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In Shipley's paper http://arxiv.org/abs/math/0209215 she proves a Quillen equivalence between the category of $H\mathbb Z$-modules and dg $\mathbb Z$-modules. So, to a chain complex $C$, she assigns a spectrum $HC$, but is it true in some sense that $\pi_i HC \cong H_i C$? I am aware there are some subtleties involved with homotopy groups of symmetric spectra, but I'm not sure how that plays out in this case.

flag	asked Sep 14 at 13:03 Justin Young 163●3

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It's true, it's not explicitly stated because Shipley may have seen it trivial, but it might be tricky if you don't know how the zig-zag of Quillen equivalences works. The 'problem' which homotopy groups of symmetric spectra is not relevant for this property. Actually, there are no problems with homotopy groups of symmetric spectra defined as homotopy classes of maps from the suspensions of the sphere spectrum. The problem is when you want to define them in the naive way, i.e. as the colimit of the homotopy groups of the components. – Fernando Muro Sep 14 at 13:34

Tyler Lawson · Accepted Answer · 2012-09-14 17:34:14Z UTC

Shipley's result (of which an alternative version can be found in EKMM, IV.2) proves that there is a string of Quillen equivalences between the derived category of chain complexes over $\mathbb{Z}$ and the category of $H\mathbb{Z}$-modules. This, in particular, gives rise to a chain of natural equivalences between their homotopy categories. We can then prove the result you're asking for by defining both homotopy and homology in the same terms.

Both homotopy categories have a suspension/shift functor, which is an autoequivalence. In both cases, this can be described as a homotopy pushout of the diagram $\ast \leftarrow X \rightarrow \ast$, where $\ast$ is the terminal object. Quillen equivalences preserves homotopy pushouts and so preserves shift (and hence, up to natural isomorphism, its inverse).

In the category of chain complexes, there is an object $\mathbb{Z}$, concentrated in degree zero, and this is explicitly carried to $H\mathbb{Z}$ under the chain of Quillen equivalences.

Finally, we can define both homotopy groups and homology groups using the derived category: $$ H_i C = [\mathbb{Z}[n], C] $$ $$ \pi_i HC = [\Sigma^n H\mathbb{Z}, HC] $$ Since all of the ingredients in this definition (shifting degree, the representing object, maps in the homotopy category) are preserved under Quillen equivalence, the two can be made naturally isomorphic.

(Of course, this really manifests that one of Shipley's string of equivalences is Dold-Kan equivalence between simplicial abelian groups and nonnegatively graded chain complexes, which takes homology groups to homotopy groups.)

Homotopy groups and homology groups for the $H\mathbb Z$ module-dg module correspondence.

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