Spectral sequence associated with a Postnikov tower (Solved by myself)

Question

Suppose $E$ is an $S^1$-spectra of simplicial Nisnevich sheaves. For any $r\in\mathbb{Z}$, we have a distinguished triangle $$E_{\geq r+1}\longrightarrow E_{\geq r}\longrightarrow F_r\longrightarrow E_{\geq r+1}[1]$$ in $SH_s^{S^1}(k)$, where $_{\geq r}$ denotes the truncation functor (homological index, which is $_{\leq -r}$ in term of cohomological index). We have $F_r\cong H(\pi_r(E))[r]$.

Let $U\in Sm/k$, $$D_{p,q}^1=[\Sigma^{\infty}U_+[p+q],E_{\geq -p}], E_{p,q}^1=[\Sigma^{\infty}U_+[p+q],F_{-p}].$$ We know that $$E_{p,q}^1=H^{-q-2p}(U,\pi_{-p}(E)).$$ What does this spectral sequence converge to and what are the $E_{p,q}^2$ terms?

Answer 1

The convergence of this spectral sequence is because the Nisnevich topology of schemes has a cohomological dimension. The corresponding filtration is given by $$F^pH_n=Im([\Sigma^{\infty}U_+[n],E_{\geq -p}]\longrightarrow [\Sigma^{\infty}U_+[n],E])$$. cf. https://pdfs.semanticscholar.org/4702/f5e28ad71b82c67e0bb9405cd3e2a1931e5f.pdf

To identify the $E_{p,q}^1$, one could prove that $$[\Sigma^{\infty}U_+,F[n]]\cong H^n(U,\pi_0(F))$$ for $F\in SH^{S^1}_s(k)^{\heartsuit}$ by using the $\delta$-functor theories. The case for $n=0$ follows from the $t$-structure. For the effaceability, one could show that $H(A)$ is stably fibrant when $A$ is an injective Nisnevich abelian sheaf. For this, one shows that $K(A,n)=\Gamma(A[-n])$ is Brown Gersten fibrant and then it's easy to prove $H(A)$ is an $\Omega$-spectrum. cf. J. F. Jardine, 'Generalized Etale Cohomology Theories'.