$$\label{thh may ss coeff} E^2_{*,*} = H_*(THH(E_0^*(I),E_0^*(M) ))\Rightarrow H_* (THH(I_0, M_0))$$ with the same differential convention. The spectral sequence has strong convergence when $\pi_k(I_i)\cong 0$ for $k<i$ and $\pi_k(M_i)\cong 0$ for $k<i$. When $M$ is also itself a cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, compatible with the $I$-module structure, the differentials will also satisfy a Leibniz rule. Full details about this construction can be found in the appendix of [@thhmay].

Decreasingly filtered commutative monoids in $\mathfrak{S}$
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In order to do computations with the THH-May spectral sequence, it is necessary to have examples of cofibrant decreasingly filtered commutative monoids in $\mathfrak{S}$. The author constructed a large class of examples by constructing a specific model for Whitehead tower associated to cofibrant connective commutative ring spectrum in a joint paper with A. Salch [@thhmay]. This model for the Whitehead tower of $j$ is used to compute mod $(p,v_1)$-homotopy of $THH(j)$ in the subsequent section, so we recall the statement of the theorem.

\[filt\] Let $R$ be a cofibrant connective commutative monoid in $\mathfrak{S}$, then there is an associated cofibrant decreasingly filtered commutative monoid in $\mathfrak{S}$, $$\xymatrix{ \dots \ar[r] &  \tau_{\ge 2}R  \ar[r] & \tau_{\ge 1}R  \ar[r]&  \tau_{\ge 0}R    }$$ where $\pi_k(\tau_{\ge 