Extension DGAs and topological Hochschild homology

In this work, we study those differential graded algebras (DGAs) that arise from ring spectra through the extension of scalars functor. Namely, we study DGAs whose corresponding Eilenberg-Mac Lane ring spectrum is equivalent to $H\mathbb{Z} \wedge E$ for some ring spectrum $E$. We call these DGAs extension DGAs. We also define and study this notion for $E_\infty$ DGAs. The topological Hochschild homology (THH) spectrum of an extension DGA splits in a convenient way. We show that formal DGAs with nice homology rings are extension and therefore their THH groups can be obtained from their Hochschild homology groups in many cases of interest. We also provide interesting examples of DGAs that are not extension. In the second part, we study properties of extension DGAs. We show that in various cases, topological equivalences and quasi-isomorphisms agree for extension DGAs. From this, we obtain that dg Morita equivalences and Morita equivalences also agree in these cases.


introduction
In [22], Stanley shows that the homotopy category of differential graded algebras is equivalent to the homotopy category of HZ-algebras.Later, Shipley improves this equivalence to a zig-zag of Quillen equivalences between the model categories of DGAs and HZ-algebras [21].This opens up a new opportunity to study DGAs, i.e. to study DGAs using ring spectra.
Dugger and Shipley use this zig-zag of Quillen equivalences to define new equivalences between DGAs called topological equivalences, see Definition 1.9 below.They show non-trivial examples of topologically equivalent DGAs and they use topological equivalences to develop a Morita theory for DGAs [7].In [2], the author uses topological equivalences to obtain classification results for DGAs.Moreover, topological equivalences for E ∞ DGAs are studied by the author in [1].
In this work, we follow this philosophy in a different way.We study what we call extension DGAs which are the DGAs that are obtained from ring spectra through the extension of scalars functor from S-algebras to HZ-algebras, i.e. the functor HZ ∧ −.More generally, we work in R-DGAs for a discrete commutative ring R.There is a zig-zag of Quillen equivalences between R-DGAs and HR-algebras [21].
Definition 1.1.An R-DGA X is R-extension if the HR-algebra corresponding to X is weakly equivalent to HR ∧ E for some cofibrant S-algebra E. For R = Z, we omit Z and write extension instead of Z-extension.
To define R-extension E ∞ R-DGAs, we use the zig-zag of Quillen equivalences between E ∞ R-DGAs and commutative HR-algebras constructed in [15].Definition 1.2.An E ∞ R-DGA X is R-extension if the commutative HR-algebra corresponding to X is weakly equivalent to HR ∧ E for some cofibrant commutative S-algebra E. For R = Z, we omit Z and write extension instead of Z-extension.
See Appendix A for a discussion on the compatibility of the two definitions above.One reason to study extension DGAs is that for an R-extension DGA X, there is the following splitting at the level of spectra THH(X) ≃ THH(HR) ∧ HR HH R (X) where HH R denotes THH HR , see [16,Theorem 1] for an instance of this splitting when X is the Eilenberg-Mac Lane spectrum of a discrete ring.If X is an R-extension E ∞ R-DGA, then this splitting is a splitting of commutative ring spectra.This follows from the fact that THH commutes with smash products and the base change formula for THH, see [12,Conventions].
This splitting simplifies THH calculations significantly in many situations.Indeed, it is an important stepping stone in many THH calculations in the literature, particularly for the case where X is a discrete ring, i.e. a DGA whose homology is concentrated in degree 0. For example, Larsen and Lindenstrauss show that this splitting exists at the level of homotopy groups for various discrete rings of characteristic p [13].Furthermore, Hesselholt and Madsen prove such a splitting for discrete rings that have a nice basis with respect to the ground ring R [9, Theorem 7.1].In the following theorem, we generalize this result to connective formal DGAs.Note that a connective DGA is a DGA whose negative homology is trivial.
Theorem 1.3.Let X be a connective formal R-DGA whose homology has a homogeneous basis as an R-module containing the multiplicative unit such that the multiplication of two basis elements is either zero or a basis element.In this situation, X is R-extension.As a result, we have the following equivalence of spectra.THH(X) ≃ THH(HR) ∧ HR HH R (X) Section 5 is devoted to the proof of this theorem.Furthermore, for a given R-DGA that satisfies the hypothesis of the theorem above, we provide an explicit description of the corresponding HR-algebra; see Proposition 5.8.The author and Moulinos show that for such HR-algebras, one often obtains non-trivial splittings at the level of topological negative cyclic homology and topological periodic homology [3, 4.8 and 6.1].Using these splittings, the author and Moulinos compute the algebraic K-theory of THH(HF p ), i.e. the algebraic K-theory of the formal DGA with homology F p [x 2 ].In a future work, the author is planning to compute the algebraic K-theory groups of various formal DGAs by using Proposition 5.8 and the splittings provided in [3].
Remark 1.4.Another way to state the hypothesis of Theorem 1.3 is the following.Let M be a monoid in the category of graded pointed sets.From M, one obtains a graded R-algebra R M whose underlying R-module is the free R-module over the graded set M − obtained by removing the based point from M. The multiplication on R M is given by the multiplication on M where the based point of M is considered as the zero element in R M .A graded R-algebra of the form R M is called a graded monoid R-algebra.With this definition, a connective formal R-DGA satisfies the hypothesis of Theorem 1.3 if and only if its homology is a graded monoid R-algebra.
Remark 1.5.We mention a few examples of graded rings that satisfy the hypothesis of the theorem above as homology of X.The polynomial algebra over R with a nonnegatively graded set S of generators R[S] satisfies the hypothesis if all the elements of S are in even degrees.The basis of R[S] is given by the monomials in S and the unit 1 ∈ R. Similarly, many examples of quotients of polynomial rings with even degree generators also satisfy this hypothesis; for example R[x]/(x 2 ), R[x, y]/(y 2 ) and R[x, y]/(x 2 y, y 3 ) with even |x| and |y|.However, there are rings that do not satisfy this hypothesis.For example for R = Z, the exterior algebra on two generators Λ[x, y] ∼ = Λ[x]⊗Λ[y] with odd |x| and |y| has a basis given by {x, y, xy}, but yx = −xy and therefore yx is not one of the basis elements.Indeed, Λ[x, y] has no basis that satisfies this hypothesis.
We prove the following non-extension results.respectively.The E ∞ DGA corresponding to X is an extension E ∞ DGA and the E ∞ DGA corresponding to Y is an E ∞ F p -DGA.Although these two E ∞ DGAs are E ∞ topologically equivalent, they are not quasi-isomorphic due to Theorem 5.3 in [1].For the associative case with p = 2, the distinction between the two DGAs corresponding to X and Y is due to Example 5.6 in [7].
In the results above, we work with (E ∞ ) DGAs in mixed characteristic, i.e. we work in (E ∞ ) Z-DGAs.A natural question to ask is if there are examples of E ∞ k-DGAs that are not k-extension for a field k.In Example 1.11 below, we show that there are E ∞ F p -DGAs that are not F p -extension.Now we discuss topological equivalences of DGAs and the properties of extension DGAs regarding topological equivalences.Definition 1.9.Two DGAs X and Y are topologically equivalent if the corresponding HZ-algebras HX and HY are weakly equivalent as S-algebras.
The definition of E ∞ topological equivalences is as follows.
