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

  • 8
    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.) – Clark Barwick Apr 22 '12 at 17:09

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

  • Thank you, but I could not locate at all. Maybe you can write the names of section/subsection/etc. (maybe there are several versions with different numbering). – Sasha Apr 23 '12 at 9:21
  • 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") – Yosemite Sam Apr 23 '12 at 9:40
  • OK, thank you. It seems that in the most updated version, one should replace your "7" by "8". – Sasha Apr 23 '12 at 11:52
  • sorry, my bad. I was clicking on the link google was providing, which was an older version of the book. – Yosemite Sam Apr 23 '12 at 15:32

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.

  • 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/… – David White Apr 24 '12 at 11:08

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