Connective modules

Proof. The adjunction between the symmetric monoidal inclusion \(C_{\geq 0}\hookrightarrow C\) and its lax symmetric monoidal right adjoint \(\tau _{\geq 0}\) induces, after passing to algebras and modules and taking the fiber over \(R\), an adjunction \[ \LMod _R(C_{\geq 0})\rightleftarrows \LMod _R(C). \] Its right adjoint is computed on underlying objects by \(\tau _{\geq 0}\). Finite limits and colimits of modules are computed on underlying objects by Corollary 19.1.17, so it remains to check the orthogonality and decomposition axioms for a t-structure. If \(X\) is connective and \(Y\) is coconnective, adjunction gives \[ \Hom _{\LMod _R(C)}(X,Y[-1]) \simeq \Hom _{\LMod _R(C_{\geq 0})}(X,\tau _{\geq 0}(Y[-1])) \simeq *. \] Finally, for an arbitrary \(R\)-module \(X\), the counit \(\tau _{\geq 0}X\to X\) in modules has connective source, and its cofiber is coconnective because this is true after forgetting to \(C\). This supplies the required decomposition. □

Proof. The sphere spectrum is connective. Given two connective spectra \(X\) and \(Y\), we need to show that \(X \otimes Y\) is again connective. If we write \(X_n := \Omega ^{\infty - n}X\) and \(Y_m := \Omega ^{\infty -m} Y\) for all \(n,m \geq 0\), then \(X_n\) is \(n\)-connective and \(Y_m\) is \(m\)-connective. The standard connectivity estimate for smash products, which follows by applying the Blakers–Massey theorem to \(X_n\vee Y_m\to X_n\times Y_m\), shows that \(X_n \wedge Y_m\) is \((n+m)\)-connective. Using the colimit formula from Subsection 4.4.3, we have \[ X \otimes Y \simeq \colim _n \colim _m (\Sigma ^{\infty }(X_n \wedge Y_m))[-(n+m)]. \] The shift-additivity ambiguity mentioned there is irrelevant here, since we use only the connectivity of the terms. Each spectrum in this colimit is connective, and \(\Sp _{\geq 0} \subseteq \Sp \) is closed under colimits, so \(X \otimes Y \in \Sp _{\geq 0}\), as desired. □

Proof. The existence of the t-structure is, in light of the previous lemma, an instance of Lemma 8.2.3. By that same lemma, an \(R\)-module is connective or coconnective precisely when its underlying spectrum is, so an infinitely connective or infinitely coconnective \(R\)-module has vanishing homotopy groups and is zero. Thus the t-structure is left and right separated. The forgetful functor \(\LMod _R \to \Sp \) preserves limits and colimits, and \(\Sp _{\geq 0}\) and \(\Sp _{\leq 0}\) are closed under countable products and countable coproducts, respectively. Lemma 6.3.31 therefore upgrades separatedness to left and right completeness. The two displayed formulas express these completeness properties. □

Proof. Under the symmetric monoidal equivalence \(\Sp _{\geq 0}\simeq \CGrp (\An )\) established in Part II (Corollary 16.6.5), this functor corresponds to the symmetric monoidal functor \(\pi _0\colon \CGrp (\An )\to \CGrp (\Set )=\Ab \) of Proposition 16.4.5. □

Proof. It is clear that \(\LMod _{R,\geq 0}\) contains \(R\) and is closed under colimits. Conversely, given a connective left \(R\)-module \(M\), we will show that \(M\) can be written as a colimit of a diagram of left \(R\)-modules \[ M(0) \to M(1) \to M(2) \to \dots \] satisfying the following properties:

(i)

The \(R\)-module \(M(0)\) is a coproduct of copies of \(R\).

(ii)

For \(i \ge 0\), there is an exact sequence \[ F[i] \to M(i) \to M(i+1), \] where \(F\) is a coproduct of copies of \(R\).

We will construct the modules \(M(i)\) inductively, together with their maps \(f_i\colon M(i) \to M\). In fact, we will inductively ensure that these maps satisfy the following third property:

(iii)

Let \(i \ge 0\), and let \(K(i)\) be the fiber of the map \(M(i) \to M\). Then \(\pi _j K(i) \simeq 0\) for \(j < i\).

For \(i = 0\), we can choose a map of left \(R\)-modules \(M(0) \to M\) where \(M(0)\) is a coproduct of copies of \(R\), such that the induced map \(\pi _0(M(0)) \to \pi _0(M)\) is surjective: for instance, we could take the coproduct indexed by the set \(\pi _0(M)\). For the induction step, we first consider the exact sequence \[ K(i) \to M(i) \to M. \] The group \(\pi _i K(i)\) is a \(\pi _0(R)\)-module, and we may choose a map of left \(R\)-modules \(g\colon F[i] \to K(i)\) from a coproduct of copies of \(R[i]\) such that the induced map \(\pi _i(F[i]) \to \pi _i(K(i))\) is surjective. Let \(\widetilde {g}\colon F[i] \to M(i)\) be the composite of \(g\) with the fiber inclusion \(K(i) \to M(i)\). We define \(M(i+1)\) to be the pushout of the diagram

so that \(M(i+1)\) is the cofiber of \(\widetilde {g}\). Since the composite \(F[i] \to K(i) \to M(i) \to M\) is canonically nullhomotopic, there is an induced map \(f_{i+1}\colon M(i+1) \to M\). The induced fiber sequence \[ F[i] \longrightarrow K(i) \longrightarrow K(i+1) \] shows that \(\pi _jK(i+1)=0\) for \(j<i\), while the surjectivity of \(\pi _i(F[i]) \to \pi _i(K(i))\) gives \(\pi _iK(i+1)=0\). This verifies the inductive step.

It remains to prove that the natural map \(\colim _i M(i) \to M\) is an isomorphism. Since filtered colimits are exact in \(\Sp \), the fiber \(K\) of this map is \(\colim _i K(i)\). By property (iii), the group \(\pi _jK(i)\) vanishes for \(i>j\), and by Lemma 4.4.28 we conclude that \(\pi _j(K)=0\) for every \(j\). Hence \(K=0\).

The same construction with right free modules proves the assertion for connective right \(R\)-modules. □

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. □