Definition 1.10.Two E ∞ DGAs X and Y are E ∞ topologically equivalent if the corresponding commutative HZ-algebras HX and HY are weakly equivalent as commutative S-algebras.
It follows from these definitions that quasi-isomorphic (E ∞ ) DGAs are (E ∞ ) topologically equivalent.However, there are examples of non-trivially topologically equivalent DGAs, i.e.DGAs that are topologically equivalent but not quasi-isomorphic [7].Furthermore, examples of non-trivially E ∞ topologically equivalent E ∞ DGAs are constructed by the author in [1].
Example 1.11.This is an example of E ∞ F p -DGAs that are not F p -extension.In Example 5.1 of [1], the author constructs non-trivially E ∞ topologically equivalent E ∞ F p -DGAs that we call X and Y , i.e.X and Y are E ∞ topologically equivalent but they are not quasi-isomorphic.Although these E ∞ F p -DGAs are E ∞ topologically equivalent, their Dyer-Lashof operations are different.
For p = 2, the homology rings of these E ∞ F p -DGAs are given by for both X and Y where |x| = 1.On the homology of X, the first Dyer-Lashof operation is trivial, i.e.Q 1 x = 0. On the other hand, we have Q 1 x = x 3 on the homology of Y .Using these properties we show (for all primes) that these E ∞ F p -DGAs are not F p -extension E ∞ F p -DGAs.See Section 3B for a proof of this fact.
By Theorem 1.6 in [1], E ∞ topological equivalences between E ∞ F p -DGAs with trivial first homology preserve Dyer-Lashof operations.We prove a stronger result for F p -extension E ∞ F p -DGAs.Theorem 1.12.Let X be an F p -extension E ∞ F p -DGA with H 1 X = 0 and let Y be an E ∞ F p -DGA.In this situation, X and Y are quasi-isomorphic if and only if they are E ∞ topologically equivalent.
In the following results, we show various situations where topological equivalences and quasi-isomorphisms agree.Theorem 1.13.Let (p = 2) p be an odd prime.Let X be an extension F p -DGA whose homology is trivial on degrees (2 r − 1) 2p r − 2 for r ≥ 1 and () 2p s − 1 for s ≥ 0 and let Y be an F p -DGA.In this situation, X and Y are quasi-isomorphic if and only if they are topologically equivalent.
For the corollary below, note that a co-connective DGA is a DGA with trivial homology in positive degrees.
Corollary 1.14.Let X be a co-connective extension F p -DGA and let Y be an F p -DGA.Then X and Y are quasi-isomorphic if and only if they are topologically equivalent.
For the theorem and the corollary below, let R = Z/(m) for some integer m = ±1.Theorem 1.15.Let X be an R-DGA whose corresponding HR-algebra is equivalent to HR∧Z for some cofibrant S-algebra Z whose underlying spectrum is equivalent to a coproduct of suspensions and/or desuspensions of the sphere spectrum.Also, let Y be an R-DGA.Then X and Y are quasi-isomorphic if and only if they are topologically equivalent.
Our main interest for this theorem is due to its corollary stated below.This follows by Proposition 5.8 which implies that an R-DGA that satisfies the hypothesis of Theorem 1.3 also satisfies the hypothesis of the theorem above.
Corollary 1.16.Let Y be an R-DGA and let X be as in Theorem 1.3.Then X and Y are quasi-isomorphic if and only if they are topologically equivalent.
Two DGAs X and Y are said to be Morita equivalent if the model categories of Xmodules and Y -modules are Quillen equivalent.There is a stronger notion of Morita equivalence for DGAs called dg Morita equivalences defined by Keller, see [11,Section 3.8] and [7, 7.6].This is a strictly stronger notion of Morita equivalence since there are examples of DGAs that are Morita equivalent but not dg Morita equivalent [7, Section 8].However in the situations where topological equivalences and quasi-isomorphisms agree, these two notions of Morita equivalences also agree, see Proposition 7.7 and Theorem 7.2 of [7].We obtain the following corollary to Theorems 1.15 and 1.13.
Corollary 1.17.Assume that X and Y are as in Theorem 1.15 or as in Theorem 1. 13.Then X and Y are Morita equivalent if and only if they are dg Morita equivalent.
Organization In Section 2, we describe the dual Steenrod algebra and the Dyer-Lashof operations on it.In Section 3, we prove Theorems 1.12, 1.13 and 1.15.Section 4 is devoted to the proof of Theorems 1.6 and 1.7.In section 5, we prove Theorem 1.3.This section is independent from Sections 2, 3 and 4 and it contains explicit descriptions of the HZ-algebras corresponding to the formal DGAs as in Theorem 1.3 which is of independent interest.We left the proof of Theorem 1.3 to the end because it uses different tools than the rest of the proof in this work.Appendix A is devoted to a discussion on the compatibility of Definitions 1.1 and 1.2.
Terminology We work in the setting of symmetric spectra in simplicial sets [10].For commutative ring spectra, we use the positive S-model structure developed in [20].When we work in the setting of associative ring spectra, we use the stable model structure of [10].Throughout this work, R denotes a general discrete commutative ring except in Section 3C where R denotes a quotient of Z.When we say (E ∞ ) DGA, we mean (E ∞ ) Z-DGA.
Acknowledgements The author would like to thank Don Stanley for suggesting to study extension DGAs and also for showing the construction of the monoid object in Construction 5.1.I also would like to thank Dimitar Kodjabachev and Tasos Moulinos for a careful reading of this work.

The dual Steenrod algebra
Here, we recall the ring structure and the Dyer-Lashof operations on the dual Steenrod algebra.Using the standard notation, we denote the dual Steenrod algebra by A * .We have π * (HF p ∧ HF p ) ∼ = A * .Milnor shows that the dual Steenrod algebra is a free graded commutative F p -algebra [14].
For an odd prime p, the following describes the dual Steenrod algebra.
In this case, we have χ(ξ r ) = ζ r and χ(τ r ) = τ r .Dyer-Lashof operations are power operations that act on the homotopy ring of H ∞ HF p -algebras [6].By forgetting structure, commutative HF p -algebras are examples of H ∞ HF p -algebras and therefore Dyer-Lashof operations are also defined on the homotopy ring of commutative HF p -algebras and maps of commutative HF p -algebras preserve these operations.For p = 2, there is a Dyer-Lashof operation denote by Q s for ever integer s where Q s increases the degree by s.For odd p, there are Dyer-Lashof operations denoted by βQ s and Q s for every integer s that increase the degree by 2s(p − 1) − 1 and 2s(p − 1) respectively.See [6, III.1.1]for further properties of these operations.
With the unit map HF p ∼ = HF p ∧ S HF p ∧ HF p , HF p ∧ HF p is a commutative HF p -algebra and therefore Dyer-Lashof operations are defined on the dual Steenrod algebra.These operations are first studied in [6,III.2].Steinberger shows that the degree one element τ 0 for odd p and ξ 1 for p = 2 generates the dual Steenrod algebra as an algebra with Dyer-Lashof operations, i.e. as an algebra over the Dyer-Lashof algebra.In particular for p = 2, we have For odd p, we have Q (p s −1)/(p−1) τ 0 = (−1) s τ s βQ (p s −1)/(p−1) τ 0 = (−1) s ζ s .

