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Hello,

Are the model categories of simplicial commutative algebras over $k$ and that of commutative differential graded algebras (in negative cohmological dimension) Quillen equivalent in char. 0 (or maybe if $k$ is a $Q$-algebra)? What would be a reference?

Thank you

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    $\begingroup$ If $k$ is a characteristic zero ring, the normalized chain complex functor is a right Quillen equivalence between commutative simplicial $k$-algebras and connective commutative differential graded $k$-algebras, each equipped with their projective model structure. The proof is described in Quillen's Rational Homotopy Theory, p. 223. See also Bousfield-Gugenheim. (This critically uses characteristic zero, of course; in characteristic $p$, the homology of a commutative simplicial algebra admits a divided power structure.) $\endgroup$
    – Clark Barwick
    Commented Apr 22, 2012 at 17:09
  • $\begingroup$ Is the only difference between the two that a cdga is strictly graded commutative whereas a commutative simplicial algebra is commutative in an $E_\infty$ sense? Or is there something more to it? $\endgroup$
    – Hari Rau-Murthy
    Commented Dec 16, 2023 at 3:05

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I think Proposition 8.1.4.11 of Lurie's Higher Algebra gives the equivalence between (negatively graded, or homologically positively graded) CDGA's and connective $E_\infty$ algebras and Proposition 8.1.4.20 does the same for simplicial algebras (search for "rational numbers").

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  • $\begingroup$ Thank you, but I could not locate 7.1.4.20 at all. Maybe you can write the names of section/subsection/etc. (maybe there are several versions with different numbering). $\endgroup$
    – Sasha
    Commented Apr 23, 2012 at 9:21
  • $\begingroup$ that's odd O_o I can see it just fine in the May 18, 2011 version. (I originally dug it up by searching "rational numbers") $\endgroup$
    – Yosemite Sam
    Commented Apr 23, 2012 at 9:40
  • $\begingroup$ OK, thank you. It seems that in the most updated version, one should replace your "7" by "8". $\endgroup$
    – Sasha
    Commented Apr 23, 2012 at 11:52
  • $\begingroup$ sorry, my bad. I was clicking on the link google was providing, which was an older version of the book. $\endgroup$
    – Yosemite Sam
    Commented Apr 23, 2012 at 15:32
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You might be interested in the paper by Mike Mandell titled : Topological Andre-Quillen Cohomology and $E_{\infty}$ Andre-Quillen Cohomology. I am pretty sure his statements do not require a characteristic 0 assumption. For example, Theorem 1.3 on page 6 might be of interest.

Sorry, I guess I forgot to mention some things. I was initially pointed to this by Shipley's paper on $H\mathbb{Z}$ algebras and DGAs. The point is that when you want to remove the characteristic zero assumption you need to remember the $E_{\infty}$ algebra structure. In characteristic zero, it has to be essentially unique. This is related to the fact that the (co)homology of the symmetric groups is usually torsion. I remember Akhil Mathew asking a question related to this a while back. Anyways, the characteristic result follows. Although, it was known prior to this, see Clark Barwick's comment.

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  • $\begingroup$ Yes, one reason you need to move to $E_\infty$ in characteristic $\neq 0$ is that you have no hope of having a model category on strictly commutative monoids unless the characteristic is 0. This means you need to use $E_\infty$ if you want homotopy theoretic meaning. See for instance: mathoverflow.net/questions/23269/… $\endgroup$
    – David White
    Commented Apr 24, 2012 at 11:08

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