[Paper Review] Correction to: HZ-algebra spectra are differential graded algebras

Brooke Shipley|ArXiv.org|Aug 9, 2007

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This correction reaffirms that the functor $ D $ in the author's 2007 paper is symmetric monoidal, validating the original proof of Theorem 1.2, which establishes that commutative $ H\mathbb{Q} $-algebra spectra are weakly equivalent to commutative differential graded algebras over $ \mathbb{Q} $. The correction resolves a confusion arising from conflating colimits of chain complexes with homotopy colimits of spaces, confirming the correctness of the original result and providing a non-natural weak equivalence via modified functors $ \overline{\Theta} $.

ABSTRACT

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.

Motivation & Objective

• To resolve a confusion in the original proof of Theorem 1.2 concerning the symmetric monoidality of the functor $ D $.
• To confirm that the original proof of the equivalence between commutative $ H\mathbb{Q} $-algebra spectra and commutative differential graded algebras (DGAs) over $ \mathbb{Q} $ is correct.
• To clarify the distinction between $ D $ defined as a colimit of chain complexes (symmetric monoidal) and $ D $ defined as a homotopy colimit of spaces (not symmetric monoidal), which had caused the initial confusion.
• To provide an alternative, non-natural weak equivalence statement using modified functors $ \overline{\Theta} $, which are constructed via cofibrant and fibrant replacement functors.
• To establish a model category structure on commutative monoids in $ Sp^\Sigma(\mathcal{C}h_\mathbb{Q}) $ with weak equivalences and fibrations detected on the underlying spectra.

Proposed method

• Reaffirming that Neil Strickland's proof establishes $ D $ as symmetric monoidal, which validates the original argument in Theorem 1.2.
• Comparing the functor $ D $ defined as a colimit of chain complexes (symmetric monoidal) with the earlier $ D $ defined as a homotopy colimit of spaces (non-symmetric monoidal), resolving the source of confusion.
• Constructing a non-natural weak equivalence via $ \overline{\Theta} = \operatorname{Ev}_0 fi\phi^*NZc $, where $ c $ and $ f $ are cofibrant and fibrant replacement functors.
• Using the symmetric monoidality of $ \operatorname{Ev}_0 $, $ i $, $ \phi^*N $, and $ Z $ to ensure the composition $ \overline{\Theta} $ is well-behaved, though not naturally symmetric monoidal due to $ c $ and $ f $.
• Applying the lifting property from [ScSh, 2.3] to construct a model category structure on commutative monoids in $ Sp^\Sigma(\mathcal{C}h_\mathbb{Q}) $, using the free commutative monoid functor $ \mathbb{P} $ and generating cofibrations $ I $, $ J $.
• Proving that $ \mathbb{P}(J) $-maps are stable equivalences via Lemma 4, which shows that orbit constructions can be replaced by homotopy orbits without changing homotopy type, and using filtration arguments from [Ma] to establish pushout stability.

Experimental results

Research questions

• RQ1 Is the functor $ D $, defined as a colimit of chain complexes in the proof of Theorem 1.2, symmetric monoidal?
• RQ2 Does the original proof of Theorem 1.2 in Shipley's 2007 paper remain valid despite the author's initial confusion about the monoidality of $ D $?
• RQ3 Can a non-natural weak equivalence be constructed between $ \overline{\Theta}C $ and a commutative differential graded $ \mathbb{Q} $-algebra for any commutative $ H\mathbb{Q} $-algebra spectrum $ C $?
• RQ4 Is there a model category structure on the category of commutative monoids in $ Sp^\Sigma(\mathcal{C}h_\mathbb{Q}) $ where weak equivalences and fibrations are detected on the underlying spectra?
• RQ5 Do pushouts of maps in $ \mathbb{P}(J) $ preserve stable equivalences and level cofibrations?

Key findings

• The original proof of Theorem 1.2 is correct: $ D $ is symmetric monoidal, as confirmed by Neil Strickland’s proof, so the main result holds as stated.
• The confusion arose from conflating $ D $ as a colimit of chain complexes (symmetric monoidal) with $ D $ as a homotopy colimit of spaces (not symmetric monoidal), a distinction now clarified.
• A non-natural weak equivalence exists between $ \overline{\Theta}C $ and a commutative differential graded $ \mathbb{Q} $-algebra, as shown by constructing $ \overline{\Theta}^\prime $ and $ \overline{\Theta}^{\prime\prime} $ via symmetric monoidal functors and fibrant replacements.
• The model category structure on commutative monoids in $ Sp^\Sigma(\mathcal{C}h_\mathbb{Q}) $ is established via the lifting criterion from [ScSh, 2.3], with weak equivalences and fibrations defined via the underlying spectra.
• Pushouts of maps in $ \mathbb{P}(J) $ are stable equivalences and level cofibrations, as shown by Lemma 5 using filtration techniques and Lemma 4 on orbit replacements.
• The map $ E\Sigma_n \otimes_{\Sigma_n} X^{(n)} \to X^{(n)}/\Sigma_n $ is a level equivalence, and the induced map on tensor products with $ Y $ is also a level equivalence, ensuring homotopy invariance under symmetric group actions.