spectral sequence and provides a large class of examples of decreasingly filtered commutative ring spectra; i.e., those that can be produced as a multiplicative model for the Whitehead tower of a connective commutative ring spectrum. In Section \[section 3\], we provide all the details needed to prove Theorem \[main thm\].

Notation and Conventions
------------------------

Throughout, let $\mathfrak{S}$ be the category of $S$-modules in symmetric spectra of pointed simplicial sets, where $S$ is a cofibrant replacement for the sphere spectrum. We equip $\mathfrak{S}$ with the flat model structure [@schwedebook]. Note that $\mathfrak{S}$ is a closed symmetric monoidal model category with $\wedge$ as symmetric monoidal product and $S$ as the unit. We will write $\Comm \mathcal{D}$, for the category of commutative monoids in a symmetric monoidal category $\mathcal{D}$. When a map $X\rightarrow Y$ between objects in $\mathfrak{S}$ is a cofibration, we will write $Y/X$ for the cofiber. We will fix a prime $p\ge 3$ throughout. This ensures that the mod $p$ Moore spectrum has a $v_1$-self map $v_1:\Sigma^{2p-2}S/p\lra S/p$ and the cofiber, denoted $V(1)$, exists.

As we observed in introduction, there is an equivalence $K(\mathbb{F}_q)_p\simeq \tau_{\ge 0} L_{K(1)}S$. There is also an equivalence $\tau_{\ge 0}L_{K(1)}S\simeq j_p$ where $j_p$ is the $p$ completion 