Proof of the results on topological equivalences and the non-extension example
In this section, we prove Theorems 1.12, 1.13 and 1.15 which provide comparison results on (E ∞ ) topological equivalences and quasi-isomorphisms of (E ∞ ) DGAs for various cases.At the end, we prove Proposition 3.2 which justifies the last claim in Example 1.11.This provides examples of E ∞ F p -DGAs that are not F p -extension.
These results are obtained using similar arguments.Therefore, we suggest the reader to go through their proof in the order presented in this section.
3A. Proof of Theorem 1.12 and Theorem 1.13.The proof of Theorem 1.12 and Theorem 1.13 are similar.Therefore, we combine them in a single proof.
In the proof of Theorems 1.12 and 1.13 and also in the proof of Theorem 1.15 and Proposition 3.2, we show that for various R-extension (E ∞ ) R-DGAs, (E ∞ ) topological equivalences and quasi-isomorphisms agree.
For this, we use the same technique to produce a quasi-isomorphism, i.e. an HRalgebra equivalence, out of a given topological equivalence, i.e. an S-algebra equivalence.We start by describing this technique.
Let us focus on the E ∞ case.Assume that we are given commutative HR-algebras Y and HR ∧ Z where Z denotes a cofibrant commutative S-algebra and assume that we are given a weak equivalence ϕ : HR ∧ Z ∼ Y of commutative S-algebras.Using ϕ, we produce a map of commutative HR-algebras through the following composite.
Here, i is the canonical map induced by the unit map S HR of HR and m is the commutative HR-algebra structure map of Y .Except Y , we provide the objects in the composite above with the commutative HR-algebra structure coming from the first HR factor.The maps i and HR ∧ ϕ are maps of commutative HR-algebras as they are obtained using the functor HR ∧ − from the category of commutative Salgebras to the category of commutative HR-algebras.Note that m is the left adjoint of the identity map of Y under the adjunction between the categories of commutative S-algebras and commutative HR-algebras whose left adjoint is given by the extension of scalars functor HR ∧− and whose right adjoint is given by the restriction of scalars functor.In particular, this shows that m is also a map of commutative HR-algebras.We deduce that ψ is a map of commutative HR-algebras as it is given by a composite of such maps.Compared to the commutative case, the definition of the map ψ is slightly more complicated in the associative case as we consider various cofibrant replacements.The results we prove in this section are obtained by showing that ψ is an equivalence under the given hypothesis.
In the proof below, we denote the category of commutative E-algebras by E-cAlg and the category of associative E-algebras by E-Alg for a given commutative ring spectrum E.
Proof of Theorem 1.12 and Theorem 1.13.First, we prove Theorem 1.12.After that, we show how this proof should be modified to obtain Theorem 1.13.
Since quasi-isomorphic E ∞ DGAs are always E ∞ topologically equivalent, we only need to show that if X and Y are E ∞ topologically equivalent then they are quasiisomorphic as E ∞ F p -DGAs.
Let HF p denote a cofibrant model of HF p in S-cAlg.The category of commutative HF p -algbera spectra is the same as the category of commutative S-algebra spectra under HF p .Therefore we have a model structure on HF p -cAlg where the cofibrations, fibrations and weak equivalences are precisely the maps that forget to cofibrations, fibrations and weak equivalences in S-cAlg.We let Y also denote the commutative HF p -algebra corresponding to the E ∞ DGA Y .Therefore π 1 (Y ) = 0. Taking a fibrant replacement, we assume Y is fibrant both in HF p -cAlg and in S-cAlg.Furthermore, we let HF p ∧ Z denote the commutative HF p -algebra corresponding to the extension E ∞ F p -DGA X where Z is a cofibrant object in S-cAlg.This ensures that HF p ∧ Z is cofibrant in HF p -cAlg.Therefore the composite S HF p HF p ∧ Z is also a cofibration in S-cAlg; this shows that HF p ∧ Z is also cofibrant in S-cAlg.To prove Theorem 1.12, we need to show that HF p ∧ Z and Y are weakly equivalent in HF p -cAlg.
Because HF p ∧ Z and Y are obtained from E ∞ topologically equivalent E ∞ DGAs, they are equivalent as commutative S-algebras.Furthermore HF p ∧Z is cofibrant and Y is fibrant, therefore there is a weak equivalence ϕ : HF p ∧ Z ∼ Y of commutative S-algebras.We consider the composite map (1) where the first map is induced by the unit map u HFp : S HF p of HF p and the last map is the HF p structure map of Y .If we consider all the objects in this composite except Y to have the HF p structure coming from the first smash factor, then all objects involved are commutative HF p -algebras and the maps involved are maps of commutative HF p -algebras.Note that i and HF p ∧ ϕ are maps of commutative HF palgebras as they are obtained via the functor HF p ∧ − : S-cAlg HF p -cAlg.The last map m is a map of commutative HF p -algebras because it is the left adjoint of the identity map of Y under the usual adjunction between S-cAlg and HF p -cAlg.Since all the maps in the composite above are maps of commutative HF p -algebras, we deduce that ψ is a map of commutative HF p -algebras.
What remains is to show that ψ is a weak equivalence.For this, we take the homotopy groups of the composite defining ψ and show that it is an isomorphism.Firstly, we have a splitting in HF p -cAlg where we consider the object on the right hand side of the equality with the HF p structure given by the first smash factor instead of the canonical one given by the smash product ∧ HFp .Because the homotopy of HF p is a field, we have π * (HF p ∧ HF p ∧ Z) ∼ = A * ⊗ π * (HF p ∧ Z), see [8,IV.4.1].With this identification, we obtain that the composite map induced in homotopy by the composite defining ψ is given by (2) ψ * : π * (HF p ∧ Z) Note that although we identify the domain of π * (HF p ∧ ϕ) as a tensor product, we do not claim that π * (HF p ∧ ϕ) splits as a tensor product of two maps.Now we state and prove the following claims.Afterwards, we combine them to prove that ψ * is an isomorphism by showing ψ * = ϕ * .
Claim 1: The composite m * • π * (HF p ∧ ϕ) maps every element of the form a ⊗ x with |a| > 0 to zero in Y * .
We have a canonical map This map is in HF p -cAlg therefore the induced map in homotopy preserves the Dyer-Lashof operations.The induced map in homotopy is given by the inclusion . Carrying x through the top row and then composing with m • h Y , we obtain the equality in our claim.
The composite of the maps below is the identity where m HFp is the multiplication map of HF p .With the identification we obtain the following composite in homotopy where π * (m HFp ∧ id) is given by the augmentation A * F p .This description of π * (m HFp ∧ id) and the fact that π * (m HFp ∧ id) • i * = id proves our claim.
Finally, we have for some a i ∈ A * with |a i | > 0. Here, the first equality follows by the definition of ψ * , the second equality follows by Claim 3 and the third follows by Claim 2 and Claim 1.This proves that ψ * is an isomorphism and therefore ψ is a weak equivalence.At this point, we are done with the proof of Theorem 1.12.
