π₀ of modules

Observation 8.2.7. Let \(R\) be a connective associative ring spectrum. The fully faithful symmetric monoidal inclusion \(\Sp _{\geq 0}\hookrightarrow \Sp \) identifies \(\LMod _R(\Sp _{\geq 0})\) with the full subcategory of \(\LMod _R\) spanned by the connective modules. By Proposition 8.2.6, extracting \(\pi _0\) therefore determines a functor \[ \pi _0\colon \LMod _{R,\geq 0} = \LMod _R(\Sp _{\geq 0}) \to \LMod _{\pi _0(R)}(\Ab ). \] Furthermore, this functor is compatible with relative tensor products: given a connective right \(R\)-module \(M\) and a connective left \(R\)-module \(N\), we have an isomorphism of abelian groups \[ \pi _0(M) \otimes ^{\heartsuit }_{\pi _0(R)} \pi _0(N) \iso \pi _0(M \otimes _R N), \qquad [x] \otimes [y] \mapsto [x \otimes y]. \] Both \(\pi _0\colon \Sp _{\geq 0}\to \Ab \) and the symmetric monoidal inclusion \(\Sp _{\geq 0}\hookrightarrow \Sp \) preserve geometric realizations. Applying Proposition 19.2.11, proved in Part II, to these two functors shows, respectively, that \(\pi _0\) preserves relative tensor products and that the relative tensor product of connective modules computed in \(\Sp _{\geq 0}\) agrees with the one computed in \(\Sp \). In particular, the relative tensor product of connective modules is connective, and this gives the asserted isomorphism. The analogous statements hold for right modules.