# Stable Dold-Kan correspondence

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.

• Careful: people mean a different thing with "stable Dold-Kan correspondence" (usually that in a stable category (co)simplicial objects are equivalent to filtered objects) – Denis Nardin Nov 13 '17 at 20:11
• ok! I was following the terminology from this ncatlab page theorem 2.2 ;link – iron feliks Nov 13 '17 at 20:21
• The Quillen equivalences were proven explicitly by Brooke Shipley and Stefan Schwede. See $H\mathbb{Z}$-algebra spectra are differential graded algebras by Shipley or Stable model categories are categories of modules by Schwede and Shipley. – Lennart Meier Nov 17 '17 at 11:00
• @LennartMeier thanks for the references and sorry for the late reply, I found what I needed in appendix of Stable model categories are categories of modules by Schwede and Shipley – iron feliks Feb 16 '18 at 12:15