Note that for Theorem 1.13, we work in the setting of associative algebras.In this case, we need to be more careful with cofibrant replacements since the forgetful functor from HF p -Alg to S-Alg does not necessarily preserve cofibrant objects.Let HF p be as before and let Z be cofibrant in S-Alg such that HF p ∧ Z is an HF palgebra that corresponds to X.By abuse of notation, let Y be a fibrant HF p -algebra corresponding to Y .Let T ∼ −։ HF p ∧ Z be a cofibrant replacement of HF p ∧ Z in S-Alg.We have the following lift (5) in S-Alg where the bottom map is given by the map Z ∼ = S ∧ Z HF p ∧ Z. Since T and Y are obtained from topologically equivalent DGAs, they are equivalent in S-Alg.Also because T is cofibrant and Y is fibrant, we have a weak equivalence ϕ : T ∼ Y of S-algebras.We obtain the following composite map of HF p -algebras where i = HF p ∧ f and m is the HF p structure map of Y .The map m is a map of HF p -algebras because it is the left adjoint of the identity map of Y under the usual adjunction between HF p -Alg and S-Alg.Note that we denote HF p ∧ f by i because the map i in the composite above should be compared to the map i in (1).
Again, what remains is to show that ψ * is an isomorphism.Note that the functor HF p ∧ − preserves weak equivalences as described in the proof of Proposition 4.7 in [20].Identifying homotopy groups of T with homotopy groups of HF p ∧ Z through the trivial fibration above and similarly identifying the homotopy groups of HF p ∧ T with those of HF p ∧ HF p ∧ Z, we obtain a description of ψ * similar to the one in (2).
It is sufficient to show that the claims above also hold in this case.Claim 1 follows by the hypothesis that π * Y is trivial at the degrees where the algebra generators of the dual Steenrod algebra are.Claim 2 follows similarly.For Claim 3, consider the following sequence of maps where m HFp is the multiplication map of HF p .Due to Diagram (5), the composite above is the identity map.Taking homotopy groups of the composite above and omitting the equivalence in the middle, one obtains (4).The rest of the proof of Claim 3 follows as before.
Remark 3.1.The proof of Theorem 1.12 is showing slightly more.For a given cofibrant Z in S-cAlg and a fibrant Y in HF p -cAlg with π 1 Y = 0 and an equivalence HF p ∧ Z ∼ Y of S-algebras, the map HF p ∧ Z Y in HF p -cAlg given by the structure map of Y on HF p and the map S ∧ Z HF p ∧ Z ∼ Y on Z is also a weak equivalence.Note that to construct this map, we use the fact that HF p ∧ Z is a coproduct of HF p and Z in S-cAlg.Proof.Recall that in Example 1.11, we provide examples of E ∞ F p -DGAs that are E ∞ topologically equivalent but not quasi-isomorphic.We prove that X is not an extension E ∞ F p -DGA.In order to show Y is not extension, it suffices to exchange the roles of X and Y in the proof below.
We assume that X is an extension E ∞ F p -DGA and obtain a contradiction by showing that X and Y are quasi-isomorphic under this assumption.This is similar to the proof of Theorem 1.12 that we assume familiarity with.Following the constructions there, we obtain a map of commutative HF p -algebras as in Diagram (1) where HF p ∧ Z denotes a commutative HF p -algebra corresponding to X and Y denotes a commutative HF p -algebra corresponding to the E ∞ F p -DGA Y by abusing notation.This is a map of commutative HF p -algebras as before.Therefore, it is sufficient to show that ψ * is an isomorphism.As in (2), we have the following description of ψ * .
By Claim 3 in the proof of Theorem 1.13, for every x ∈ π * (HF p ∧ Z) we have for some a i ∈ A * with |a i | > 0 and x i ∈ π * (HF p ∧ Z).
For p = 2, π * (HF p ∧ Z) ∼ = F 2 [x]/(x 4 ) with |x| = 1.By degree reasons, we either have i * (x) = 1 ⊗ x or i * (x) = 1 ⊗ x + ξ 1 ⊗ 1.Since (1 ⊗ x + ξ 1 ⊗ 1) 4 = 0 but x 4 = 0, the second option is not possible.Therefore we have i * (x) = 1 ⊗ x.Since i is a map of ring spectra, i * is multiplicative so we have i * (x l ) = 1 ⊗ x l for every l.By Claim 2 in the proof of Theorem 1.12, this shows that ψ * is an isomorphism.This provides a contradiction as X and Y are not quasi-isomorphic as E ∞ F 2 -DGAs.
For odd p, we have with |x| = 1, |y| = 2p − 2. By ( 6) above, we have either i * (y) = 1 ⊗ y or i * (y) = cξ 1 ⊗ 1 + 1 ⊗ y for some unit c ∈ F p .However, y 2 = 0 but (cξ 1 ⊗ 1 + 1 ⊗ y) 2 = 0 so only the first option is possible.This shows that ψ * (y) = y due to Claim 2 in the proof of Theorem 1.12.The 2p − 2 Postnikov sections of Y and HF p ∧ Z agrees with that of HF p ∧ HF p in commutative HF p -algebras, see Example 5.1 in [1].Using this together with the fact that βQ 1 τ 0 = −ζ 1 in the dual Steenrod algebra, we obtain that we have βQ 1 x = y up to a unit both in π * (HF p ∧ Z) and in π * Y .Because ψ is a map of commutative HF p -algebras, ψ * preserves Dyer-Lashof operations.Since ψ * (y) = y, we obtain that ψ * (x) = x up to a unit of F p .Because ψ * is a ring map, we deduce that ψ * is indeed an isomorphism.Therefore ψ is a weak equivalence of commutative HF p -algebras between the commutative HF p -algebras corresponding to the E ∞ F p -DGAs X and Y .This contradicts the fact that X and Y are not quasi-isomorphic as E ∞ F p -DGAs and finishes our proof.
3C. Proof of Theorem 1.15.In the proof below, R denotes Z/(m) for some m = ±1.Furthermore, X denotes an R-DGA whose corresponding HR-algebra is equivalent to HR ∧ Z where Z is a cofibrant S-algebra whose underlying spectrum is weakly equivalent to a coproduct of suspensions and/or desuspensions of the sphere spectrum.For every R-DGA Y , we need to show that X and Y are quasi-isomorphic if and only if they are topologically equivalent.
Proof of Theorem 1.15.Let HR be cofibrant as a commutative S-algebra.This guarantees that HR ∧ − preserves weak equivalences, see [20, proof of Proposition 4.7].
Since HR ∧ − preserves weak equivalences, we can further assume Z to be fibrant.
Let Y be an R-DGA.Since quasi-isomorphic R-DGAs are always topologically equivalent, we only need to show that X and Y are quasi-isomorphic if they are topologically equivalent.Abusing notation, we also let Y denote a fibrant HR-algebra corresponding to the R-DGA Y .We assume that X and Y are topologically equivalent, i.e.HR ∧ Z and Y are equivalent as S-algebras.Using this, we are going to show that there is a weak equivalence Let g : T ∼ −։ HR ∧ Z be a cofibrant replacement of HR ∧ Z in S-algebras.As in Diagram ( 5), there exists a map f : Z T such that the following diagram commutes. (7) Here, h Z denotes the canonical map Since X and Y are topologically equivalent, T and Y are equivalent as S-algebras.Furthermore, T is cofibrant and Y is fibrant, therefore we have a weak equivalence ϕ : T ∼ Y of S-algebras.
We obtain the composite map algebras where m denotes the HR structure map of Y .Note that the last map above is a map of HR-algebras as it is the left adjoint of the identity map of Y under the usual adjunction between the categories of HR-algebras and S-algebras.Since ψ is a map of HR-algebras, it is sufficient to show that ψ induces an isomorphism in homotopy.
