Lemma 8.2.9. Let \(R\) be a connective associative ring spectrum. Then the functor \(\pi _0\) from Observation 8.2.7 restricts to equivalences \[ \LMod _R^{\heartsuit } \iso \LMod _{\pi _0(R)}(\Ab ), \qquad \RMod _R^{\heartsuit } \iso \RMod _{\pi _0(R)}(\Ab ) \] between the hearts of the module categories and the abelian categories of discrete modules over \(\pi _0(R)\).
Proof. We prove the statement for left modules; the proof for right modules is identical. For essential surjectivity, write a discrete \(\pi _0(R)\)-module \(M\) as a coequalizer of free modules. The two maps between the free modules lift to maps between coproducts of copies of \(R\), since maps out of \(R\) are classified by elements of \(\pi _0\). Form their coequalizer \(\widetilde M\) in \(\LMod _{R,\geq 0}\) and then apply \(\tau _{\leq 0}\). Since \(\pi _0\) preserves colimits of connective modules, we have \(\pi _0(\widetilde M)\cong M\), and hence \(\tau _{\leq 0}\widetilde M\) is a discrete \(R\)-module whose image under \(\pi _0\) is \(M\).
We will now show that the functor is fully faithful. Let \(N \in \LMod _R^{\heartsuit }\), and consider the subcategory \(C \subseteq \LMod _{R,\geq 0}\) spanned by those left \(R\)-modules \(M\) such that the map \[ \pi _0 \Hom _{\LMod _R}(M,N) \to \Hom _{\LMod _{\pi _0(R)}(\Ab )}(\pi _0(M),\pi _0(N)) \] is an isomorphism. The subcategory \(C\) contains \(R\). It is also closed under colimits: for connective \(M\) and discrete \(N\), the mapping spectrum \(\hom _R(M,N)\) is coconnective, and \(\pi _0\colon \Sp _{\leq 0}\to \Ab \) preserves limits. Thus both sides of the displayed map carry colimits in \(M\) to limits. It follows from Proposition 8.2.8 that \(C = \LMod _{R,\geq 0}\), as desired. □
Generated from the authoritative LaTeX source.