This correction article is actually unnecessary. The proof of Theorem 1.2, concerning commutative HQ-algebra spectra and commutative differential graded algebras, in the author's paper [American Journal of Mathematics vol. 129 (2007) 351-379 (arxiv:math/0209215v4)] is correct as originally stated. Neil Strickland carefully proved that D is symmetric monoidal; so Proposition 4.7 and hence also Theorem 1.2 hold as stated. Strickland's proof will appear in joint work with Stefan Schwede; see related work in Strickland's [arxiv:0810.1747]. Note here D is defined as a colimit of chain complexes; in contrast, non-symmetric monoidal functors analogous to D are defined as homotopy colimits of spaces in previous work of the author.
In the author's paper [S1], the proof of Theorem 1.2 is correct as stated; the functor D is symmetric monoidal. The author's confusion about this fact came from the comparison of this functor D, which is defined as a colimit of chain complexes, with the functor D in [S4] which is defined as a homotopy colimit of spaces. See also the discussion of commutative I-monoids in section 2.2 of [Sc]. In the topological case D is not symmetric monoidal; in the algebraic case though the functor D is symmetric.
Since the paper [S1] is mainly concerned with associative algebras, the only place this issue arises is in the proof of Theorem 1.2. As stated in Remark 2.11 in [S1], the main theorems (Theorem 1.1, Corollary 2.15 and Corollary 2.16 in [S1]) would also hold with the “three step” functors H and Θ replaced by the “four step” functors H = U Lc C 0 f F 0 c and Θ = Ev 0 f iφ * N Zc where c and f are the appropriate cofibrant and fibrant replacement functors. Since here the functors Ev 0 , i, φ * N and Z are symmetric monoidal, we have the following non-natural version of Theorem 1.2 from [S1] with Θ replaced by Θ. This statement first appeared as Theorem 1.3 in [S3].
Theorem 1. For C any commutative HQ-algebra, ΘC is weakly equivalent to a commutative differential graded Q-algebra.
Proof. As noted in the proof of Theorem 1.2 from [S1], the reason Θ is not symmetric monoidal is because the cofibrant and fibrant replacement functors involved in Θ are not symmetric monoidal. This is why ΘC is only weakly equivalent and not isomorphic to a commutative dg Q algebra.
The method for dealing with the cofibrant replacement functor in Θ proceeds as in [S1]. As proved there, a natural zig-zag of weak equivalences exists between Zc and the symmetric monoidal functor
Next we need to consider the fibrant replacement functor f which appears in Θ ′ (and Θ). This is the fibrant replacement functor in the model category of monoids in Sp Σ (Ch Q ). As in [S3], we exchange f for the fibrant replacement functor f ′ in the model category of commutative monoids in Sp Σ (Ch Q ) as established below in Proposition 3. For any commutative monoid A in Sp Σ (Ch Q ), we thus have two weak equivalences A -→ f A and A -→ f ′ A. Since f ′ A is also fibrant as a monoid and A -→ f A is a trivial cofibration of monoids, lifting provides a weak equivalence
Since ΘC is weakly equivalent to Θ ′ C and Θ ′′ C is a commutative differential graded Q-algebra, this completes the proof.
Remark 2. Although Theorem 1 does not give a natural identification of ΘC with a commutative DGA, for any small, fixed I-diagram D of commutative HQ-algebra spectra there will be a map of I-diagrams from ΘD to an I-diagram of commutative DGAs which is given by a variant of Θ ′′ D with f ′ replaced by the fibrant replacement functor in the model category of I-diagrams of commutative monoids in Sp Σ (Ch Q ) given by [Hi,11.6.1].
Proposition 3. There is a model category structure on the category of commutative monoids in Sp Σ (Ch Q ) in which a map is a weak equivalence or fibration if and only if the underlying map in Sp Σ (Ch Q ) is so.
Let S Q denote the unit and let ⊗ S denote the monoidal product in Sp Σ (Ch Q ). To establish this model category we use the lifting property from [ScSh,2.3(i)] applied to the free commutative monoid functor P which is left adjoint to the forgetful functor from commutative monoids in
Let I denote the generating cofibrations and J denote the generating trivial cofibrations in Sp Σ (Ch Q ); see [Ho,7]. To establish the lifting criterion in [ScSh,2.3], we first show that applying P to any map in J produces a stable equivalence. We do this by showing that in the source and target the orbit constructions can be replaced by homotopy orbits without changing the homotopy type.
Lemma 4. Let X, Y be in Sp Σ (Ch Q ) and n ≥ 1.
(1) The map
is also a level equivalence.
Proof. The first statement follows directly from the fact that given any Σ
The second statement follows as well by extending the Σ n -action trivially to Y and shifting the parentheses.
Next we show that pushouts of maps in P(J) are stable equivalences and level cofibrations. Since directed colimits of such maps are again stable equivalences, Proposition 3 then follows from Lemmas 4 and 5 by [ScSh,2.3].
Lemma 5. Let f : T -→ U be a cofibration in Sp Σ (Ch Q ) and V be a PT -module. Then the map q : V -→ V ⊗ PT PU is a level cofibration. If f is a trivial cofibration, then q is a stable equivalence.
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