Closure properties of locally nilpotent modules

Proof. Throughout we use that for a homogeneous element \(s \in \pi _d(R)\) the operation \(x \mapsto s \cdot x\) is a natural transformation \(\pi _k(-) \to \pi _{k+d}(-)\) of functors \(\LMod _R \to \Ab \), and that \(\pi _*\) carries exact sequences of left \(R\)-modules to long exact sequences of graded \(\pi _*(R)\)-modules.

Both subcategories evidently contain \(0\) and are closed under shifts. For closure under cofibers, consider an exact sequence \(M' \to M \to M''\) in \(\LMod _R\).

If \(M'\) and \(M\) are \(S\)-local, then in the long exact sequence \[ \pi _*(M') \to \pi _*(M) \to \pi _*(M'') \to \pi _{*-1}(M') \to \pi _{*-1}(M) \] multiplication by \(s\) acts as an isomorphism on four of the five terms, so the five lemma shows that it is an isomorphism on \(\pi _*(M'')\) as well; hence \(M''\) is \(S\)-local. The same argument applied to the rotated sequence shows closure under fibers, so \(\LMod _R^{\Loc (S)}\) is stable.

If \(M'\) and \(M\) are \(S\)-nilpotent, let \(x \in \pi _n(M'')\). Its image \(\partial (x) \in \pi _{n-1}(M')\) is annihilated by some \(s \in S\), so \(\partial (s\cdot x) = s \cdot \partial (x) = 0\) and therefore \(s\cdot x\) lifts to an element \(y \in \pi _{n+\abs {s}}(M)\). Choosing \(t \in S\) with \(t\cdot y = 0\), we obtain \((ts)\cdot x = t\cdot (s \cdot x) = 0\), and \(ts \in S\) since \(S\) is multiplicative. Hence \(M''\) is \(S\)-nilpotent.

Similarly, suppose that \(M'\) and \(M''\) are \(S\)-nilpotent and let \(x\in \pi _n(M)\). Choose \(s\in S\) which annihilates the image of \(x\) in \(\pi _n(M'')\). Then \(s\cdot x\) is the image of an element \(y\in \pi _{n+\abs {s}}(M')\). Choosing \(t\in S\) with \(t\cdot y=0\) gives \((ts)\cdot x=0\). Thus \(M\) is \(S\)-nilpotent as well, so \(\LMod _R^{S\dnil }\) is stable and closed under extensions.

Being stable, both subcategories are in particular closed under finite coproducts: a biproduct \(M_1 \oplus M_2\) sits in an exact sequence \(M_1 \to M_1 \oplus M_2 \to M_2\), and both subcategories are closed under extensions. Consequently both are closed under all finite colimits.

Next, both are closed under filtered colimits, since \(\pi _*\) preserves these by Lemma 4.4.28. Indeed, a filtered colimit of isomorphisms is an isomorphism, which handles the \(S\)-local case; and in the \(S\)-nilpotent case every element of \(\pi _n(\colim _i M_i) \cong \colim _i \pi _n(M_i)\) is the image of some \(x \in \pi _n(M_i)\), and any \(s \in S\) annihilating \(x\) also annihilates its image. Since an arbitrary coproduct is the filtered colimit of its finite subcoproducts, both subcategories are closed under small coproducts. As an \(\infty \)-category admitting pushouts and small coproducts admits all small colimits, this completes the proof. □