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A bit of light googling turns up several sources asserting that the model structure on simplicial commutative algebras over a ring is left proper (for example, 2.9 in Charles Rezk's paper Every homotopy theory of simplicial algebras admits a proper model). Does a proof of this fact occur anywhere in the literature?

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    $\begingroup$ I don't know where it "appears", though of course it is not hard to prove; it amounts to the fact that polynomial algebras are flat. $\endgroup$ Sep 27, 2012 at 2:48

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For simplicial commutative rings this is proved in Lemma 3.1.2 of Schwede's “Spectra in model categories and applications to the algebraic cotangent complex”, and the proof there immediately extends to algebras.

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  • $\begingroup$ Charles was, I suppose, too modest to mention that it is also a special case of the paper I referred to in the question, but this is an earlier reference. $\endgroup$ Jun 30, 2015 at 10:29

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