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?



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


