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Nanjun Yang
Nanjun Yang
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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: The spectral sequence strongly converges to $$[\Sigma^{\infty}U_+[p+q],E]$$.

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: The spectral sequence strongly converges to $$[\Sigma^{\infty}U_+[p+q],E]$$.

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?

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Nanjun Yang
Nanjun Yang
  • 918
  • 4
  • 11

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

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

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

Answer: The spectral sequence strongly converges to $$[\Sigma^{\infty}U_+[p+q],E]$$.

Spectral sequence associated with a Postnikov tower

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

Let $U\in Sm/k$, $$D_{p,q}^1=[\Sigma^{\infty}U_+[p+q],E_{\leq -p}], E_{p,q}^1=[\Sigma^{\infty}U_+[p+q-1],F_{-p+1}].$$ We know that $$E_{p,q}^1=H^{2-q-2p}(U,\pi_{1-p}(E)).$$ 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)

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: The spectral sequence strongly converges to $$[\Sigma^{\infty}U_+[p+q],E]$$.

Source Link
Nanjun Yang
Nanjun Yang
  • 918
  • 4
  • 11

Spectral sequence associated with a Postnikov tower

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

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