We have the following commuting diagram where the vertical maps are the canonical maps induced by the unit map u R : S HR.This shows that the composite map starting from T ∼ = S ∧ T and ending in Y is given by ϕ and therefore is a weak equivalence.In particular, π * (m • (HR ∧ ϕ)) is an isomorphism when it is restricted to the image of the Hurewicz map of T π * h T : π * (S ∧ T ) π * (HR ∧ T ).
Therefore in order to prove that ψ * is an isomorphism, it is sufficient to show that the map π * (HR ∧ f ) : is injective and its image agrees with the image of π * h T .For this, it is sufficient to prove that the corresponding statements are true after composing with the isomorphism of HR ∧ Z.Note that h HR∧Z is induced by the unit map of HR as usual.Therefore, it is sufficient to show that the image of π * (HR ∧ h Z ) agrees with the image of π * (h HR∧Z ).The map HR ∧ h Z is the canonical map This is the same as the composite ( 8) where τ is the transposition map of the monoidal structure.Since the map h HR∧Z in the middle of the composite in (8) induces π * (h HR∧Z ), it is sufficient to show that π * (τ ∧ id) is the identity map on the image of π * (h HR∧Z ).By hypothesis, the underlying spectrum of Z is a wedge of suspensions of the sphere spectrum.Let E = ∨ a∈A Σ |a| S be weakly equivalent to Z as a spectrum where A is a graded set.Since E is cofibrant and Z is fibrant, there is a weak equivalence of spectra E ∼ Z.
This equivalence induces the horizontal maps in the following commuting diagram of S-modules.
Here, h HR∧E denotes the canonical map that induces the Hurewicz map of HR ∧ E in homotopy.In order to show that π * (τ ∧ id) (of the bottom row) is the identity map on the image of π * (h HR∧Z ), it is sufficient to show that π * (τ ∧ id) (of the top row) is given by the identity map on the image of π * (h HR∧E ).For this, it is sufficient to show that the composite of the maps on the top row is given by π * (h HR∧E ) in homotopy.
Note that the canonical R-module basis elements of are also abelian group generators because R = Z/(m) for some integer m.Therefore, it is sufficient to show that for every canonical basis element x.Such an x is represented by a map where i a is the inclusion of the cofactor corresponding to an a ∈ A.
In other words, it is sufficient to show that the composite To see this, note that the composite maps above are of the form υ ∧ i a and ν ∧ i a respectively where υ and ν are S-algebra maps from S to HR ∧ HR.Since S is the initial object in the category of S-algebras, we deduce that υ = ν.Therefore, the two composites above agree as claimed.

E-infinity F p -DGAs are not extension
This section is devoted to the proof of Theorems 1.6 and 1.7.We restate these theorems below.
Theorem 4.2.(Theorem 1.7) Let X be a DGA.If X is quasi-isomorphic to an F 2 -DGA then X is not an extension DGA.
In the proof of these theorems, we use the ring structure and the Dyer-Lashof operations on π * (HF p ∧ HZ) = HF p * HZ.For odd p, the ring structure is given by where the degrees of the generators are the same as those of the dual Steenrod algebra.Note that HF p * HZ has the same generators as the dual Steenrod algebra except that HF p * HZ does not contain the degree 1 generator τ 0 .Indeed, the map HF p * HZ HF p * HF p = A * induced by HZ HF p is the canonical inclusion [17,II.10.26].This inclusion is induced by a map of commutative HF p -algebras and therefore it preserves the Dyer-Lashof operations.Therefore through this map, the Dyer-Lashof operations on the dual Steenrod algebra determine the Dyer-Lashof operations on HF p * HZ, see [6,III.2].
For p = 2, we have Again, the canonical map HF 2 * HZ HF 2 * HF 2 = A * is the canonical inclusion and this determines the Dyer-Lashof operations on HF 2 * HZ.
For the rest of this section, we assume that HZ is cofibrant as a commutative Salgebra and HF p is cofibrant as a commutative HZ-algebra in the model structure developed in [20].Since the category of commutative HZ-algebras is the same as the category of commutative S-algebras under HZ, cofibrations of commutative HZalgebras forget to cofibrations of commutative S-algebras.Therefore, HZ HF p is also a cofibration of commutative S-algebras.This ensures that HF p is also cofibrant as a commutative S-algebra and therefore the functor HF p ∧ − preserves all weak equivalences, see the proof of Proposition 4.7 in [20].
We start by proving the following lemma.This lemma is obvious if one assumes that for a map of discrete commutative rings R R ′ , the Quillen equivalences of ( [15]) [21] are compatible with the restriction of scalars functors from (E ∞ ) R ′ -DGAs to (E ∞ ) R-DGAs and from (commutative) HR ′ -algebras to (commutative) HR-algebras.However, there is no such compatibility result available in the literature and proving it is beyond the scope of this work.
In this situation, there is a map of (commutative) HZ-algebras c(HF p ) HX where HX denotes a fibrant (commutative) HZ-algebra corresponding to the (E ∞ ) DGA X.Furthermore, c(HF p ) denotes a cofibrant replacement of HF p in (commutative) HZ-algebras.In the commutative case, HF p is cofibrant in commutative HZ-algebras due to our standing assumptions and therefore the cofibrant replacement above may be omitted.
Proof.We only prove the E ∞ case; the associative case follows in a similar manner.Assume that we are using a unital E ∞ operad, i.e. an operad given by the monoidal unit F p in operadic degree zero.Barratt-Eccles operad is an example of a unital E ∞ -operad [4].In this situation, F p is the free E ∞ F p -DGA generated by the trivial F p -chain complex 0. Therefore, F p is the initial object in E ∞ F p -DGAs.This, together with the fact that X is quasi-isomorphic to an E ∞ F p -DGA implies that there is a map F p X in the homotopy category of E ∞ DGAs.The equivalence of categories between the homotopy categories of commutative HZalgebras and E ∞ DGAs imply that there is also a map HF p HX in the homotopy category of commutative HZ-algebras.Since HX is fibrant in commutative HZalgebras, there is a map c(HF p ) HX of commutative HZ-algebras as desired.
The following starts with the proof of Theorem 1.6 and at the end, we mention how this also shows Theorem 1.7.
Proof of Theorems 1.6 and 1.7.Assume to the contrary that there is an extension where ϕ HFp denotes the canonical map.Since X is a Z-extension E ∞ DGA, there is a cofibrant commutative S-algebra Y such that HZ ∧ Y is weakly equivalent to HX in commutative HZ-algebras.
Note that HZ∧Y is cofibrant as a commutative HZ-algebra; this is the case because HZ ∧ − is a left Quillen functor from commutative S-algebras to commutative HZalgebras and therefore it preserves cofibrant objects.
Since HX is fibrant and HZ ∧ Y is cofibrant, there is a weak equivalence of commutative HZ-algebras ψ : HZ ∧ Y ∼ HX.Because ψ is a map of commutative HZ-algebras, we obtain a commutative diagram Let p denote an odd prime, we discuss the case p = 2 at the end of this proof.We have βQ 1 τ 0 = ζ 1 (up to a unit we are going to omit) in HF p * HF p .Note that f (ζ 1 ) = ζ 1 ⊗ 1.This follows by considering the composite in (10), Equation ( 9) and by noting that HF p * ϕ HFp is the canonical inclusion.Since f preserves Dyer-Lashof operations, we obtain the following.
