are connective.