Criterion for local modules

Proof. Let \(s \in S\) be an element of degree \(d\). Applying the mapping spectrum functor \(\hom _{\LMod _R}(-,N)\) to the exact sequence \[ R[d] \xrightarrow {\cdot s} R \to R/Rs \] and using the identifications \(\hom _{\LMod _R}(R,N) \simeq N\) and \(\hom _{\LMod _R}(R[d],N)\simeq N[-d]\) gives an exact sequence of spectra \[ \hom _{\LMod _R}(R/Rs,N)\longrightarrow N \xrightarrow {\ s\cdot -\ }N[-d]. \] The second map is left multiplication by \(s\): precomposition with right multiplication \(\cdot s\colon R[d]\to R\) sends a map \(f\colon R \to N\) to the map determined by \(f(s) = s \cdot f(1)\).

Condition (1) says precisely that the second map is an isomorphism for every \(s\in S\), which by exactness is equivalent to the vanishing of \(\hom _{\LMod _R}(R/Rs,N)\). Since a spectrum vanishes if and only if all the animae \(\Omega ^{\infty }(-[-n])\) do, this is in turn equivalent to \(\Hom _{\LMod _R}((R/Rs)[n],N)\) being contractible for all \(n\). This proves \((1) \Leftrightarrow (2)\).

To prove the implication \((3) \Rightarrow (2)\), we observe that since \(S\) satisfies the left Ore condition, the module \(R/Rs\) is \(S\)-nilpotent for every \(s \in S\), as established in Lemma 8.4.14. As \(\LMod _R^{S\dnil }\) is closed under shifts (Observation 8.4.10), condition (3) applied to \(M = (R/Rs)[n]\) immediately yields condition (2).

We now prove the implication \((2) \Rightarrow (3)\). Let \(M\) be an \(S\)-nilpotent left \(R\)-module, and write it as the colimit of a sequence \[ 0 = M_0 \to M_1 \to M_2 \to \cdots \] whose successive fibers are coproducts of modules \((R/Rs_\alpha )[n_\alpha ]\), as provided by Lemma 8.4.15. Since hom animae turn colimits in the first variable into limits, we have \[ \Hom _{\LMod _R}(M,N) \iso \lim _{i} \Hom _{\LMod _R}(M_i,N). \] It will therefore suffice to show that each hom anima \(\Hom _{\LMod _R}(M_i,N)\) is contractible. We proceed by induction on \(i\). The case \(i=0\) is clear, since \(M_0=0\). For the inductive step, assume that \(\Hom _{\LMod _R}(M_i,N)\) is contractible. Let \(K_i\) denote the fiber of the map \(M_i \to M_{i+1}\). By construction, we have an exact sequence of modules \(K_i \to M_i \to M_{i+1}\). Applying the functor \(\Hom _{\LMod _R}(-,N)\) yields a fiber sequence of animae \[ \Hom _{\LMod _R}(M_{i+1},N) \to \Hom _{\LMod _R}(M_i,N) \to \Hom _{\LMod _R}(K_i,N). \] By the inductive hypothesis, the middle term is contractible. It therefore suffices to show that the right-hand term, \(\Hom _{\LMod _R}(K_i,N)\), is also contractible. By construction, \(K_i\) is a coproduct of modules of the form \((R/Rs_\alpha )[n_\alpha ]\). Since hom animae send coproducts in the first variable to products, we have \[ \Hom _{\LMod _R}(K_i,N) \iso \prod _\alpha \Hom _{\LMod _R}((R/Rs_\alpha )[n_\alpha ],N). \] By condition (2), each factor in this product is contractible. It follows that the entire product is contractible, which completes the inductive step and the proof. □