We obtain a contradiction by showing that there is no z in HF p * HZ ⊗ HF p * Y that satisfies βQ 1 z = ζ 1 ⊗ 1, i.e. there is no candidate for f (τ 0 ).For an element of the form 1 ⊗ y ∈ HF p * HZ ⊗ HF p * Y , βQ 1 (1 ⊗ y) = 1 ⊗ βQ 1 y does not contain ζ 1 ⊗ 1 as a summand.Now consider an element of the form a ⊗ y ∈ HF p * HZ ⊗ HF p * Y with |a| > 0. By the Cartan formula and the fact that the Bockstein operation is a derivation, βQ 1 (a ⊗ y) is a sum of elements of the form a ′ ⊗ y ′ where a ′ is obtained by applying a Dyer-Lashof operation to a.In particular, |a ′ | > |a| ≥ |ζ 1 |, therefore βQ 1 (a ⊗ y) does not contain ζ 1 ⊗ 1 as a summand neither.We deduce that βQ 1 z does not contain ζ 1 ⊗ 1 as a summand for all z ∈ HF p * HZ ⊗ HF p * Y .
For p = 2, we do not need to use the Dyer-Lashof operations.In this case, we have due to the composite in (10).We obtain that f (ζ 1 ) 2 = ζ 2 1 ⊗1.However, there is no element in HF 2 * HZ ⊗ HF 2 * Y that squares to ζ 2 1 ⊗ 1.Since this does not use Dyer-Lashof operations, these arguments also work for DGAs and HZ-algebras and provide a proof of Theorem 1.7.

Formal DGAs to HZ-algebras
This section is devoted to the proof of Proposition 5.8 which provides an explicit description of the HR-algebra corresponding to a formal R-DGA whose homology satisfies the hypothesis of Theorem 1.3.This description provides Theorem 1.3.Recall that we also use Proposition 5.8 to obtain Corollary 1.16.
We work in several different monoidal categories in this section.When we work in the category of chain complexes or in the category of differential graded algebras, we denote the monoidal product by ⊗.In all the other cases, we let ∧ denote the monoidal product.In particular, we contradict the notation we used in the previous sections and denote the monoidal product of HR-modules by ∧ instead of ∧ HR .In this section, HR denotes the Eilenberg Mac Lane spectrum of a general discrete commutative ring as in [10, 1.2.5].
Let X be an R-DGA satisfying the hypothesis of Theorem 1.3.Recall from Remark 1.4 that there is a monoid M in graded pointed sets for which H * (X) ∼ = R M as R-algebras where the underlying R-module of R M is the free graded R-module over the graded set M − obtained by removing the base point of M. Furthermore, the multiplication on R M is the canonical one induced by that of M. For the rest of this section, let M denote a monoid in non-negatively graded pointed sets.5A.A monoid object corresponding to M. Here, we construct a monoid in a general monoidal category by using M. Furthermore, we show that this construction is preserved by strong monoidal Quillen Pairs.
We start by explaining a notation we use for the symmetric monoidal pointed model categories we consider in this section.For a cofibrant C, ΣC denotes the pushout of the diagram * C * where * is obtained by a factorization C * ∼ −։ * of the map C * by a cofibration followed by a trivial fibration and * denotes the final object.For the unit I of the monoidal structure, Σ k I denotes (ΣI) ∧k for k > 0 and denotes I for k = 0. Construction 5.1.Let (C, ∧, I) denote a pointed cofibrantly generated closed symmetric monoidal model category whose unit I is cofibrant.Furthermore, assume that C satisfies the monoid axiom and the smallness axioms of [19].This implies that the category of modules over a monoid in C carries an induced model structure where the fibrations and the weak equivalences are precisely those of C [19, 4.1].For a given M as above, we construct a monoid structure on where ∨ denotes the coproduct in C. The multiplication map (11) is given (on the cofactor corresponding to (m, n) ∈ M × M) by the inclusion of the cofactor corresponding to mn ∈ M if mn = 0 and given by the zero map if mn = 0. Note that in a pointed model category, there is a unique zero map between every pair of objects which is defined to be the map that factors through the point object.One easily checks that the multiplication above is associative and unital.
If E is a commutative monoid in C, then the category of E-modules is also a symmetric monoidal model category [19, 4.1].We let ∨ m∈M − Σ |m| E denote the monoid we obtain by applying the construction above in the category of E-modules.In particular, ∨ m∈M − Σ |m| E is an E-algebra.
Using the construction above, we obtain an HR-algebra ∨ m∈M − Σ |m| HR.In order to prove Theorem 1.3, we go through the zig-zag of Quillen equivalences between the model categories of R-DGAs and HR-algebras to show that the HR-algebra corresponding to the formal R-DGA with homology R M is given by ∨ m∈M − Σ |m| HR.We deduce that the formal R-DGA with homology R M is R-extension by showing that ∨ m∈M − Σ |m| HR is weakly equivalent to HR ∧ c(∨ m∈M − Σ |m| S) in HR-algebras where c denotes the cofibrant replacement functor in S-algebras.For this, we start with the following lemmas.Proof.Since F is a strong monoidal functor, we have a natural isomorphism F (X) ∧ F (Y ) ∼ = F (X ∧ Y ) and an isomorphism F (I C ) ∼ = I D .This isomorphism provides the weak equivalence υ in the hypothesis of Lemma 5.2.Therefore, there is a weak equivalence ϕ : Using ϕ, we produce a weak equivalence of monoids: ∼ Here, Φ is the coproduct of maps given by the isomorphism F (I C ) ∼ = I D for |m| = 0 and the map for |m| > 0. Here, the first and the last equalities follow by our definition of Σ k − for k > 0 and the second isomorphism comes from the strong monoidal structure of F .Also, note that ϕ ∧|m| is a weak equivalence because it is a smash product of weak equivalences between cofibrant objects.Since Φ is a coproduct of weak equivalences between cofibrant objects, it is a weak equivalence by Lemma 4.7 of [23].
It is clear that Φ is a map of monoids by the definition of the monoidal structure on both sides and from the fact that left adjoint functors between pointed categories preserve the zero maps.This shows that Φ is a weak equivalence of monoids between Therefore, in order to finish the proof of the lemma, it is sufficient to show that the monoids F c(∨ m∈M − Σ |m| I C ) and F (∨ m∈M − Σ |m| I C ) are weakly equivalent.Since c is the cofibrant replacement functor in the category of monoids, there is a weak equivalence of monoids ∼ By Theorem 4.1 of [19], the source of f is cofibrant in C.This means that f is a weak equivalence between cofibrant objects and therefore F (f ) is a weak equivalence.Furthermore, F (f ) is a weak equivalence of monoids because a strong monoidal functor preserves maps of monoids.Therefore, the monoids F c(∨ m∈M − Σ |m| I C ) and F (∨ m∈M − Σ |m| I C ) are weakly equivalent as desired.
5B.From DGAs to HZ-algebras.Here, we carry out our discussion for the case R = Z.The case of general discrete commutative ring R follows similarly.
