n}R)\cong \pi_k(R)$ for $k\ge n$ and $\pi_k(\tau_{\ge n}R)\cong 0$ for $k<n$, equipped with structure maps $$\rho_{i,j}:\tau_{\ge i}R \wedge \tau_{\ge j}R  \rightarrow \tau_{\ge i+j}R$$ satisfying commutativity, associativity, unitality, and compatibility.

See the author’s joint paper with A. Salch [@thhmay] for a detailed proof.

The fact that we can construct an object in $\mathfrak{S}^{\mathbb{N}^{\op}}$ with these properties is well known. The main thrust of the proof in [@thhmay] is that $\tau_{\ge \bullet}R$ can be built with multiplicative structure; i.e. $\tau_{\ge \bullet}R$ can be constructed as an object in $\Comm \mathfrak{S}^{\mathbb{N}^{\op}}$.

 \[exm J\]Assume a prime $p\ge5$ is fixed and $q$ is a prime power that topologically generates $\mathbb{Z}_p^{\times}$. Recall that $j$ is a cofibrant replacement in $\Comm \mathfrak{S}$ for the connective commutative ring spectrum $\hat{K}(\mathbb{F}_q)_p$. Theorem \[filt\] produces a decreasingly filtered commutative monoid in $\mathfrak{S}$, denoted $\mathbb{J}$. The associated graded $E_0^*\mathbb{J}$ is additively equivalent to $$H\pi_0j\vee\Sigma^{2p-3}H\pi_{2p-3}j\vee \Sigma^{4p-5}H\pi_{4p-5}j\vee ...$$ or more succinctly $H\pi_*(j)$. Its homotopy groups $\pi_*(E_0^*\mathbb{J})$ are isomorphic to $\pi_*(j)$, but it is a generalized Eilenberg-Maclane spectrum. In other words, we have killed off all the Postnikov $k$-invariants. After smashing with $S/p$, there is an equivalence $$S/p \wedge E_0^*\mathbb{J}\simeq H\mathbb{F}_p \vee \bigvee_{i\ge 1} \Sigma^{(2p-2)i-1}H\mathbb{F}_p\vee \Sigma^{(2p-2)i}H\mathbb{F}_p$$ and $\pi_*(S/p \wedge E_0^*\mathbb{J})\cong P(v_1)\otimes E(\alpha_1)$. In particular, 