There exists a Quillen equivalence between $HRModSpectra$ (homotopy category of ring spectra over Eilenberg-MacLane spectra $EM(R)$, where $R$ is a commutative ring, with stable model structure) and $HoCh$ (homotopy category of unbounded chain complexes of $R$-modules).

I was wondering what the Quillen functors are that give the above Quillen equivalence.

One can start with an unbounded chain complex $X$ and apply Dold-Kan functor $\Gamma$ to chain complex $X_{\geq 0}$ to get a simplicial abelian group $\Gamma(X_{\geq 0})$, and then consider $\Gamma(X[-n]_{\geq0})$ (shifting $X$ to the left by n places  , truncating and then applying $\Gamma$). This way one gets an $\Omega$- spectrum $Y$ = {${Y_{0}, Y_{1},...}$}, with $Y_{n} = \Gamma(X[-n]_{\geq0})$.

How does then one proceed to prove that $Y$ is a symmetric spectrum? For that one needs an action of symmetric group $S_{n}$ on $Y_{n}$. Now each $Y_{i}$ is in fact as a simplicial set equivalent to $\prod K(\pi_{k}(Y_{n}), k)$ and $K(\pi_{n}(Y_{n}), n)$ has an $S_{n}$ action, though I'm not sure if this action will satisfy the compatibility conditions that are required of a symmetric spectrum.

There exists a Quillen equivalence between $HRModSpectra$ (model category of ring spectra over Eilenberg-MacLane spectra $EM(R)$, where $R$ is a commutative ring, with stable model structure) and $Ch$ (model category of unbounded chain complexes of $R$-modules).

I was wondering what the Quillen functors are that give the above Quillen equivalence.

One can start with an unbounded chain complex $X$ and apply the Dold-Kan functor $\Gamma$ to chain complex $X_{\geq 0}$ to get a simplicial abelian group $\Gamma(X_{\geq 0})$, and then consider $\Gamma(X[-n]_{\geq0})$ (shifting $X$ to the left by n places, truncating and then applying $\Gamma$). This way one gets an $\Omega$-spectrum $Y$ = {${Y_{0}, Y_{1},...}$}, with $Y_{n} = \Gamma(X[-n]_{\geq0})$.

How does then one proceed to prove that $Y$ is a symmetric spectrum? For that one needs an action of symmetric group $S_{n}$ on $Y_{n}$. Now each $Y_{i}$ is in fact as a simplicial set equivalent to $\prod K(\pi_{k}(Y_{n}), k)$ and $K(\pi_{n}(Y_{n}), n)$ has an $S_{n}$ action, though I'm not sure if this action will satisfy the compatibility conditions that are required of a symmetric spectrum.

There exists a Quillen equivalence between $HRModSpectra$ (homotopy category of ring spectra over Eilenberg-MacLane spectra EM(R), where R is a commutative ring, with stable model structure  ) and $HoCh$ (homotopy category of unbounded chain complexes of R-modules  ).

I was wondering what are the Quillen functors which give the above Quillen equivalence.

One can start with an unbounded chain complex $X$ and apply Dold-Kan functor $\Gamma$ to chain complex $X_{\geq 0}$ to get a simplicial abelian group $\Gamma(X_{\geq 0})$, and then consider $\Gamma(X[-n]_{\geq0})$ (shifting $X$ to the left by n places , truncating and then applying $\Gamma$). This way one gets an $\Omega$- spectra $Y$ = {${Y_{0}, Y_{1},...}$}, with $Y_{n} = \Gamma(X[-n]_{\geq0}) $

How does then one proceed to prove that $Y$ is a symmetric spectra. For that one needs an action of symmetric group $S_{n}$ on $Y_{n}$. Now each $Y_{i}$ is in fact as a simplicial set equivalent to $\prod K(\pi_{k}(Y_{n}), k)$ and $K(\pi_{n}(Y_{n}), n)$ has an $S_{n}$ action, though I'm not sure if this action will satisfy the compatibility conditions that are required of a symmetric spectra.

There exists a Quillen equivalence between $HRModSpectra$ (homotopy category of ring spectra over Eilenberg-MacLane spectra $EM(R)$, where $R$ is a commutative ring, with stable model structure) and $HoCh$ (homotopy category of unbounded chain complexes of $R$-modules).

I was wondering what the Quillen functors are that give the above Quillen equivalence.

One can start with an unbounded chain complex $X$ and apply Dold-Kan functor $\Gamma$ to chain complex $X_{\geq 0}$ to get a simplicial abelian group $\Gamma(X_{\geq 0})$, and then consider $\Gamma(X[-n]_{\geq0})$ (shifting $X$ to the left by n places , truncating and then applying $\Gamma$). This way one gets an $\Omega$- spectrum $Y$ = {${Y_{0}, Y_{1},...}$}, with $Y_{n} = \Gamma(X[-n]_{\geq0})$.

How does then one proceed to prove that $Y$ is a symmetric spectrum? For that one needs an action of symmetric group $S_{n}$ on $Y_{n}$. Now each $Y_{i}$ is in fact as a simplicial set equivalent to $\prod K(\pi_{k}(Y_{n}), k)$ and $K(\pi_{n}(Y_{n}), n)$ has an $S_{n}$ action, though I'm not sure if this action will satisfy the compatibility conditions that are required of a symmetric spectrum.

There exists a Quillen equivalence between $HRModSpectra$ (homotopy category of ring spectra over Eilenberg-MacLane spectra EM(R), where R is a commutative ring, with stable model structure ) and $HoCh$ (homotopy category of unbounded chain complexes of R-modules ).

I was wondering what are the Quillen functors which give the above Quillen equivalence.

One can start with an unbounded chain complex $X$ and apply Dold-Kan functor $\Gamma$ to chain complex $X_{\geq 0}$ to get a simplicial abelian group $\Gamma(X_{\geq 0})$, and then consider $\Gamma(X[-n]_{\geq0})$ (shifting $X$ to the left by n places , truncating and then applying $\Gamma$). This way one gets an $\Omega$- spectra $Y$ ={$\Gamma(X_{\geq 0})$, $\Gamma(X[-1]_{\geq0})$ , ...}.

How does then one proceed to prove that $Y$ is a symmetric spectra.

There exists a Quillen equivalence between $HRModSpectra$ (homotopy category of ring spectra over Eilenberg-MacLane spectra EM(R), where R is a commutative ring, with stable model structure ) and $HoCh$ (homotopy category of unbounded chain complexes of R-modules ).

I was wondering what are the Quillen functors which give the above Quillen equivalence.

One can start with an unbounded chain complex $X$ and apply Dold-Kan functor $\Gamma$ to chain complex $X_{\geq 0}$ to get a simplicial abelian group $\Gamma(X_{\geq 0})$, and then consider $\Gamma(X[-n]_{\geq0})$ (shifting $X$ to the left by n places , truncating and then applying $\Gamma$). This way one gets an $\Omega$- spectra $Y$ = {${Y_{0}, Y_{1},...}$}, with $Y_{n} = \Gamma(X[-n]_{\geq0}) $

How does then one proceed to prove that $Y$ is a symmetric spectra. For that one needs an action of symmetric group $S_{n}$ on $Y_{n}$. Now each $Y_{i}$ is in fact as a simplicial set equivalent to $\prod K(\pi_{k}(Y_{n}), k)$ and $K(\pi_{n}(Y_{n}), n)$ has an $S_{n}$ action, though I'm not sure if this action will satisfy the compatibility conditions that are required of a symmetric spectra.