The DGA corresponding to an HZ-algebra is obtained using the following zig-zag of monoidal Quillen equivalences of [21] HZ-Mod Sp Σ (sAB) where the left adjoints are the top arrows and the pairs (Z, U) and (D, R) are both strong monoidal Quillen equivalences.The pair (L, φ * N) is a weak monoidal Quillen equivalence.See [18, 3.6] for the definitions of strong monoidal Quillen equivalences and weak monoidal Quillen equivalences.We often use the fact that the model categories in the zig-zag above are pointed.
Since each Quillen equivalence in the zig-zag is a monoidal Quillen equivalence, there is an induced zig-zag of Quillen equivalences of the corresponding model categories of monoids.This gives the induced derived functors H : DGA HZ-Alg and Θ : HZ-Alg DGA in Theorem 1.1 of [21].We have where L mon is the induced left adjoint at the level of monoids and c denotes the cofibrant replacement functors in the corresponding model category of monoids.See Section 3.3 of [18] for a definition of the induced left adjoint at the level of monoids.
In the lemmas below, I 1 and I 2 denote the monoidal units of Sp Σ (sAB) and Sp Σ (Ch + ) respectively.Note that the units of the monoidal model categories in the zig-zag above are all cofibrant.By Construction 5.1, we have the monoids ∨ m∈M − Σ |m| I 1 and ∨ m∈M − Σ |m| I 2 in Sp Σ (sAB) and Sp Σ (Ch + ) respectively.Therefore, it is sufficient to show that ⊕ m∈M − Σ |m| Z is quasi-isomorphic to the formal DGA with homology Z M .Let 0 denote the chain complex consisting of Z in degrees 0 and 1 and the trivial module in the rest of the degrees; its differentials are trivial except degree 1 where the differential is the identity.There is a factorization Z 0 ∼ −։ 0 of the trivial map Z 0 as a cofibration followed by a trivial fibration.Let σZ denote the chain complex consisting of Z in degree 1 and the trivial module in rest of the degrees.This is the pushout of the diagram 0 Z 0. Note that due to our conventions, ΣZ is the pushout of the diagram 0 Z 0. Since the category of chain complexes is left proper, there is a weak equivalence ϕ : ΣZ ∼ σZ.Let σ n Z denote (σ 1 Z) ⊗n .Following Construction 5.1, we obtain a formal DGA ⊕ m∈M − σ |m| Z. Similar to the map Φ in the proof of Lemma 5.3, we obtain a quasi-isomorphism of DGAs Φ : given by the identity map for |m| = 0 and given by ϕ |m| for |m| > 0. This shows that ⊕ m∈M − Σ |m| Z and ⊕ m∈M − σ |m| Z are quasi-isomorphic as DGAs where the latter is the formal DGA with homology Z M .We state and prove the following two lemmas that we use in the proof of Lemma 5.7.
Proof.The category of symmetric spectra in a closed symmetric monoidal model category C is the category of modules over a monoid in symmetric sequences in C, see Definition 2.7 in [21].Since symmetric sequences in C is a diagram category in C, the colimits in symmetric sequences are levelwise.Furthermore, the forgetful functor from modules over a monoid to the underlying closed monoidal category preserves colimits.Therefore colimits of symmetric spectra in C are also levelwise.
Here, N is the normalization functor sAB Ch + of the Dold-Kan correspondence, an equivalence of categories, applied levelwise.Therefore it preserves colimits.Furthermore, φ * is the restriction of scalars functor between the categories of modules over two monoids induced by a map of these monoids in symmetric sequences in Ch + , see Page 358 of [21].Therefore φ * is the identity functor identity on the underlying symmetric sequences and therefore it also preserves colimits.
Lemma 5.6.For every cofibrant A in Sp Σ (Ch + ) and every B in Sp Σ (sAB), a map L(A) B is a weak equivalence if and only if its adjoint A φ * N(B) is a weak equivalence.
Proof.This follows from the fact that φ * N preserves all weak equivalences.Let B ∼ f B be a fibrant replacement of B. Adjoint of the composite L(A) B ∼ f B is given by the composite A φ * N(B) ∼ φ * N(f B) whose first map is the adjoint of the map L(A) B. Because (L, φ * N) is a Quillen equivalence, the first composite is a weak equivalence if and only if the second composite is a weak equivalence.The result follows by the 2-out-of-3 property of weak equivalences.
The following lemma takes care of the middle step in the zig-zag of Quillen equivalences between the model categories of HZ-algebras and DGAs.Note that since (L, φ * N) is a weak monoidal Quillen pair, φ * N is a lax monoidal functor, see Definition 3.3 of [18].Therefore, φ * N carries monoids to monoids.In particular, φ * N(∨ m∈M − Σ |m| I 1 ) is a monoid.followed by the inclusion of the cofactor corresponding to mn ∈ M if mn = 0 and given by the zero map if mn = 0. Note that the map above is the lax monoidal structure map of φ * N and the equality above follows by our definition of Σ k −.Furthermore, one checks using this definition that the isomorphism in ( 12) is an isomorphism of monoids.Therefore, in order to prove the lemma, it is sufficient to show that there is an isomorphism of monoids between ∨ m∈M − φ * N(Σ |m| I 1 ) and ∨ m∈M − Σ |m| I 2 .
There is a weak equivalence L(I 2 ) ∼ I 1 , see [18, 3.6].Therefore, there is also a weak equivalence ϕ : L(ΣI 2 ) ∼ ΣI 1 by Lemma 5. where the equalities follow by our definition of Σ ℓ and the second map is obtained by successive applications of the transformation φ * N(−) ∧ φ * N(−) φ * N(− ∧ −) that is a part of the lax monoidal structure of φ * N, see [18, 3.3].Now we define a map of monoids as the coproduct of ψ |m| over m ∈ M − .By the associativity and the unitality of the lax monoidal structure on φ * N and by the fact that right adjoint functors preserve the zero maps between pointed categories, Ψ is a map of monoids, see [5, 6.4.1].Finally, we need to show that Ψ is a weak equivalence.By Lemmas 5.5 and 5.6, it is sufficient to show that the adjoint of Ψ is a weak equivalence.Since both φ * N and L preserve coproducts and since Ψ is a coproduct of maps ψ |m| , the adjoint of Ψ is a coproduct of the adjoints of the maps ψ |m| .Note that a coproduct of weak equivalences of cofibrant objects is again a weak equivalence by [23, 4.7].Since the adjoint of ψ ℓ is a map between cofibrant objects, it is sufficient to show that the adjoint of ψ ℓ is a weak equivalence for each ℓ ≥ 0.
For the case ℓ = 0, we have that the adjoint of ψ 0 is the weak equivalence L(I 2 ) ∼ I 1 mentioned above.For ℓ = 1, the adjoint of ψ 1 is the map ϕ above which is also a weak equivalence.
We show the ℓ = 2 case and the rest follows similarly.In particular, we show that the adjoint to the composite defining ψ 2 The first map in this composite is the comonoidal map induced by the lax monoidal structure of φ * N and by Definition 3.6 of [18], this is a weak equivalence.Furthermore, the second map in the composite is a smash product of weak equivalences between cofibrant objects; and therefore, it is also a weak equivalence.This shows that the composite is a weak equivalence.
To show that ψ 2 is the adjoint to this composite, first note that by the discussion on Equation (3.4) in [18], the comonoidal map c L is the adjoint of the composite map where the first map is induced by the unit of the adjunction and the second map comes from the lax monoidal structure on φ * N. Considering the adjoint of the composite (13) as the adjoint of the first map c L in the composite followed by φ * N(ϕ ∧ ϕ), we obtain that the adjoint of ( 13) is given by the composite HR-algebras and R-DGAs in [21] is compatible with the zig-zag of Quillen equivalences between commutative HR-algebras and E ∞ R-DGAs in [15].In conclusion, if we assume that these Quillen equivalences are compatible, then the Definitions 1.1 and 1.2 are also compatible in the sense described above.

Remark 1 . 8 .
These theorems should be compared with the two commutative HZalgebras X and Y obtained from HZ ∧ HF p through the structure maps HZ ∼ = HZ ∧ S HZ ∧ HF p and HZ ∼ = S ∧ HZ S ∧ HF p HZ ∧ HF p and this shows that Dyer-Lashof operations on this subset of A * ⊗ π * (HF p ∧ Z) are given by the action of the Dyer-Lashof operations on the dual Steenrod algebra i.e.Q s (a ⊗ 1) = (Q s a) ⊗ 1.Let p be an odd prime.Since π 1 (Y ) is trivial, m * • π * (HF p ∧ ϕ)(τ 0 ⊗ 1) = 0.Because the dual Steenrod algebra is generated with the Dyer-Lashof operations by τ 0 , this shows that m * • π * (HF p ∧ ϕ)(a ⊗ 1) = 0 for all a ∈ A * with |a| > 0. Since all maps involved are ring maps and a ⊗ x = (a ⊗ 1)(1 ⊗ x), this finishes the proof of our claim.Note that for p = 2, one uses ξ 1 instead of τ 0 .Claim 2: We have m * • π * (HF p ∧ ϕ)(1 ⊗ x) = ϕ * (x) for every x ∈ π * (HF p ∧ Z).We consider the following commutative diagram.(3)

3B. Example 1 . 11 .
Here, we show that the E ∞ F p -DGAs provided in Example 1.11 are not F p -extension.Proposition 3.2.Let X and Y be as in Example 1.11.As E ∞ F p -DGAs, X and Y are not F p -extension.

π
* (HR ∧ g) : π * (HR ∧ T ) ∼ = − π * (HR ∧ HR ∧ Z).In other words, it is sufficient to show thatπ * (HR ∧ g) • π * (HR ∧ f )is injective and the image of this map agrees with the image of the map π * (HR ∧ g)• π * h T .Due to Diagram (7), g • f = h Z .Therefore, it is sufficient to show that π * (HR ∧ h Z ) is injective in homotopy and its image agrees with the image of π * (HR ∧ g) • π * h T .The following composite is the identity map:HR ∧ Z HR∧h Z −−−− HR ∧ HR ∧ Z m∧id − −− HR ∧ Z, where m denotes the multiplication map of HR and id denotes the identity map of Z. From this, we deduce that π * (HR ∧ h Z ) is injective in homotopy as desired.What remains to prove is that the image of π * (HR ∧ h Z ) agrees with the image of π * (HR ∧ g) • π * h T .Due to the commuting diagram: S ∧ T S ∧ HR ∧ Z HR ∧ T HR ∧ HR ∧ Z, the map π * (HR ∧ g) • π * h T is given by the image of the Hurewicz map π * (h HR∧Z ) : π * (S ∧ HR ∧ Z) π * (HR ∧ HR ∧ Z) DGA.It follows by Lemma 4.3 that there is a map HF p HX of commutative HZ-algebras where HX denotes a fibrant commutative HZ-algebra corresponding to the E ∞ DGA X.In particular, the HZ-structure map HZ HX of HX factors as HZ ϕ HFp − −− HF p HX composite on the right from HZ to HX is the composite given above.The map ϕ HZ∧Y is the HZ-structure map of HZ ∧ Y which is given by HZ ∼ = HZ ∧ S HZ ∧ Y .Applying the homology functor HF p * to this diagram and inverting HF p * ψ, we obtain the following.HF p * HZ HF p * HF p HF p * HZ ⊗ HF p * Y HF p * HX HFp * ϕ HFp HFp * ϕ HZ∧Y ∼ = (HFp * ψ) −1 By the Künneth spectral sequence in [8, IV.4.1],HF p * (HZ ∧ Y ) ∼ = HF p * HZ ⊗ HF p * Y and the morphism on the left is given by (9) HF p * ϕ HZ∧Y (a) = a ⊗ 1.Since the diagram above commutes, we obtain that HF p * ϕ HZ∧Y factors as (10) HF p * ϕ HZ∧Y : HF p * HZ HFp * ϕ HFp − −−−−− HF p * HF p f − HF p * HZ ⊗ HF p * Y where the second map f is the composite in the triangle above starting from HF p * HF p and ending in the bottom left corner.Both maps in the composite above are ring maps that preserve the Dyer-Lashof operations.

Lemma 5 . 2 .∼Lemma 5 . 3 .
Assume that (C, ∧, I C ) and (D, ∧, I D ) are pointed and closed symmetric monoidal model categories with cofibrant units.Furthermore, let C D F G be a Quillen pair where F denotes the left adjoint.If there is a weak equivalence υ : F (I C ) ∼ I D , then there exists a weak equivalence ϕ : F (ΣI C ) ∼ ΣI D .Proof.By factoring the map I C * by a cofibration followed by a trivial fibration, we obtain a factorization F (I C ) F ( * ) ∼ F ( * ) ∼ = * .Note that the isomorphism follows by the fact that F is a left adjoint functor between pointed categories.To see that the second map is a weak equivalence, note that * is cofibrant in the pointed model category C and that F preserves all weak equivalences between cofibrant objects.Similarly, we have a factorization I D * ∼ −։ * consisting of a cofibration followed by a trivial fibration.We use the equivalence υ : F (I C ) ∼ I D and the lift in the following squareF (I C )This in turn gives a map ϕ of the corresponding pushouts of these diagrams.This is a weak equivalence because these are diagrams consisting only of cofibrations between cofibrant objects; therefore their pushout is the homotopy pushout.Since the pushout of the diagram on the left hand side is F (ΣI C ) and the pushout of the diagram on the right hand side is ΣI D , we obtain the weak equivalence ϕ : F (ΣI C ) ∼ ΣI D we wanted to construct.Assume that (C, ∧, I C ) and (D, ∧, I D ) are pointed and closed symmetric monoidal model categories with cofibrant units as in Construction 5.1.Furthermore, let C D F G be a Quillen pair where the left adjoint F is a strong monoidal functor.In this situation, F c(∨ m∈M − Σ |m| I C ) and ∨ m∈M − Σ |m| I D are weakly equivalent as monoids in D where c denotes the cofibrant replacement functor in the model category of monoids in C [19, 4.1].

Lemma 5 . 4 .
In Sp Σ (sAB), Zc(∨ m∈M − Σ |m| HZ) and ∨ m∈M − Σ |m| I 1 are weakly equivalent as monoids.In Ch, Dc(∨ m∈M − Σ |m| I 2 ) and the formal DGA with homology Z M are quasi-isomorphic as DGAs.Proof.The first statement is a direct consequence of Lemma 5.3.We prove the second statement of the lemma.It again follows by Lemma 5.3 that Dc(∨ m∈M − Σ |m| I 2 ) and ⊕ m∈M − Σ |m| Z are quasi-isomorphic as DGAs (i.e.weakly equivalent as monoids in Ch).